
// Make newform 756.3.bk.c in Magma, downloaded from the LMFDB on 15 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_756_bk();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_756_bk_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_756_3_bk_c();".
// 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_756_3_bk_c();".
// 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, -1, 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_756_bk();"
function MakeCharacter_756_bk()
    N := 756;
    order := 6;
    char_gens := [379, 29, 325];
    v := [6, 3, 4];
    // 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_756_bk_Hecke();"
function MakeCharacter_756_bk_Hecke( : Kf := false)
    N := 756;
    order := 6;
    char_gens := [379, 29, 325];
    char_values := [[1, 0], [-1, 0], [-1, 1]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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<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]];
    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], [0, 0], [8, -5], [0, 0], [-22, 0], [0, 0], [37, -37], [0, 0], [0, 0], [0, 13], [-26, 26], [0, 0], [-61, 0], [0, 0], [0, 0], [0, 0], [121, -121], [0, -122], [0, 0], [0, -143], [142, -142], [0, 0], [0, 0], [167, 0], [0, 0], [-194, 194], [0, 0], [0, -71], [0, 0], [107, 0], [0, 0], [0, 0], [-22, 0], [0, 0], [0, -59], [0, 118], [-299, 299], [0, 0], [0, 0], [0, 0], [-1, 0], [0, 0], [0, 382], [0, 0], [0, 277], [-253, 0], [338, 0], [0, 0], [409, -409], [0, 0], [0, 0], [0, 193], [0, 0], [0, 0], [0, 0], [0, 0], [-299, 299], [0, -407], [0, 0], [0, 517], [0, 0], [611, 0], [0, 0], [457, -457], [0, 0], [-299, 299], [167, 0], [0, 0], [671, 0], [0, 0], [0, 0], [0, -227], [121, -121], [-694, 0], [0, 0], [0, 0], [-431, 431], [0, 0], [0, -626], [0, 0], [839, 0], [0, 0], [359, 0], [94, -94], [0, 0], [0, 0], [-767, 767], [0, 0], [923, 0], [0, 0], [0, 0], [0, 613], [0, 0], [-851, 851], [0, 0], [0, 0], [0, 0], [982, -982], [793, -793], [-1093, 0], [0, 0], [0, 0], [0, 0], [0, -1067], [0, -962], [0, 0], [0, 0], [0, 0], [1199, 0], [373, -373], [0, -983], [0, 0], [0, 214], [587, 0], [0, 0], [-1237, 0], [0, 0], [0, 0], [0, 0], [0, 1201], [-1177, 0], [0, 0], [0, 0], [-1019, 1019], [0, 0], [-1391, 1391], [0, 0], [947, 0], [1417, -1417], [0, -1331], [0, 0], [1381, -1381], [-673, 0], [0, 0], [671, 0], [0, 0], [0, 613], [0, 0], [0, 0], [1514, 0], [0, 0], [0, 1621], [0, 0], [0, -1151], [0, 0], [-1177, 0], [0, 0], [1549, -1549], [0, 0], [-1727, 1727], [0, 0], [1259, 0], [0, 0], [0, 214], [0, 0], [1837, -1837], [0, 0], [-1198, 0], [0, 0], [0, 0], [0, 0], [-1534, 0], [0, 0], [0, 0], [0, 0], [0, 46], [0, -407]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_756_bk_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_756_3_bk_c();".
// 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_756_3_bk_c(:prec:=2)
    chi := MakeCharacter_756_bk();
    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_756_3_bk_c();".
// 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_756_3_bk_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_756_bk();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![0, 1]>,<13,R![22, 1]>],Snew);
    return Vf;
end function;