
// Make newform 2304.3.h.c in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2304_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2304_h_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_2304_3_h_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_2304_3_h_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, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 2, 0], [0, 3, 0, 3], [0, 3, 0, -3]];
    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_2304_h();"
function MakeCharacter_2304_h()
    N := 2304;
    order := 2;
    char_gens := [1279, 2053, 1793];
    v := [2, 1, 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_2304_h_Hecke();"
function MakeCharacter_2304_h_Hecke(Kf)
    N := 2304;
    order := 2;
    char_gens := [1279, 2053, 1793];
    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], [0, 0, 0, 1], [-4, 0, 0, 0], [0, 0, 0, 4], [0, 4, 0, 0], [0, 0, 3, 0], [0, 8, 0, 0], [0, 0, 4, 0], [0, 0, 0, 1], [-44, 0, 0, 0], [0, 17, 0, 0], [0, 0, 11, 0], [0, -20, 0, 0], [0, 0, -20, 0], [0, 0, 0, -9], [0, 0, 0, 8], [0, 25, 0, 0], [0, -4, 0, 0], [0, 0, 12, 0], [16, 0, 0, 0], [76, 0, 0, 0], [0, 0, 0, -28], [0, 0, 3, 0], [176, 0, 0, 0], [0, 0, 0, -7], [-28, 0, 0, 0], [0, 0, 0, 0], [0, 28, 0, 0], [0, 0, 37, 0], [-92, 0, 0, 0], [0, 0, 0, 40], [0, 0, 37, 0], [0, 76, 0, 0], [0, 0, 0, -65], [-148, 0, 0, 0], [0, -41, 0, 0], [0, -28, 0, 0], [0, 0, -8, 0], [0, 0, 0, -41], [0, 0, 0, 48], [0, 116, 0, 0], [0, 0, -8, 0], [206, 0, 0, 0], [0, 0, 0, -39], [20, 0, 0, 0], [0, -148, 0, 0], [436, 0, 0, 0], [0, 0, 0, -4], [0, -4, 0, 0], [0, 0, -3, 0], [0, 0, 32, 0], [32, 0, 0, 0], [0, 0, 0, 12], [0, 0, 43, 0], [0, 0, -88, 0], [0, 0, 0, -81], [-380, 0, 0, 0], [0, 164, 0, 0], [0, 0, 67, 0], [0, -104, 0, 0], [0, 0, 0, 103], [0, 260, 0, 0], [0, 0, 88, 0], [94, 0, 0, 0], [0, 0, 0, 79], [0, 268, 0, 0], [-208, 0, 0, 0], [0, 0, 0, 68], [0, -119, 0, 0], [0, 0, 53, 0], [0, 0, 132, 0], [-284, 0, 0, 0], [0, 95, 0, 0], [0, -80, 0, 0], [0, 0, -64, 0], [0, 0, 0, 95], [0, 73, 0, 0], [0, 0, -77, 0], [-368, 0, 0, 0], [0, 0, 0, -92], [0, 20, 0, 0], [0, 0, -36, 0], [542, 0, 0, 0], [-4, 0, 0, 0], [0, 0, 0, 76], [0, 0, -51, 0], [400, 0, 0, 0], [0, 0, 0, -71], [604, 0, 0, 0], [0, 0, 0, -84], [0, 0, 124, 0], [596, 0, 0, 0], [0, 0, 0, -64], [0, -112, 0, 0], [0, 0, -204, 0], [0, 0, 0, 113], [0, 0, -123, 0], [0, -368, 0, 0], [0, -404, 0, 0], [0, -268, 0, 0], [0, 0, 0, -39], [0, 0, 0, 76], [0, 0, 37, 0], [0, 184, 0, 0], [-142, 0, 0, 0], [0, 0, 0, 88], [0, 0, -261, 0], [0, 0, -188, 0], [-158, 0, 0, 0], [-332, 0, 0, 0], [0, -289, 0, 0], [0, 0, -13, 0], [0, 448, 0, 0], [20, 0, 0, 0], [0, 0, -61, 0], [0, -364, 0, 0], [0, 0, 108, 0], [0, 0, 0, 71], [0, 0, 0, 248], [0, -31, 0, 0], [-670, 0, 0, 0], [0, 0, 0, 305], [0, 0, 0, -132], [0, 20, 0, 0], [0, 0, 0, 225], [0, -484, 0, 0], [0, 0, 276, 0], [-508, 0, 0, 0], [0, -572, 0, 0], [0, 152, 0, 0], [0, 0, -200, 0], [-188, 0, 0, 0], [0, 620, 0, 0], [0, 0, -37, 0], [-910, 0, 0, 0], [0, 0, 0, 327], [0, 680, 0, 0], [0, 0, 0, -25], [0, 0, -261, 0], [0, -80, 0, 0], [0, 0, 0, -103], [332, 0, 0, 0], [0, 0, 0, 24], [0, 316, 0, 0], [0, 0, -172, 