// Make newform 400.3.p.g in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_400_p();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_400_p_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_400_3_p_g();". // 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_400_3_p_g();". // 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, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]]; 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_400_p();" function MakeCharacter_400_p() N := 400; order := 4; char_gens := [351, 101, 177]; v := [4, 4, 3]; // 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_400_p_Hecke();" function MakeCharacter_400_p_Hecke(Kf) N := 400; order := 4; char_gens := [351, 101, 177]; char_values := [[1, 0], [1, 0], [0, 1]]; 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], [3, -3], [0, 0], [-3, -3], [-12, 0], [12, -12], [-12, -12], [0, -20], [3, -3], [0, -30], [8, 0], [48, 48], [-48, 0], [-27, 27], [27, 27], [12, -12], [0, 60], [32, 0], [-3, -3], [48, 0], [12, -12], [0, -40], [93, -93], [0, 30], [-12, -12], [-78, 0], [93, -93], [27, 27], [0, 160], [72, -72], [117, 117], [-132, 0], [168, 168], [0, 100], [0, 0], [248, 0], [-72, -72], [93, -93], [-3, -3], [-168, 168], [0, 300], [142, 0], [-192, 0], [132, -132], [-132, -132], [0, -160], [28, 0], [-117, 117], [-93, -93], [0, -370], [252, -252], [0, -360], [32, 0], [-252, 0], [-192, -192], [333, -333], [0, 480], [88, 0], [288, 288], [-288, 0], [-117, 117], [-168, 168], [-243, -243], [-552, 0], [-48, 48], [228, 228], [148, 0], [-192, -192], [117, 117], [0, -130], [-288, 288], [0, 120], [-213, -213], [-168, 168], [0, -20], [123, -123], [0, 0], [108, 108], [-18, 0], [0, 80], [0, 540], [-608, 0], [-312, 0], [252, -252], [0, -40], [213, -213], [0, 480], [-432, -432], [222, 0], [213, -213], [-3, -3], [0, 240], [627, 627], [588, 0], [0, -460], [-627, 627], [0, 450], [-558, 0], [123, -123], [542, 0], [147, 147], [288, 288], [-477, 477], [0, -240], [-692, 0], [168, 168], [-213, -213], [312, -312], [0, -240], [-608, 0], [267, 267], [-228, 228], [348, 348], [0, -940], [808, 0], [-768, 0], [-477, 477], [627, 627], [12, -12], [0, 540], [352, 0], [732, -732], [108, 108], [933, -933], [68, 0], [192, 0], [0, 50], [0, -840], [-963, -963], [72, -72], [0, -20], [243, -243], [-1072, 0], [408, 408], [1362, 0], [0, 370], [132, -132], [-93, -93], [228, 228], [0, -750], [-412, 0], [672, 0], [-717, 717], [-123, -123], [0, 1280], [0, -1560], [372, -372], [-552, -552], [0, 620], [123, -123], [1128, 1128], [912, 0], [-957, 957], [-483, -483], [1077, 1077], [1128, 0], [0, 1600], [0, 960], [-492, -492], [-738, 0], [237, 237], [-648, 648], [627, 627], [708, 0], [-612, -612], [-627, 627], [1168, 0], [108, 108]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_400_p_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_400_3_p_g();". // 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_400_3_p_g(:prec:=2) chi := MakeCharacter_400_p(); 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_400_3_p_g();". // 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_400_3_p_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_400_p(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign))); Vf := Kernel([<3,R![18, -6, 1]>,<7,R![18, 6, 1]>],Snew); return Vf; end function;