
// Make newform 720.3.j.c in Magma, downloaded from the LMFDB on 13 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_720_j();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_720_j_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_720_3_j_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_720_3_j_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, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 3]];
    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_720_j();"
function MakeCharacter_720_j()
    N := 720;
    order := 2;
    char_gens := [271, 181, 641, 577];
    v := [1, 2, 2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_720_j_Hecke();"
function MakeCharacter_720_j_Hecke( : Kf := false)
    N := 720;
    order := 2;
    char_gens := [271, 181, 641, 577];
    char_values := [[-1, 0], [1, 0], [1, 0], [-1, 0]];
    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], [4, 1], [0, 0], [0, 0], [0, -8], [0, -10], [0, 0], [0, 0], [40, 0], [0, 0], [0, 8], [80, 0], [0, 0], [0, 0], [0, -30], [0, 0], [22, 0], [0, 0], [0, 0], [0, 32], [0, 0], [0, 0], [160, 0], [0, -48], [40, 0], [0, 0], [0, 0], [-182, 0], [0, -10], [0, 0], [0, 0], [0, -70], [0, 0], [-280, 0], [0, 0], [0, 88], [0, 0], [0, 0], [0, 110], [0, 0], [38, 0], [0, 0], [0, -112], [0, 130], [0, 0], [0, 0], [0, 0], [0, 0], [-442, 0], [0, -70], [0, 0], [418, 0], [0, 0], [0, 170], [0, 0], [-520, 0], [0, 0], [0, -168], [320, 0], [0, 0], [0, 190], [0, 0], [0, 0], [0, 208], [0, -50], [0, 0], [0, 192], [0, 0], [-598, 0], [0, -150], [0, 0], [0, 0], [0, 168], [0, 0], [0, 0], [680, 0], [0, -152], [-80, 0], [-782, 0], [0, 0], [58, 0], [0, 0], [0, -272], [0, 0], [0, 0], [560, 0], [0, 112], [760, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-440, 0], [880, 0], [0, 0], [-682, 0], [0, 0], [0, 110], [0, 0], [1040, 0], [0, 0], [0, -32], [0, 0], [0, 310], [0, 0], [1102, 0], [0, 0], [0, 408], [0, 70], [0, 0], [0, 0], [-400, 0], [0, 0], [0, 0], [0, 210], [0, 0], [-1178, 0], [0, 368], [0, -450], [0, 0], [0, 0], [-520, 0], [518, 0], [0, 0], [0, 0], [0, 72], [0, 0], [0, 0], [0, 0], [0, 312], [-1520, 0], [962, 0], [0, 130], [0, 0], [0, 370], [-560, 0], [0, 0], [1400, 0], [0, 0], [0, 0], [1258, 0], [0, 0], [0, -552], [0, 550], [0, 0], [0, 0], [0, -232], [-1600, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-1840, 0], [0, 608], [-1160, 0], [0, 0], [0, -410], [0, 0], [0, 0], [0, 630], [0, 0], [0, 0], [0, 248]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_720_j_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_720_3_j_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_720_3_j_c(:prec:=2)
    chi := MakeCharacter_720_j();
    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_720_3_j_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_720_3_j_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_720_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![0, 1]>,<11,R![0, 1]>,<29,R![-40, 1]>],Snew);
    return Vf;
end function;