
// Make newform 2304.3.h.e 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_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_2304_3_h_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, 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, -7], [0, 0, 0, 0], [0, 0, 0, 0], [0, 12, 0, 0], [0, 0, -7, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 41], [0, 0, 0, 0], [0, 35, 0, 0], [0, 0, -31, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -17], [0, 0, 0, 0], [0, 11, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-96, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 41, 0], [144, 0, 0, 0], [0, 0, 0, -79], [0, 0, 0, 0], [0, 0, 0, 0], [0, -60, 0, 0], [0, 0, 127, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -17, 0], [0, 0, 0, 0], [0, 0, 0, -89], [0, 0, 0, 0], [0, 85, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -113], [0, 0, 0, 0], [0, -180, 0, 0], [0, 0, 0, 0], [190, 0, 0, 0], [0, 0, 0, -223], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -60, 0, 0], [0, 0, 103, 0], [0, 0, 0, 0], [-240, 0, 0, 0], [0, 0, 0, 0], [0, 0, -223, 0], [0, 0, 0, 0], [0, 0, 0, -329], [0, 0, 0, 0], [0, 252, 0, 0], [0, 0, -391, 0], [0, 0, 0, 0], [0, 0, 0, -353], [0, 0, 0, 0], [0, 0, 0, 0], [-50, 0, 0, 0], [0, 0, 0, -233], [0, 0, 0, 0], [576, 0, 0, 0], [0, 0, 0, 0], [0, 299, 0, 0], [0, 0, -497, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -275, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 151], [0, -325, 0, 0], [0, 0, -439, 0], [-240, 0, 0, 0], [0, 0, 0, 0], [0, -420, 0, 0], [0, 0, 0, 0], [-290, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 631, 0], [-336, 0, 0, 0], [0, 0, 0, 641], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -679], [0, 0, -161, 0], [0, 0, 0, 0], [0, 420, 0, 0], [0, 0, 0, 0], [0, 0, 0, -367], [0, 0, 0, 0], [0, 0, 751, 0], [0, 0, 0, 0], [-1150, 0, 0, 0], [0, 0, 0, 0], [0, 0, 833, 0], [0, 0, 0, 0], [-1102, 0, 0, 0], [0, 0, 0, 0], [0, -35, 0, 0], [0, 0, 713, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 809, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -257], [0, 0, 0, 0], [0, -589, 0, 0], [770, 0, 0, 0], [0, 0, 0, -727], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -391], [0, 660, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 108, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 468, 0, 0], [0, 0, -799, 0], [962, 0, 0, 0], [0, 0, 0, 943], [0, 0, 0, 0], [0, 0, 0, -17], [0, 0, -1039, 0], [0, 0, 0, 0], [0, 0, 0, -271], [0, 0, 0, 0], [0, 0, 0, 0], [0, -540, 0, 0], [0, 0, 0, 0], [0, -205, 0, 0], [0, 0, 593, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -805, 0, 0], [0, 0, 1169, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 1049, 0], [-430, 0, 0, 0], [0, 0, 0, 161], [0, 0, 0, 0], [0, 0, -113, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -1193, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 925, 0, 0], [-1680, 0, 0, 0], [0, 0, 0, -967], [0, 0, 0, 0], [0, -779, 0, 0], [0, 0, 0, 0], [-384, 0, 0, 0], [0, 0, 0, 0], [0, 0, -679, 0], [0, 0, 0, 0], [0, 0, 0, -1481], [0, 0, 0, 0], [0, -731, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 132, 0, 0], [0, 0, -343, 0], [0, 0, 0, 0], [0, 0, 0, 1241], [0, -1092, 0, 0], [0, 0, 0, 0], [-2160, 0, 0, 0], [0, 0, 0, 0], [-1056, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -1471], [0, 0, 0, 0], [0, 0, 1687, 0], [98, 0, 0, 0], [0, -1188, 0, 0], [0, 0, -1697, 0], [0, 0, 0, 0], [0, 0, 0, -1361], [0, 0, 0, 0], [0, -612, 0, 0], [1920, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 1783], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -1721, 0], [0, 0, 0, 0], [-2590, 0, 0, 0], [0, 0, 0, -1351], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-2542, 0, 0, 0], [0, 0, 0, 0], [0, 0, 679, 0], [0, 0, 0, 0], [0, 0, 0, -1217], [0, 1020, 0, 0], [0, 0, 0, 0], [0, 0, 1241, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 371, 0, 0], [0, 0, -713, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 228, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 151, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-2640, 0, 0, 0], [0, 0, 0, 1927], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 901, 0, 0], [0, 0, 977, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 1428, 0, 0], [0, 0, -1519, 0], [0, 0, 0, 0], [480, 0, 0, 0], [0, 0, 0, 2263], [0, 0, 0, 0], [0, 780, 0, 0], [0, 0, 0, 0], [0, 0, 0, 1327], [-1870, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -1140, 0, 0], [0, 1045, 0, 0], [0, 0, -1337, 0], [0, 0, 0, 0], [0, 0, 0, 799], [0, 0, 2359, 0], [0, 0, 0, 0], [0, 0, 0, -137], [0, 59, 0, 0], [0, 0, 0, 0], [-3456, 0, 0, 0], [0, 0, 0, 0], [-2496, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 1739, 0, 0], [-1298, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 61, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [3696, 0, 0, 0], [0, 0, 0, -2633], [0, 0, 0, 0], [0, 0, 2671, 0], [0, 0, 0, -2369], [0, 0, 0, 0], [0, 0, 1097, 0], [0, 0, 0, 0], [0, -1595, 0, 0], [0, 0, 0, 2609], [0, 0, 0, 0], [0, 0, 0, 833], [0, 0, 0, 0], [0, 0, 0, 0], [3410, 0, 0, 0], [0, 0, 0, -2287], [0, 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_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_2304_3_h_e(: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_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_2304_3_h_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2304_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![-98, 0, 1]>,<7,R![0, 1]>],Snew);
    return Vf;
end function;