
// Make newform 576.3.g.j in Magma, downloaded from the LMFDB on 06 May 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_j();".
// 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_j();".
// 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, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 0, 4], [0, 8, 0, -4], [-4, 0, 8, 0]];
    Rf_basisdens := [1, 1, 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, 0, 0], [1, 0, 0, 0], [1, 0, 0, 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], [0, 0, 0, 0], [-2, 0, 1, 0], [0, 1, 0, -1], [0, -2, 0, -1], [-10, 0, -2, 0], [6, 0, 2, 0], [0, -6, 0, -1], [0, -2, 0, 0], [-10, 0, -1, 0], [0, 1, 0, 5], [-18, 0, 4, 0], [22, 0, -2, 0], [0, -10, 0, 5], [0, -14, 0, -2], [6, 0, -1, 0], [0, -8, 0, -11], [14, 0, 0, 0], [0, 8, 0, -7], [0, -10, 0, 12], [-30, 0, 8, 0], [0, -19, 0, -3], [0, 14, 0, 1], [78, 0, -4, 0], [-62, 0, -12, 0], [166, 0, -1, 0], [0, -1, 0, -3], [0, -12, 0, 13], [6, 0, 2, 0], [-82, 0, 12, 0], [0, 3, 0, 23], [0, 16, 0, -1], [-26, 0, 26, 0], [0, 16, 0, 15], [62, 0, 9, 0], [0, -7, 0, 11], [-194, 0, 4, 0], [0, -18, 0, 15], [0, 56, 0, -2], [46, 0, 17, 0], [0, -24, 0, -13], [54, 0, -14, 0], [0, -8, 0, -8], [82, 0, -8, 0], [-106, 0, -21, 0], [0, 43, 0, -15], [0, 56, 0, 5], [0, -17, 0, 11], [0, 58, 0, 1], [-26, 0, 30, 0], [302, 0, 12, 0], [0, 4, 0, 56], [-190, 0, 20, 0], [0, -18, 0, 11], [-130, 0, 32, 0], [0, -64, 0, -2], [-138, 0, -9, 0], [0, -81, 0, -25], [22, 0, 34, 0], [-194, 0, 0, 0], [0, 64, 0, 11], [-26, 0, 71, 0], [0, -24, 0, 45], [0, -68, 0, 14], [-46, 0, -56, 0], [-154, 0, 35, 0], [0, -20, 0, 27], [-94, 0, 8, 0], [0, -58, 0, 43], [-146, 0, -32, 0], [-242, 0, 12, 0], [0, -42, 0, 4], [0, -31, 0, -55], [62, 0, -40, 0], [0, -46, 0, -3], [0, -64, 0, -48], [-98, 0, -63, 0], [174, 0, 24, 0], [-74, 0, 54, 0], [98, 0, 60, 0], [0, -94, 0, 41], [278, 0, 34, 0], [0, -18, 0, -54], [210, 0, -24, 0], [0, -97, 0, 1], [0, -106, 0, 35], [198, 0, -86, 0], [610, 0, -12, 0], [494, 0, 57, 0], [0, 103, 0, 15], [0, 18, 0, 53], [0, 14, 0, -34], [0, 153, 0, -17], [0, 4, 0, -35], [0, -32, 0, 53], [0, -50, 0, 72], [-346, 0, -53, 0], [-458, 0, -42, 0], [0, -14, 0, 57], [38, 0, -6, 0], [0, -110, 0, -13], [798, 0, -11, 0], [0, 14, 0, -7], [182, 0, 46, 0], [0, 140, 0, -1], [834, 0, 8, 0], [0, 92, 0, -99], [-18, 0, 76, 0], [0, 50, 0, -108], [-542, 0, 0, 0], [0, 203, 0, -41], [-738, 0, 0, 0], [-2, 0, -56, 0], [0, -56, 0, -113], [0, -183, 0, -45], [422, 0, 98, 0], [0, 122, 0, 3], [0, 222, 0, -12], [862, 0, 61, 0], [0, -164, 0, 63], [238, 0, 52, 0], [-686, 0, 16, 0], [-818, 0, 53, 0], [0, 42, 0, 147], [0, 254, 0, -21], [198, 0, 3, 0], [-122, 0, 30, 0], [0, -270, 0, 38], [0, -7, 0, -5], [662, 0, -106, 0], [0, -116, 0, -27], [0, -172, 0, -38], [0, -91, 0, 93], [-122, 0, -18, 0], [-698, 0, -6, 0], [242, 0, 112, 0], [150, 0, -29, 0], [0, 46, 0, 83], [62, 0, -11, 0], [102, 0, -118, 0], [0, 54, 0, 5], [-874, 0, -5, 0], [0, 61, 0, 39], [0, 156, 0, 89], [6, 0, 50, 0], [0, -38, 0, -128], [302, 0, 156, 0], [118, 0, -146, 0], [0, -22, 0, -211], [0, -24, 0, 64], [-274, 0, -136, 0], [-450, 0, 104, 0], [0, -42, 0, -29], [0, 112, 0, -30], [0, -190, 0, -55], [0, -148, 0, -168], [0, 189, 0, 95], [310, 0, 222, 0], [162, 0, -48, 0], [718, 0, -63, 0], [0, -56, 0, -189], [-234, 0, -26, 0], [0, 7, 0, 29], [0, 2, 0, -141], [662, 0, -2, 0], [0, 200, 0, 34], [0, 45, 0, -175], [-290, 0, -24, 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_j();".
// 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_j(:prec:=4)
    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_j();".
// 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_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_576_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![-44, 4, 1]>],Snew);
    return Vf;
end function;