
// Make newform 6240.2.a.bn in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6240_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_6240_2_a_bn();".
// 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_6240_2_a_bn();".
// 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 := [-2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 2]];
    Rf_basisdens := [1, 1];
    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_6240_a();"
function MakeCharacter_6240_a()
    N := 6240;
    order := 1;
    char_gens := [1951, 2341, 2081, 2497, 5761];
    v := [1, 1, 1, 1, 1];
    // 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;

function MakeCharacter_6240_a_Hecke(Kf)
    return MakeCharacter_6240_a();
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<X, Y> := 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<T> := 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 := 2;
    raw_aps := [[0, 0], [-1, 0], [1, 0], [0, 1], [-4, 0], [-1, 0], [2, -1], [0, -1], [0, 1], [-2, 1], [4, -2], [2, -2], [6, 2], [-4, 2], [-4, -2], [2, -4], [-4, 0], [-2, 0], [0, 0], [-8, 0], [6, -1], [0, -2], [-4, -4], [-2, 2], [-10, 3], [-2, 3], [-8, 2], [4, 0], [-6, 3], [-6, -3], [8, -2], [-8, -1], [2, 4], [-8, 2], [-10, -2], [4, 6], [10, 4], [0, 6], [-4, -4], [-14, 0], [0, -3], [14, 2], [-8, -4], [-2, -7], [-2, 4], [0, -6], [-16, 0], [0, -1], [-12, 2], [-6, -1], [18, 3], [0, 6], [10, 4], [-8, 1], [-6, 1], [0, -5], [-10, 3], [-4, -2], [10, -8], [-10, -6], [-20, -2], [-10, 0], [16, 0], [-8, -8], [-6, 6], [-2, 4], [-16, 3], [18, 2], [-4, 2], [18, -1], [-14, 4], [0, 4], [-8, 2], [2, 4], [0, -7], [12, 8], [-18, -5], [10, -2], [-2, -10], [-6, -2], [-16, 3], [2, 1], [-8, 6], [-14, -2], [8, 0], [-4, 6], [-10, -2], [14, -5], [-34, 0], [8, -5], [-4, -4], [16, -2], [-8, 1], [-16, 3], [-16, 1], [24, 3], [-10, 4], [-30, 2], [-20, 8], [-14, 9], [12, -4], [6, -12], [-20, 2], [-14, 10], [-24, 6], [22, -3], [-12, 2], [10, -2], [32, 0], [-14, 4], [0, -2], [-30, 6], [-6, 8], [32, 1], [4, 0], [18, -8], [8, -8], [-16, -9], [10, 10], [-24, 7], [2, 1], [10, 0], [-22, 10], [4, -12], [-32, -1], [-18, 3], [10, -13], [32, 2], [-8, 8], [18, -6], [8, -11], [-12, 0], [40, 4], [34, -4], [-2, 6], [2, 12], [-42, -4], [48, -2], [10, -8], [-14, -4], [-16, 11], [14, -10], [0, -2], [20, 4], [30, 4], [8, -6], [18, 6], [-6, 1], [-8, 0], [20, 0], [42, -4], [-6, 6], [-20, 0], [8, 13], [44, 2], [8, -6], [0, 8], [-2, -10], [10, 0], [-34, 2], [12, 0], [-6, 19], [16, -7], [8, 1], [2, 14], [4, -8], [-8, -2], [-14, 4], [-22, 12], [-22, 12], [28, -2], [42, -5], [-40, -2], [-18, 5], [0, 14], [-22, -10], [0, -1], [-10, -6], [56, 0], [30, 10], [24, -11], [-24, 3], [2, -4], [-14, -6], [-36, 8], [22, -11], [-30, 0], [0, 12], [-22, 2], [32, -10], [-6, -16], [4, 6], [-8, -4], [-18, 2], [28, 2], [-30, -11], [42, 6], [-6, 0], [-22, -10], [32, 3], [-18, -4], [16, -4], [-6, -4], [-6, -16], [28, 