// Make newform 900.3.c.n in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_900_c();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_900_c_Hecke();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_900_3_c_n();". // This may take a long time! To see verbose output, uncomment the SetVerbose lines below. // The precision argument determines an initial guess on how many Fourier coefficients to use. // This guess is increased enough to uniquely determine the newform. // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_900_3_c_n();". // This may take a long time! To see verbose output, uncomment the SetVerbose line below. // The default sign is -1. You can change this with the optional parameter "sign". function ConvertToHeckeField(input: pass_field := false, Kf := []) if not pass_field then poly := [1, -12, 55, -84, 50, 0, -7, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [-5669, 20620, -20680, 291, 3897, -16, -599, -73], [3582, -21000, 27350, -14163, -381, 1968, 32, -291], [-1399, 17810, -30040, 17961, 157, -2506, -24, 347], [-1399, 16780, -30040, 17961, 157, -2506, -24, 347], [2372, -19070, 34050, -23728, 914, 3308, -143, -486], [-5953, 44700, -72710, 45672, -606, -6302, 132, 924], [-5363, 45870, -85630, 57082, -1446, -8062, 162, 1134]]; Rf_basisdens := [1, 515, 515, 515, 515, 515, 515, 515]; Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf); Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]]; inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf); return Eltseq(inp_vec); end function; // To make the character of type GrpDrchElt, type "MakeCharacter_900_c();" function MakeCharacter_900_c() N := 900; order := 2; char_gens := [451, 101, 577]; v := [1, 2, 2]; // chi(gens[i]) = zeta^v[i] assert UnitGenerators(DirichletGroup(N)) eq char_gens; F := CyclotomicField(order); chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]); return MinimalBaseRingCharacter(chi); end function; // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_900_c_Hecke();" function MakeCharacter_900_c_Hecke(Kf) N := 900; order := 2; char_gens := [451, 101, 577]; char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0]]; assert UnitGenerators(DirichletGroup(N)) eq char_gens; values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field F := Universe(values);// the Hecke field chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values); return chi; end function; function ExtendMultiplicatively(weight, aps, character) prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec primes := PrimesUpTo(prec); prime_powers := primes; assert #primes eq #aps; log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1) F := Universe(aps); FXY := PolynomialRing(F, 2); // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ... R := PowerSeriesRing(FXY : Precision := log_prec + 1); recursion := Coefficients(1/(1 - X*T + Y*T^2)); coeffs := [F!0: i in [1..(prec+1)]]; coeffs[1] := 1; //a_1 for i := 1 to #primes do p := primes[i]; coeffs[p] := aps[i]; b := p^(weight - 1) * F!character(p); r := 2; p_power := p * p; //deals with powers of p while p_power le prec do Append(~prime_powers, p_power); coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]); p_power *:= p; r +:= 1; end while; end for; Sort(~prime_powers); for pp in prime_powers do for k := 1 to Floor(prec/pp) do if GCD(k, pp) eq 1 then coeffs[pp*k] := coeffs[pp]*coeffs[k]; end if; end for; end for; return coeffs; end function; function qexpCoeffs() // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" weight := 3; raw_aps := [[0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, -1, -1], [1, 1, 2, 2, -1, 1, 1, 0], [2, -1, 0, 4, 1, 1, 0, -1], [2, -2, 5, -1, 6, 2, 2, 1], [-1, -1, 2, 2, -7, -4, 1, -2], [-1, -1, -4, -4, 5, -5, 1, 4], [6, -2, 