0], [0, -223, 0, 0], [0, 0, -101, 0], [0, 364, 0, 0], [0, 0, -216, 0], [0, -455, 0, 0], [0, 0, 219, 0], [0, -532, 0, 0], [0, 0, 328, 0], [0, -884, 0, 0], [0, 0, -56, 0], [380, 0, 0, 0], [0, 0, -157, 0], [178, 0, 0, 0], [0, 0, 0, -103], [0, 0, 0, -424], [0, 0, 309, 0], [1700, 0, 0, 0], [0, 0, 0, 108], [0, 0, -179, 0], [0, 0, 248, 0], [772, 0, 0, 0], [0, -97, 0, 0], [560, 0, 0, 0], [0, 0, 0, -335], [0, 0, 0, -272], [0, -473, 0, 0], [0, 0, 64, 0], [544, 0, 0, 0], [-1412, 0, 0, 0], [0, 0, -445, 0], [0, -416, 0, 0], [0, 0, 0, 303], [-196, 0, 0, 0], [0, 439, 0, 0], [-1508, 0, 0, 0], [0, 0, 0, -4], [0, -868, 0, 0], [0, 0, 51, 0], [0, 0, 176, 0], [0, 0, 0, -271], [0, 388, 0, 0], [0, 776, 0, 0], [-512, 0, 0, 0], [0, 0, 24, 0], [1280, 0, 0, 0], [0, 0, 0, -368], [0, 584, 0, 0], [0, 0, 0, -551], [0, 0, 0, -216], [0, 0, -19, 0], [-46, 0, 0, 0], [0, -524, 0, 0], [0, 0, -91, 0], [0, 0, 240, 0], [0, 0, 0, -313], [148, 0, 0, 0], [0, -364, 0, 0], [1616, 0, 0, 0], [0, 0, 0, -228], [0, 0, 0, -33], [1708, 0, 0, 0], [0, 0, 0, 196], [0, 0, 349, 0], [0, -500, 0, 0], [-430, 0, 0, 0], [0, 0, 0, -287], [428, 0, 0, 0], [0, 0, 0, -32], [0, 0, 148, 0], [1426, 0, 0, 0], [1900, 0, 0, 0], [0, 0, 77, 0], [0, 0, 192, 0], [0, 0, 0, 311], [0, -940, 0, 0], [620, 0, 0, 0], [0, 0, 19, 0], [1852, 0, 0, 0], [0, 0, 0, 220], [0, 1361, 0, 0], [0, 0, -35, 0], [0, 0, 516, 0], [2012, 0, 0, 0], [0, 0, 0, -344], [0, -500, 0, 0], [0, -892, 0, 0], [748, 0, 0, 0], [0, 0, 653, 0], [0, 772, 0, 0], [0, 0, -620, 0], [-496, 0, 0, 0], [0, 0, 0, 159], [0, 0, 0, -412], [0, 0, -468, 0], [0, 0, 0, 284], [0, 316, 0, 0], [-1468, 0, 0, 0], [0, 1063, 0, 0], [0, 0, -533, 0], [0, 0, -340, 0], [-164, 0, 0, 0], [0, 0, 0, -440], [0, 220, 0, 0], [0, 0, 456, 0], [0, 940, 0, 0], [0, 0, 555, 0], [0, 0, 124, 0], [1120, 0, 0, 0], [0, 0, 0, -65], [0, 0, 0, 72], [0, 404, 0, 0], [0, -272, 0, 0], [0, 0, 0, -393], [2194, 0, 0, 0], [-1364, 0, 0, 0], [0, 0, 0, -304], [0, 68, 0, 0], [0, 343, 0, 0], [0, 0, 301, 0], [0, 92, 0, 0], [0, 0, 0, -105], [0, 0, 781, 0], [0, 472, 0, 0], [0, 0, 0, -17], [0, -887, 0, 0], [0, -1288, 0, 0], [-896, 0, 0, 0], [-908, 0, 0, 0], [1376, 0, 0, 0], [-1780, 0, 0, 0], [0, 0, 0, 404], [0, -839, 0, 0], [2878, 0, 0, 0], [0, 0, 0, -208], [0, 0, -556, 0], [1484, 0, 0, 0], [0, 0, 572, 0], [0, 1631, 0, 0], [0, -248, 0, 0], [0, 0, -384, 0], [128, 0, 0, 0], [0, 0, 0, -209], [3308, 0, 0, 0], [0, 0, -171, 0], [0, 0, 0, 103], [0, 0, 0, -744], [0, 0, -157, 0], [0, 0, 0, -592], [0, 1303, 0, 0], [0, 0, 0, 233], [1732, 0, 0, 0], [0, 0, 0, 521], [0, 0, 0, 84], [0, 776, 0, 0], [514, 0, 0, 0], [0, 0, 0, -279], [700, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2304_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2304_3_h_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_2304_3_h_c(:prec:=4)
    chi := MakeCharacter_2304_h();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_2304_3_h_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_2304_3_h_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2304_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![-18, 0, 1]>,<7,R![4, 1]>],Snew);
    return Vf;
end function;