
// Make newform 2304.3.b.h in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2304_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2304_b_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_b_h();".
// 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_b_h();".
// 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, 2]];
    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_2304_b();"
function MakeCharacter_2304_b()
    N := 2304;
    order := 2;
    char_gens := [1279, 2053, 1793];
    v := [1, 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_2304_b_Hecke();"
function MakeCharacter_2304_b_Hecke(Kf)
    N := 2304;
    order := 2;
    char_gens := [1279, 2053, 1793];
    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], [0, 2], [0, 2], [16, 0], [0, -1], [-24, 0], [-24, 0], [0, -16], [0, -22], [0, -26], [0, -9], [8, 0], [56, 0], [0, 16], [0, -18], [-32, 0], [0, -31], [-80, 0], [0, 64], [66, 0], [0, -10], [16, 0], [-144, 0], [94, 0], [0, -10], [0, 22], [-96, 0], [0, -33], [80, 0], [0, 66], [128, 0], [248, 0], [16, 0], [0, 58], [0, -110], [0, -143], [168, 0], [0, -80], [0, -122], [-96, 0], [0, -135], [0, 112], [178, 0], [0, 46], [0, -82], [-80, 0], [0, 106], [-400, 0], [0, -55], [16, 0], [0, 16], [110, 0], [-432, 0], [32, 0], [0, -128], [0, 66], [0, -6], [0, -103], [-352, 0], [352, 0], [0, -130], [48, 0], [0, 224], [94, 0], [0, -166], [-32, 0], [110, 0], [176, 0], [0, -191], [-176, 0], [0, 0], [0, 182], [0, -137], [632, 0], [0, -208], [0, 74], [0, 225], [376, 0], [-110, 0], [-272, 0], [0, 265], [0, -48], [66, 0], [0, 94], [176, 0], [-312, 0], [-526, 0], [0, 158], [0, -398], [-336, 0], [0, 320], [0, -150], [-512, 0], [-736, 0], [0, -352], [0, 50], [-88, 0], [-200, 0], [0, -289], [824, 0], [0, -522], [-464, 0], [616, 0], [272, 0], [-462, 0], [128, 0], [432, 0], [0, -176], [-994, 0], [0, 290], [0, 231], [32, 0], [-800, 0], [0, -102], [680, 0], [-200, 0], [0, -48], [0, 190], [704, 0], [0, 119], [-350, 0], [0, -550], [912, 0], [616, 0], [0, 450], [0, 617], [0, -288], [0, -614], [0, -385], [-592, 0], [0, -80], [0, 254], [0, -199], [-1384, 0], [-910, 0], [0, 246], [1496, 0], [0, -418], [312, 0], [648, 0], [0, -482], [0, 650], [1056, 0], [0, -465], [0, 128], [0, 151], [584, 0], [584, 0], [0, 528], [0, 689], [-480, 0], [-792, 0], [0, 416], [488, 0], [0, -496], [0, 426], [952, 0], [750, 0], [0, -386], [-1312, 0], [264, 0], [0, -106], [-880, 0], [-1384, 0], [0, 352], [0, -546], [0, 599], [318, 0], [0, 154], [1088, 0], [0, -367], [0, -496], [-814, 0], [0, -418], [-704, 0], [704, 0], [0, 330], [0, 462], [0, 913], [0, 166], [1168, 0], [0, -295], [672, 0], [0, -208], [0, 58], [0, -737], [-224, 0], [1090, 0], [0, 96], [126, 0], [-608, 0], [-192, 0], [0, 1078], [960, 0], [64, 0], [-110, 0], [0, -545], [1760, 0], [0, 960], [0, -506], [0, -374], [0, -55], [974, 0], [-720, 0], [0, 42], [0, -54], [-1744, 0], [1168, 0], [-2552, 0], [1554, 0], [0, -950], [0, 1166], [-1760, 0], [0, 544], [334, 0], [0, 1078], [-1784, 0], [0, -48], [0, 374], [0, -551], [0, -242], [-1640, 0], [0, 786], [464, 0], [0, 215], [-1520, 0], [0, 672], [0, -602], [2560, 0], [0, -737], [600, 0], [0, 466], [352, 0], [616, 0], [0, 1408], [1150, 0], [0, 330], [-176, 0], [0, 624], [-1232, 0], [192, 0], [0, -1382], [0, -1023], [-2624, 0], [0, -1216], [0, -354], [352, 0], [-160, 0], [0, 0], [0, -337], [-1008, 0], [0, -880], [-2222, 0], [0, -974], [-416, 0], [0, -1047], [-1232, 0], [0, -890], [1710, 0], [0, -1510], [160, 0], [0, -967], [0, -783], [856, 0], [-1216, 0], [0, 1278], [2992, 0], [-88, 0], [0, -518], [0, 209], [-1392, 0], [370, 0], [0, -370], [94, 0], [0, 510], [-1424, 0], [0, 497], [-834, 0], [1184, 0], [0, 1568], [0, 286], [0, 1072], [0, 455], [-2552, 0], [0, -64], [3246, 0], [0, -382], [0, 242], [1136, 0], [0, 1070], [2464, 0], [496, 0], [-160, 0], [0, -335], [0, 1294], [0, -1582], [0, -1322], [-3600, 0], [-584, 0], [-1538, 0], [0, 1158], [0, -1314]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2304_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2304_3_b_h();".
// 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_b_h(:prec:=2)
    chi := MakeCharacter_2304_b();
    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_b_h();".
// 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_b_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2304_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![16, 0, 1]>,<7,R![16, 0, 1]>,<11,R![-16, 1]>,<17,R![24, 1]>,<19,R![24, 1]>],Snew);
    return Vf;
end function;