7, 1, 2, 2, 0, 5], [-4, -4, -4, -4, -4, -7, 3, 3], [-24, 2, -10, 2, -2, -2, 0, -8], [4, -2, -5, 9, 6, 2, 2, -9], [3, 3, 2, 2, 5, -8, 3, 4], [-2, -2, -8, -8, 10, 12, 4, 10], [38, 0, -1, -7, 8, 0, 4, -5], [-4, -4, 0, 0, -12, -2, 4, 0], [24, -7, 14, -2, 23, 7, 8, -1], [1, 1, 14, 14, -25, 14, 1, -12], [-5, -5, 0, 0, -15, 1, 3, -2], [20, 10, -18, -14, -18, -10, -4, -4], [2, 2, 12, 12, -18, -6, 0, -10], [4, 4, -4, -4, 20, 18, 12, 20], [-16, 0, 4, 4, -8, 0, -4, 8], [35, -8, 4, 12, 24, 8, 8, -12], [-18, 4, -7, -25, 12, -4, 8, -11], [0, 0, -20, -20, 40, 8, -14, 6], [-7, -7, 2, 2, -25, -33, 5, -4], [-58, -7, 14, -2, 23, 7, 8, -1], [-16, 4, -8, -16, 4, -4, 4, -8], [-10, -10, 0, 0, -30, -18, 18, 8], [-7, -7, -2, -2, -17, -33, 13, 8], [-86, -8, -6, 30, 16, 8, 4, -18], [-8, -8, -8, -8, -8, 60, -2, -2], [-44, 2, -6, -10, 6, -2, 4, -8], [-12, -12, 16, 16, -68, -45, -1, -29], [-16, -17, 46, 6, 33, 17, 8, 21], [17, 17, -2, -2, 55, -20, 1, 20], [1, 1, 24, 24, -45, 47, 5, -18], [34, 2, -13, 29, -26, -2, -12, 1], [6, 6, 8, 8, 2, -4, 10, 8], [-28, 3, -28, 16, -3, -3, 0, -25], [-1, -1, 0, 0, -3, 151, -7, -8], [-49, -10, 34, 14, 2, 10, -4, 28], [32, -10, -14, 62, 2, 10, -4, -20], [-10, -10, -16, -16, 2, -23, -3, 3], [-17, -17, 2, 2, -55, 64, 5, -14], [8, 8, -8, -8, 40, 47, -13, 3], [19, 19, 26, 26, 5, -47, -29, -36], [-10, 5, -2, -2, -21, -5, -8, 11], [-204, 18, -15, -13, -62, -18, -22, 25], [-5, -5, 28, 28, -71, 53, 11, -22], [-129, -2, 0, 24, -14, 2, -8, 6], [-6, -6, 24, 24, -66, -150, 26, -4], [-92, 28, -34, -46, -60, -28, -16, 10], [-9, -9, 24, 24, -75, -147, -1, -34], [90, 6, -33, 25, -22, -6, -8, -19], [-4, -4, 20, 20, -52, -136, 2, -22], [-4, 5, -16, -36, 27, -5, 16, -27], [-210, 14, 15, -83, -2, -14, 6, 23], [-11, -11, 6, 6, -45, -92, -13, -30], [120, 2, 24, -8, -26, -2, -12, 38], [25, 25, 30, 30, 15, 180, -17, -22], [-8, -8, 16, 16, -56, 66, -18, -42], [-7, 4, -6, -2, -12, -4, -4, 2], [-16, 4, 22, -46, 4, -4, 4, 22], [10, 10, 28, 28, -26, 218, 6, -12], [-303, 18, -26, 26, -90, -18, -36, 28], [-20, -20, -60, -60, 60, -90, -12, 28], [6, -8, -52, 84, 8, 8, 0, -60], [-10, -42, 73, 99, 38, 42, -2, 33], [-1, -1, -72, -72, 141, -75, -13, 58], [44, 44, 36, 36, 60, -67, 15, 23], [-88, 3, 0, 4, -19, -3, -8, 11], [15, 15, 34, 34, -23, 126, -1, -20], [2, 2, 28, 28, -50, -30, 6, -20], [-314, -4, 31, -31, 20, 4, 8, 19], [108, -1, -44, 64, -15, 1, -8, -37], [-204, 30, -91, 39, -98, -30, -34, -27], [-155, 12, -2, -6, -52, -12, -20, 30], [-23, -23, 6, 6, -81, 15, 9, -20], [-76, 6, 14, 42, -86, -6, -40, 60], [17, 17, -8, -8, 67, 165, 43, 68], [-329, -22, -6, 54, 62, 22, 20, -48], [-2, -2, -64, -64, 122, 23, -37, 25], [-42, -42, -28, -28, -70, -54, 14, 0], [268, 20, 18, -42, -76, -20, -28, 66], [-12, 10, -72, 0, 22, -10, 16, -78], [38, 8, -53, -19, 32, -8, 20, -65], [36, 36, 84, 84, -60, 168, -2, -50], [-21, -21, 6, 6, -75, -237, 59, 32], [2, 2, -4, -4, 14, 312, 0, 6], [-12, -12, -48, -48, 60, -327, -23, 13], [28, 28, 28, 28, 28, -172, 36, 36], [47, 47, 66, 66, 9, -16, -89, -108], [19, 19, -44, -44, 145, -103, 27, 90], [-252, -50, 54, 74, 122, 50, 36, -32], [-282, -26, 79, 21, 30, 26, 2, 51], [-21, -21, -54, -54, 45, 26, -43, -10], [16, 35, -36, -56, -83, -35, -24, 23], [0, 0, -40, -40, 80, 96, 86, 126], [96, 22, -76, 4, -38, -22, -8, -46], [-55, -55, -30, -30, -105, 177, 33, 8], [314, 42, -69, -63, -78, -42, -18, -9], [33, 33, -50, -50, 199, 114, -23, 60], [-171, 36, -8, -136, -36, -36, 0, 28], [45, 45, 70, 70, -5, 131, 45, 20], [-192, 58, -137, -75, -78, -58, -10, -69], [-14, -14, 48, 48, -138, -14, 50, -12], [-21, 52, -30, -74, -156, -52, -52, 74], [42, 42, 28, 28, 70, 138, -28, -14], [124, -54, 128, -88, 230, 54, 88, -14], [68, 34, -133, 97, -134, -34, -50, -49], [-17, -17, -102, -102, 153, 208, 1, 86], [50, 50, 68, 68, 