
// Make newform 576.3.g.e in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_576_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_576_g_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_576_3_g_e();".
// 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_576_3_g_e();".
// 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;
    Rf_num := [[1, 0], [-4, 8]];
    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_576_g();"
function MakeCharacter_576_g()
    N := 576;
    order := 2;
    char_gens := [127, 325, 65];
    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_576_g_Hecke();"
function MakeCharacter_576_g_Hecke(Kf)
    N := 576;
    order := 2;
    char_gens := [127, 325, 65];
    char_values := [[-1, 0], [1, 0], [1, 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<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 := 3;
    raw_aps := [[0, 0], [0, 0], [-2, 0], [0, -1], [0, 1], [-2, 0], [-10, 0], [0, -3], [0, 4], [-26, 0], [0, 1], [-26, 0], [-58, 0], [0, 7], [0, -10], [-74, 0], [0, -13], [-26, 0], [0, -1], [0, 0], [-46, 0], [0, 17], [0, 7], [-82, 0], [2, 0], [-74, 0], [0, -11], [0, 3], [46, 0], [110, 0], [0, -21], [0, 17], [-10, 0], [0, -7], [-2, 0], [0, -13], [214, 0], [0, -3], [0, -14], [334, 0], [0, -27], [-2, 0], [0, 32], [290, 0], [-26, 0], [0, 57], [0, 35], [0, -49], [0, -41], [142, 0], [-82, 0], [0, -56], [-46, 0], [0, 21], [254, 0], [0, -22], [262, 0], [0, 3], [-290, 0], [-226, 0], [0, -43], [-362, 0], [0, -21], [0, 34], [-478, 0], [-170, 0], [0, -59], [338, 0], [0, 29], [-506, 0], [-178, 0], [0, -24], [0, 29], [310, 0], [0, 63], [0, 88], [-578, 0], [-26, 0], [-250, 0], [290, 0], [0, -49], [-674, 0], [0, -78], [-334, 0], [0, 17], [0, -11], [-394, 0], [-478, 0], [142, 0], [0, 91], [0, 3], [0, -106], [0, 15], [0, -133], [0, 11], [0, 84], [-842, 0], [326, 0], [0, -45], [-530, 0], [0, 49], [766, 0], [0, 71], [422, 0], [0, 41], [-46, 0], [0, 91], [-82, 0], [0, -8], [-334, 0], [0, -53], [214, 0], [1118, 0], [0, 97], [0, -21], [-10, 0], [0, -175], [0, 48], [670, 0], [0, -119], [1222, 0], [-334, 0], [1006, 0], [0, -27], [0, -143], [-1034, 0], [-530, 0], [0, 102], [0, 35], [-194, 0], [0, 195], [0, -98], [0, -95], [1006, 0], [758, 0], [2, 0], [262, 0], [0, 209], [-866, 0], [-10, 0], [0, 63], [838, 0], [0, 127], [0, -105], [-1298, 0], [0, -28], [-506, 0], [998, 0], [0, -73], [0, -24], [646, 0], [-898, 0], [0, 105], [0, -122], [0, -197], [0, 56], [0, 87], [-1594, 0], [674, 0], [430, 0], [0, -11], [-730, 0], [0, 133], [0, 213], [-346, 0], [0, -106], [0, 141], [-458, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_576_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_576_3_g_e();".
// 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_576_3_g_e(:prec:=2)
    chi := MakeCharacter_576_g();
    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_576_3_g_e();".
// 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_576_3_g_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_576_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![2, 1]>],Snew);
    return Vf;
end function;