
// Make newform 2304.3.h.g in Magma, downloaded from the LMFDB on 25 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_g();".
// 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_g();".
// 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, 1, 0, 1], [0, 1, 0, -1]];
    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], [8, 0, 0, 0], [0, 0, 0, -8], [0, 4, 0, 0], [0, 0, 9, 0], [0, 16, 0, 0], [0, 0, -24, 0], [0, 0, 0, -31], [-40, 0, 0, 0], [0, -13, 0, 0], [0, 0, -47, 0], [0, -8, 0, 0], [0, 0, -8, 0], [0, 0, 0, 23], [0, 0, 0, -16], [0, 27, 0, 0], [0, -40, 0, 0], [0, 0, 56, 0], [-96, 0, 0, 0], [104, 0, 0, 0], [0, 0, 0, -72], [0, 0, -55, 0], [-80, 0, 0, 0], [0, 0, 0, -87], [-72, 0, 0, 0], [0, 0, 0, 128], [0, 44, 0, 0], [0, 0, -97, 0], [56, 0, 0, 0], [0, 0, 0, 176], [0, 0, -33, 0], [0, -8, 0, 0], [0, 0, 0, -129], [168, 0, 0, 0], [0, 5, 0, 0], [0, 40, 0, 0], [0, 0, -208, 0], [0, 0, 0, 39], [0, 0, 0, -96], [0, 4, 0, 0], [0, 0, 48, 0], [-258, 0, 0, 0], [0, 0, 0, -263], [88, 0, 0, 0], [0, 184, 0, 0], [-104, 0, 0, 0], [0, 0, 0, -120], [0, 172, 0, 0], [0, 0, -57, 0], [0, 0, 192, 0], [-272, 0, 0, 0], [0, 0, 0, -24], [0, 0, -31, 0], [0, 0, 144, 0], [0, 0, 0, -33], [-264, 0, 0, 0], [0, 20, 0, 0], [0, 0, -135, 0], [0, 112, 0, 0], [0, 0, 0, -249], [0, -216, 0, 0], [0, 0, -144, 0], [14, 0, 0, 0], [0, 0, 0, -417], [0, -8, 0, 0], [128, 0, 0, 0], [0, 0, 0, 248], [0, -5, 0, 0], [0, 0, 175, 0], [0, 0, 232, 0], [696, 0, 0, 0], [0, -227, 0, 0], [0, 32, 0, 0], [0, 0, 256, 0], [0, 0, 0, -273], [0, 331, 0, 0], [0, 0, 121, 0], [176, 0, 0, 0], [0, 0, 0, 56], [0, 244, 0, 0], [0, 0, 216, 0], [478, 0, 0, 0], [392, 0, 0, 0], [0, 0, 0, -24], [0, 0, -89, 0], [16, 0, 0, 0], [0, 0, 0, 201], [-568, 0, 0, 0], [0, 0, 0, 296], [0, 0, 24, 0], [-424, 0, 0, 0], [0, 0, 0, -512], [0, -96, 0, 0], [0, 0, -312, 0], [0, 0, 0, 177], [0, 0, 111, 0], [0, -288, 0, 0], [0, 268, 0, 0], [0, 72, 0, 0], [0, 0, 0, 377], [0, 0, 0, -408], [0, 0, 703, 0], [0, -208, 0, 0], [834, 0, 0, 0], [0, 0, 0, -48], [0, 0, -159, 0], [0, 0, 488, 0], [626, 0, 0, 0], [-232, 0, 0, 0], [0, 333, 0, 0], [0, 0, 201, 0], [0, 384, 0, 0], [472, 0, 0, 0], [0, 0, -263, 0], [0, -120, 0, 0], [0, 0, 504, 0], [0, 0, 0, 823], [0, 0, 0, 144], [0, -509, 0, 0], [-382, 0, 0, 0], [0, 0, 0, -175], [0, 0, 0, -120], [0, 520, 0, 0], [0, 0, 0, 465], [0, 476, 0, 0], [0, 0, 136, 0], [-648, 0, 0, 0], [0, -604, 0, 0], [0, -656, 0, 0], [0, 0, 432, 0], [632, 0, 0, 0], [0, -420, 0, 0], [0, 0, 369, 0], [130, 0, 0, 0], [0, 0, 0, -201], [0, -432, 0, 0], [0, 0, 0, -537], [0, 0, 641, 0], [0, -736, 0, 0], [0, 0, 0, 841], [-664, 0, 0, 0], [0, 0, 0, -176], [0, 140, 0, 0], [0, 0, 392, 0], [0, -381, 0, 0], [0, 0, -1087, 0], [0, -776, 0, 0], [0, 0, 16, 0], [0, 43, 0, 0], [0, 0, 465, 0], [0, 248, 0, 0], [0, 0, 720, 0], [0, 440, 0, 0], [0, 0, 1104, 0], [264, 0, 0, 0], [0, 0, 201, 0], [-1070, 0, 