14, -93, -47, -65], [-96, 8, -22, -18, 0, -8, 4, -18], [-24, -24, 104, 104, -280, -116, 98, -30], [-40, -40, 40, 40, -200, 108, -32, -112], [432, -58, -4, 132, 162, 58, 52, -114], [17, 17, 46, 46, -41, 19, -79, -108], [232, 50, -104, -144, -2, -50, 24, -78], [-420, 30, -42, -166, 58, -30, 44, -56], [-138, 58, -133, -75, -82, -58, -12, -63], [68, 68, 12, 12, 180, -84, -88, -32], [46, 46, 60, 60, 18, 26, -42, -56], [278, 90, -129, -79, -242, -90, -76, 37], [274, -83, 134, 70, 211, 83, 64, -13], [69, 69, -40, -40, 287, 153, -1, 108], [-12, -12, -168, -168, 300, -141, -65, 91], [622, 24, -4, -28, -88, -24, -32, 52], [-28, -28, -32, -32, -20, -264, 40, 44], [-44, -44, 44, 44, -220, -192, 116, 28], [22, 22, -32, -32, 130, 46, 110, 164], [-164, -55, 126, -50, 199, 55, 72, -1], [-452, 40, -120, -40, -40, -40, 0, -80], [347, -44, 154, 62, 4, 44, -20, 130], [690, -28, 57, 71, 12, 28, -8, 37], [13, 13, 62, 62, -85, 212, -83, -132], [136, 8, -6, 54, -88, -8, -40, 42], [740, 42, -77, -31, -102, -42, -30, -5], [29, 29, 30, 30, 27, 130, 75, 74], [364, -30, 128, 48, -26, 30, -28, 126], [-46, -46, -124, -124, 110, 89, -93, -15], [-3, -3, 38, 38, -85, 371, 45, 4], [-232, 66, -116, -68, -146, -66, -40, -10], [-35, -35, 124, 124, -353, -355, -9, -168], [246, 33, -126, 154, -193, -33, -80, -13], [-264, 72, -54, -170, -136, -72, -32, 50], [-42, -42, -92, -92, 58, 114, -46, 4], [-30, -30, -100, -100, 110, -66, -90, -20], [-330, 61, -54, -62, -189, -61, -64, 71], [144, -36, 134, -94, 140, 36, 52, 46], [77, 77, 38, 38, 155, -150, -33, 6], [70, 70, 20, 20, 170, 80, -36, 14], [44, 44, 16, 16, 100, -16, 52, 80], [18, 18, 124, 124, -194, -64, 104, -2], [28, 28, 76, 76, -68, 483, 53, 5], [406, -62, 217, -109, 202, 62, 70, 85], [105, 48, -186, 82, -136, -48, -44, -94], [-216, -58, 104, 40, 146, 58, 44, 2], [31, 31, 94, 94, -95, -5, 99, 36], [-600, 64, -4, -12, -304, -64, -120, 180], [-14, -14, 124, 124, -290, -406, 12, -126], [78, 78, 96, 96, 42, 272, -90, -108], [-622, -12, 66, 94, -100, 12, -56, 110], [-20, -20, -100, -100, 140, 72, -92, -12], [-134, -134, -124, -124, -154, -97, 13, 3], [-262, -16, -108, 92, 96, 16, 40, -164]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_900_c_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_900_3_c_n();". // This may take a long time! To see verbose output, uncomment the SetVerbose lines below. // The precision argument determines an initial guess on how many Fourier coefficients to use. // This guess is increased enough to uniquely determine the newform. function MakeNewformModFrm_900_3_c_n(:prec:=8) chi := MakeCharacter_900_c(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 3)); S := BaseChange(S, Kf); maxprec := NextPrime(997) - 1; while true do trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]); B := Basis(S, prec + 1); S_basismat := Matrix([AbsEltseq(g): g in B]); if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then S_basismat := ChangeRing(S_basismat,Kf); f_lincom := Solution(S_basismat,trunc_vec); f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]]; return f; end if; error if prec eq maxprec, "Unable to distinguish newform within newspace"; prec := Min(Ceiling(1.25 * prec), maxprec); end while; end function; // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_900_3_c_n();". // This may take a long time! To see verbose output, uncomment the SetVerbose line below. // The default sign is -1. You can change this with the optional parameter "sign". function MakeNewformModSym_900_3_c_n( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_900_c(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign))); Vf := Kernel([<7,R![961, 0, 9532, 0, 6470, 0, 188, 0, 1]>,<13,R![-1559, 3916, -418, -4, 1]>,<17,R![132400, -1920, -832, 0, 1]>],Snew); return Vf; end function;