Magma V2.24-6 Sun Jan 12 2020 21:03:52 on stan [Seed = 2480289706] Type ? for help. Type -D to quit. Computing space of Newforms.. 0.000 ..done 0.281 Starting elimination.. 0.282 ++++++++++++++++++++++ 0.282 working with form number => 12 The coefficient field Kf contains K so the compositum is Kf first bound obtained with auxiliary prime q= 13 trying auxiliary prime q = 17 { 3, 5, 7 } trying auxiliary prime q = 29 { 7 } { 7 } ++++++++++++++++++++++ 0.325 working with form number => 16 The coefficient field Kf contains K so the compositum is Kf first bound obtained with auxiliary prime q= 11 trying auxiliary prime q = 13 { 2, 13 } trying auxiliary prime q = 17 { 2, 13 } trying auxiliary prime q = 29 { 2, 13 } { 2, 13 } **** not eliminated ++++++++++++++++++++++ 0.343 working with form number => 17 The coefficient field Kf contains K so the compositum is Kf first bound obtained with auxiliary prime q= 5 trying auxiliary prime q = 11 { 2, 3, 5 } trying auxiliary prime q = 13 { 3 } { 3 } ++++++++++++++++++++++ 0.344 working with form number => 18 The coefficient field Kf contains K so the compositum is Kf first bound obtained with auxiliary prime q= 5 trying auxiliary prime q = 11 { 2, 3, 5, 7 } trying auxiliary prime q = 13 { 2, 3, 7 } { 2, 3, 7 } ++++++++++++++++++++++ 0.344 working with form number => 19 The coefficient field Kf contains K so the compositum is Kf first bound obtained with auxiliary prime q= 5 trying auxiliary prime q = 11 { 2, 3, 5, 7 } trying auxiliary prime q = 13 { 2, 3, 7 } { 2, 3, 7 } ++++++++++++++++++++++ 0.345 working with form number => 20 The coefficient field Kf contains K so the compositum is Kf first bound obtained with auxiliary prime q= 11 trying auxiliary prime q = 13 { 3, 7, 13, 41, 71 } trying auxiliary prime q = 17 { 3, 7 } { 3, 7 } ++++++++++++++++++++++ 0.349 working with form number => 21 The coefficient field Kf contains K so the compositum is Kf first bound obtained with auxiliary prime q= 11 trying auxiliary prime q = 13 { 3, 7, 13, 41, 71 } trying auxiliary prime q = 17 { 3, 7 } { 3, 7 } ++++++++++++++++++++++ 0.352 working with form number => 22 The coefficient field Kf contains K so the compositum is Kf first bound obtained with auxiliary prime q= 11 trying auxiliary prime q = 13 { 3, 7, 13, 29 } trying auxiliary prime q = 17 { 3, 7, 13 } trying auxiliary prime q = 29 { 3, 7, 13 } { 3, 7, 13 } **** not eliminated ++++++++++++++++++++++ 0.388 working with form number => 23 The coefficient field Kf contains K so the compositum is Kf first bound obtained with auxiliary prime q= 11 trying auxiliary prime q = 13 { 3, 7, 13, 29 } trying auxiliary prime q = 17 { 3, 7, 13 } trying auxiliary prime q = 29 { 3, 7, 13 } { 3, 7, 13 } **** not eliminated ++++++++++++++++++++++ 0.438 working with form number => 24 The coefficient field Kf contains K so the compositum is Kf first bound obtained with auxiliary prime q= 5 trying auxiliary prime q = 11 { 2, 3, 5 } trying auxiliary prime q = 13 { 3 } { 3 } ++++++++++++++++++++++ 0.440 working with form number => 26 The coefficient field Kf contains K so the compositum is Kf first bound obtained with auxiliary prime q= 5 trying auxiliary prime q = 11 { 2, 3, 5 } trying auxiliary prime q = 13 { 2 } { 2 } ++++++++++++++++++++++ 0.442 working with form number => 28 The coefficient field Kf contains K so the compositum is Kf first bound obtained with auxiliary prime q= 13 trying auxiliary prime q = 17 { 3, 5, 7 } trying auxiliary prime q = 29 { 7 } { 7 } ++++++++++++++++++++++ 0.505 working with form number => 33 The coefficient field Kf contains K so the compositum is Kf first bound obtained with auxiliary prime q= 5 trying auxiliary prime q = 11 { 2, 3, 5, 31 } trying auxiliary prime q = 13 { 2, 3 } { 2, 3 } ++++++++++++++++++++++ 0.507 working with form number => 38 The coefficient field Kf contains K so the compositum is Kf first bound obtained with auxiliary prime q= 5 trying auxiliary prime q = 11 { 2, 7, 11, 17 } trying auxiliary prime q = 13 { 2, 7 } { 2, 7 } ++++++++++++++++++++++ 0.510 working with form number => 41 The coefficient field Kf contains K so the compositum is Kf