8], [-14, 10], [28, -2], [-4, -6], [-10, -14], [-16, -4], [18, -10], [22, 5], [-16, 16], [52, 4], [-8, 2], [-38, -12], [24, 8], [-14, 14], [12, -20], [38, -4], [6, 2], [44, 0], [6, 14], [16, -15], [20, 10], [-10, 2], [50, -1], [56, -2], [-40, -8], [36, 0], [10, -16], [0, -8], [-12, 10], [-6, 6], [60, -2], [4, 0], [2, -6], [-34, 8], [0, 5], [-16, 6], [-28, -12], [16, -16], [-24, 16], [26, 13], [-30, 10], [-16, -16], [-8, 11], [28, 10], [0, -1], [-40, 7], [2, 18], [22, 6], [-4, 18], [-6, -8], [-14, -4], [-28, -14], [-10, 12], [16, 14], [-46, 12], [30, -1], [8, -16], [-12, 20], [26, 3], [10, 12], [-54, -2], [-40, 6], [14, 4], [30, -2], [-16, -8], [18, 10], [14, 6], [32, -6], [-34, -11], [-32, 0], [10, 0], [-16, -7], [36, 6], [-6, 5], [2, 22], [8, -3], [-8, 15], [-44, 8], [24, -1], [2, -5], [48, 8], [-24, 10], [10, 14], [54, -4], [20, 4], [26, 16], [-50, -11], [44, -4], [42, -6], [12, -6], [10, 8], [-26, -9], [0, 14], [42, 2], [-48, 7], [56, 2], [-22, -10], [-10, 24], [40, -6], [-52, 10], [-32, -8], [-34, -15], [-12, 4], [14, -10], [32, -14], [-22, -4], [-24, -7], [-2, 8], [-6, 2], [12, -4], [-20, 2], [-6, 0], [36, 4], [0, -16], [-50, -1], [-30, -4], [-32, 14], [-34, -5], [-82, 3], [48, -5], [26, 14], [26, -16], [40, 9], [32, -12], [32, 15], [58, -4], [10, -15], [-38, -2], [-16, 24], [-28, -10], [-32, -5], [36, -10], [-30, -11], [50, -8], [10, 18], [-32, -2], [-46, -8], [26, -11], [14, 10], [-48, -2], [-10, -20], [-56, -13], [30, 4], [40, 12], [-32, -16], [42, -14], [0, -19], [22, 9], [22, 26], [-24, -4], [62, 6], [18, 15], [32, 18], [28, 12], [-78, 3], [76, 2], [50, 12], [-38, -4], [32, 3], [-36, 10], [-4, -30], [-30, -22], [54, 4], [-8, -17], [-6, 16], [-16, 11], [32, 6], [-4, 28], [14, -27], [-48, -2], [-6, -4], [-12, 18], [-56, 16], [-74, 5], [34, -2], [-46, 14], [14, -6], [-14, -12], [-56, -13], [-14, 2], [24, -3], [-92, -2], [4, -20], [18, -16], [56, 8], [-40, 7], [-6, 24], [14, -20], [-12, -28], [12, 8], [72, 8], [18, 6], [-4, -26], [10, -4], [-8, 16], [-26, 12], [66, 1], [2, -27], [24, 23], [-14, 22], [46, -8], [24, 0], [18, 18], [-58, -10], [-56, 10], [84, 2], [58, -10], [66, 0], [-20, 18], [-16, -18], [18, 16], [-58, 5], [8, -6], [0, -10], [10, 10], [-24, 7], [-58, 13], [18, 20], [-28, -6], [8, 1], [-34, -13], [54, 16], [20, -12], [38, 2], [-8, -3], [24, -16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6240_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6240_2_a_bn();".
// 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_6240_2_a_bn(:prec:=2)
    chi := MakeCharacter_6240_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_6240_2_a_bn();".
// 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_6240_2_a_bn( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6240_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-8, 0, 1]>,<11,R![4, 1]>,<17,R![-4, -4, 1]>,<19,R![-8, 0, 1]>],Snew);
    return Vf;
end function;