0, 0], [0, 0, 0, -407], [0, 0, 0, -432], [0, 0, 735, 0], [696, 0, 0, 0], [0, 0, 0, -1112], [0, 0, 359, 0], [0, 0, -336, 0], [-520, 0, 0, 0], [0, -243, 0, 0], [-560, 0, 0, 0], [0, 0, 0, 929], [0, 0, 0, -96], [0, -27, 0, 0], [0, 0, -512, 0], [512, 0, 0, 0], [520, 0, 0, 0], [0, 0, -103, 0], [0, -320, 0, 0], [0, 0, 0, -561], [8, 0, 0, 0], [0, 277, 0, 0], [968, 0, 0, 0], [0, 0, 0, 392], [0, -148, 0, 0], [0, 0, -983, 0], [0, 0, -1440, 0], [0, 0, 0, 833], [0, 468, 0, 0], [0, 80, 0, 0], [496, 0, 0, 0], [0, 0, -1296, 0], [-1760, 0, 0, 0], [0, 0, 0, -32], [0, -688, 0, 0], [0, 0, 0, -791], [0, 0, 0, -848], [0, 0, -1065, 0], [1570, 0, 0, 0], [0, 628, 0, 0], [0, 0, -289, 0], [0, 0, -928, 0], [0, 0, 0, 1287], [-424, 0, 0, 0], [0, -1100, 0, 0], [-576, 0, 0, 0], [0, 0, 0, 712], [0, 0, 0, -1473], [-344, 0, 0, 0], [0, 0, 0, -392], [0, 0, 935, 0], [0, 952, 0, 0], [-1630, 0, 0, 0], [0, 0, 0, -1119], [-216, 0, 0, 0], [0, 0, 0, -192], [0, 0, 776, 0], [-1518, 0, 0, 0], [1448, 0, 0, 0], [0, 0, -1385, 0], [0, 0, 1024, 0], [0, 0, 0, 855], [0, 340, 0, 0], [1576, 0, 0, 0], [0, 0, -1143, 0], [1672, 0, 0, 0], [0, 0, 0, 1224], [0, 1027, 0, 0], [0, 0, -297, 0], [0, 0, 1128, 0], [2120, 0, 0, 0], [0, 0, 0, -464], [0, -692, 0, 0], [0, -664, 0, 0], [-2648, 0, 0, 0], [0, 0, 663, 0], [0, -152, 0, 0], [0, 0, 8, 0], [-1264, 0, 0, 0], [0, 0, 0, -609], [0, 0, 0, -840], [0, 0, 1912, 0], [0, 0, 0, 1224], [0, -552, 0, 0], [504, 0, 0, 0], [0, 309, 0, 0], [0, 0, -1775, 0], [0, 0, -1544, 0], [-1976, 0, 0, 0], [0, 0, 0, 1008], [0, 152, 0, 0], [0, 0, -1328, 0], [0, -612, 0, 0], [0, 0, -1039, 0], [0, 0, -1896, 0], [-864, 0, 0, 0], [0, 0, 0, -385], [0, 0, 0, 240], [0, 1476, 0, 0], [0, -928, 0, 0], [0, 0, 0, -201], [-1486, 0, 0, 0], [40, 0, 0, 0], [0, 0, 0, 96], [0, -1148, 0, 0], [0, -827, 0, 0], [0, 0, 1047, 0], [0, 1432, 0, 0], [0, 0, 0, 151], [0, 0, -425, 0], [0, -80, 0, 0], [0, 0, 0, 207], [0, 971, 0, 0], [0, -80, 0, 0], [-3200, 0, 0, 0], [1176, 0, 0, 0], [2368, 0, 0, 0], [2280, 0, 0, 0], [0, 0, 0, -1832], [0, 347, 0, 0], [1070, 0, 0, 0], [0, 0, 0, -1760], [0, 0, 648, 0], [-3608, 0, 0, 0], [0, 0, -616, 0], [0, 45, 0, 0], [0, -1264, 0, 0], [0, 0, -768, 0], [-1520, 0, 0, 0], [0, 0, 0, -593], [-472, 0, 0, 0], [0, 0, -305, 0], [0, 0, 0, -585], [0, 0, 0, -1584], [0, 0, 489, 0], [0, 0, 0, -352], [0, -75, 0, 0], [0, 0, 0, -1479], [3448, 0, 0, 0], [0, 0, 0, 441], [0, 0, 0, -1704], [0, -1264, 0, 0], [530, 0, 0, 0], [0, 0, 0, -1543], [-1912, 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_g();".
// 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_g(: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_g();".
// 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_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2304_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![-2, 0, 1]>,<7,R![-8, 1]>],Snew);
    return Vf;
end function;