first bound obtained with auxiliary prime q= 5 trying auxiliary prime q = 11 { 2, 3, 5, 7, 17 } trying auxiliary prime q = 13 { 2, 3, 7 } { 2, 3, 7 } ++++++++++++++++++++++ 0.512 working with form number => 42 The coefficient field Kf contains K so the compositum is Kf first bound obtained with auxiliary prime q= 5 trying auxiliary prime q = 11 { 2, 3, 7, 11 } trying auxiliary prime q = 13 { 2, 3, 7 } { 2, 3, 7 } ++++++++++++++++++++++ 0.515 working with form number => 45 The coefficient field Kf contains K so the compositum is Kf first bound obtained with auxiliary prime q= 5 trying auxiliary prime q = 11 { 2, 3, 7, 11, 17, 137 } trying auxiliary prime q = 13 { 2, 3, 7 } { 2, 3, 7 } ++++++++++++++++++++++ 0.519 working with form number => 46 The coefficient field Kf contains K so the compositum is Kf first bound obtained with auxiliary prime q= 5 trying auxiliary prime q = 11 { 2, 3, 7, 19, 31 } trying auxiliary prime q = 13 { 3, 7 } { 3, 7 } ++++++++++++++++++++++ 0.522 working with form number => 47 The coefficient field Kf contains K so the compositum is Kf first bound obtained with auxiliary prime q= 5 trying auxiliary prime q = 11 { 2, 3, 7, 19, 31 } trying auxiliary prime q = 13 { 3, 7 } { 3, 7 } ++++++++++++++++++++++ 0.526 working with form number => 48 The coefficient field Kf contains K so the compositum is Kf first bound obtained with auxiliary prime q= 5 trying auxiliary prime q = 11 { 2, 7 } { 2, 7 } ++++++++++++++++++++++ 0.529 working with form number => 51 The coefficient field Kf contains K so the compositum is Kf first bound obtained with auxiliary prime q= 5 trying auxiliary prime q = 11 { 2, 7 } { 2, 7 } ++++++++++++++++++++++ 0.531 working with form number => 57 The coefficient field Kf contains K so the compositum is Kf first bound obtained with auxiliary prime q= 5 trying auxiliary prime q = 11 { 2, 3, 5, 7, 11, 17 } trying auxiliary prime q = 13 { 3, 5, 7 } trying auxiliary prime q = 17 { 5, 7 } trying auxiliary prime q = 29 { 7 } { 7 } ++++++++++++++++++++++ 0.711 working with form number => 58 The coefficient field Kf contains K so the compositum is Kf first bound obtained with auxiliary prime q= 5 trying auxiliary prime q = 11 { 2, 3, 5, 7, 11, 17 } trying auxiliary prime q = 13 { 3, 5, 7 } trying auxiliary prime q = 17 { 5, 7 } trying auxiliary prime q = 29 { 7 } { 7 } Up to this point we have eliminated all but the forms indexed by i=60,61 and those indexed by i=16,22,23 for exponent p=13 ++++++++++++++++++++++ 0.906 We now apply refined elimination for the latter three forms using auxiliary prime q=29 Applying refined elimination with form i=16 and exponent p=13 using auxiliary prime q=29 Degree of coefficient field is 3 Degree of compositum of K and coefficient field is 3 Eliminated form i=16 completely. ++++++++++++++++++++++ 1.000 Applying refined elimination with form i=22 and exponent p=13 using auxiliary prime q=29 Degree of coefficient field is 3 Degree of compositum of K and coefficient field is 3 Eliminated form i=22 completely. ++++++++++++++++++++++ 1.105 Applying refined elimination with form i=23 and exponent p=13 using auxiliary prime q=29 Degree of coefficient field is 3 Degree of compositum of K and coefficient field is 3 Eliminated form i=23 completely. ++++++++++++++++++++++ 1.221 Up to this point we have eliminated all but forms i=60,61. These have field of coefficients of degree 54 which causes the computations to take much longer for each of them, but the elimination procedure is exactly the same. Starting elimination of forms i=60,61 ++++++++++++++++++++++ 1.221 working with form number => 60 first bound obtained with auxiliary prime q= 5 trying auxiliary prime q = 11 { 2, 5, 7, 11, 23, 47 } trying auxiliary prime q = 13 { 2, 7 } { 2, 7 } ++++++++++++++++++++++ 3.503 working with form number => 61 first bound obtained with auxiliary prime q= 5 trying auxiliary prime q = 11 { 2, 5, 7, 11, 23, 47 } trying auxiliary prime q = 13 { 2, 7 } { 2, 7 } We have eliminated all the forms, hence completing the proof of Theorem 6.25 ++++++++++++++++++++++ 5.795 Total time: 20862.120 seconds, Total memory usage: 1954.75MB