
// Make newform 576.4.d.e in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_576_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_576_d_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_4_d_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_4_d_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, 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, 2], [-8, 0, 16, 0], [0, 32, 0, -16]];
    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_d();"
function MakeCharacter_576_d()
    N := 576;
    order := 2;
    char_gens := [127, 325, 65];
    v := [2, 1, 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_d_Hecke();"
function MakeCharacter_576_d_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 := 4;
    raw_aps := [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, -1], [0, 21, 0, 0], [0, 0, -3, 0], [6, 0, 0, 0], [0, 47, 0, 0], [0, 0, 0, -5], [0, 0, -17, 0], [0, 0, 0, 4], [0, 0, -1, 0], [54, 0, 0, 0], [0, -221, 0, 0], [0, 0, 0, -2], [0, 0, 5, 0], [0, 69, 0, 0], [0, 0, -31, 0], [0, -89, 0, 0], [0, 0, 0, -31], [434, 0, 0, 0], [0, 0, 0, 6], [0, -135, 0, 0], [1182, 0, 0, 0], [-1238, 0, 0, 0], [0, 0, -127, 0], [0, 0, 0, -37], [0, -651, 0, 0], [0, 0, -19, 0], [-1650, 0, 0, 0], [0, 0, 0, -40], [0, 105, 0, 0], [-522, 0, 0, 0], [0, 275, 0, 0], [0, 0, -27, 0], [0, 0, 0, 53], [0, 0, 121, 0], [0, -289, 0, 0], [0, 0, 0, -43], [0, 0, 187, 0], [0, -1959, 0, 0], [0, 0, 259, 0], [0, 0, 0, -112], [-1846, 0, 0, 0], [0, 0, 33, 0], [0, 0, 0, 87], [0, 1495, 0, 0], [0, 0, 0, -196], [0, -1647, 0, 0], [0, 0, -137, 0], [-4818, 0, 0, 0], [0, 0, 0, 206], [-550, 0, 0, 0], [0, 3333, 0, 0], [-4962, 0, 0, 0], [0, 0, 0, 201], [0, 0, 75, 0], [0, 0, 0, 38], [0, 0, 355, 0], [3102, 0, 0, 0], [0, 2051, 0, 0], [0, 0, 281, 0], [0, -2369, 0, 0], [0, 0, 0, -169], [9466, 0, 0, 0], [0, 0, -297, 0], [0, 4187, 0, 0], [-3566, 0, 0, 0], [0, -1947, 0, 0], [0, 0, -367, 0], [-1698, 0, 0, 0], [0, 0, 0, -91], [0, 0, 0, -118], [0, 0, 27, 0], [0, -53, 0, 0], [0, 0, 0, 464], [0, 0, -391, 0], [0, 0, -1019, 0], [6918, 0, 0, 0], [-7270, 0, 0, 0], [0, -5367, 0, 0], [0, 0, -1129, 0], [0, 0, 0, -586], [-10982, 0, 0, 0], [0, 0, 0, -615], [0, 1341, 0, 0], [-14346, 0, 0, 0], [-7894, 0, 0, 0], [0, 0, -261, 0], [0, 0, 0, 246], [0, -2367, 0, 0], [0, 0, 0, -444], [0, 0, 0, -141], [0, 3309, 0, 0], [0, 8503, 0, 0], [0, 0, 0, -425], [0, 0, 671, 0], [246, 0, 0, 0], [0, -7109, 0, 0], [0, 0, -727, 0], [0, -10729, 0, 0], [0, 0, -789, 0], [0, -11103, 0, 0], [-2202, 0, 0, 0], [0, -2845, 0, 0], [18466, 0, 0, 0], [0, -2475, 0, 0], [-6834, 0, 0, 0], [0, 0, 0, -53], [-2014, 0, 0, 0], [0, 0, 0, 164], [0, 0, 2087, 0], [-11730, 0, 0, 0], [0, -9157, 0, 0], [0, 0, 0, -167], [-18570, 0, 0, 0], [0, 13511, 0, 0], [0, 0, 0, -103], [0, 0, -2037, 0], [0, 4281, 0, 0], [0, 0, 1163, 0], [-25430, 0, 0, 0], [0, 0, 1153, 0], [0, -4179, 0, 0], [0, -11449, 0, 0], [0, 0, 639, 0], [0, 0, -1065, 0], [0, 0, 0, 474], [0, 0, 0, -603], [0, 0, 2273, 0], [0, 7375, 0, 0], [0, 0, 0, 77], [0, 0, 0, -190], [0, 0, -1949, 0], [-20538, 0, 0, 0], [-10678, 0, 0, 0], [0, 0, 2009, 0], [0, -6409, 0, 0], [0, 0, 487, 0], [-29130, 0, 0, 0], [0, -14309, 0, 0], [0, 0, -875, 0], [0, 0, 0, 1545], [0, -20787, 0, 0], [0, 0, 1617, 0], [0, 0, 0, 1129], [0, 0, 307, 0], [41862, 0, 0, 0], [0, -5381, 0, 0], [0, 0, 0, 68], [0, 0, 2717, 0], [-4722, 0, 0, 0], [0, 15695, 0, 0], [0, 0, 0, -1425], [0, -2621, 0, 0], [0, 0, 0, 1018], [0, 0, 0, 69], [20118, 0, 0, 0], [-3950, 0, 0, 0], [0, 0, 835, 0], [0, 23073, 0, 0], [47718, 0, 0, 0], [0, 0, 0, -1241], [0, -1059, 0, 0], [-8826, 0, 0, 0], [0, 0, 0, 843], [0, 0, 0, -604], [0, 0, -945, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_576_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_576_4_d_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_4_d_e(:prec:=4)
    chi := MakeCharacter_576_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_4_d_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_4_d_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_576_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![192, 0, 1]>,<7,R![-768, 0, 1]>,<17,R![-6, 1]>],Snew);
    return Vf;
end function;