
// Make newform 2304.3.g.s in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2304_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2304_g_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_g_s();".
// 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_g_s();".
// 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], [0, 16, 0, -8], [-16, 0, 32, 0]];
    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_g();"
function MakeCharacter_2304_g()
    N := 2304;
    order := 2;
    char_gens := [1279, 2053, 1793];
    v := [1, 2, 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_g_Hecke();"
function MakeCharacter_2304_g_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, 0], [0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 0], [0, 0, 0, -1], [0, 0, 0, 0], [0, 0, 0, 0], [0, -23, 0, 0], [0, 0, -5, 0], [0, 0, 0, 0], [0, 0, 0, 3], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 7, 0], [0, 0, 0, -2], [0, 0, 0, 0], [-46, 0, 0, 0], [0, -71, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [2, 0, 0, 0], [0, 0, 0, 0], [0, -97, 0, 0], [0, 0, 0, 0], [0, 0, -3, 0], [0, 0, 0, 0], [0, -73, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -10], [0, 0, 0, 0], [0, -143, 0, 0], [0, 0, -21, 0], [0, 0, 0, 7], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 13, 0], [0, 0, 0, 0], [-382, 0, 0, 0], [0, 0, 0, 0], [0, -193, 0, 0], [0, 0, 0, 14], [0, -169, 0, 0], [0, 0, 0, 0], [0, 0, 33, 0], [0, 0, 0, 0], [0, 0, 0, 0], [286, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -121, 0, 0], [0, 0, -39, 0], [0, 0, 0, 0], [0, 0, 0, -12], [0, 0, 0, 0], [0, 0, 0, -18], [0, 0, 0, 0], [-142, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 20], [482, 0, 0, 0], [0, 0, 0, 0], [0, 0, 35, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -359, 0, 0], [0, 0, 19, 0], [0, 0, 0, 11], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 51, 0], [0, 0, 0, 0], [626, 0, 0, 0], [0, 0, 0, 0], [0, 0, -55, 0], [0, 0, 0, 0], [862, 0, 0, 0], [0, 47, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-814, 0, 0, 0], [0, 0, 0, 0], [0, 263, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 481, 0, 0], [0, 0, 0, 0], [0, 0, 0, 36], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -13], [0, 0, -23, 0], [0, 0, 0, -35], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -26], [-962, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [526, 0, 0, 0], [0, 407, 0, 0], [0, 0, 35, 0], [0, 0, 0, 0], [0, 0, 0, -44], [0, 337, 0, 0], [0, 0, 0, 0], [0, 0, 0, 45], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 95, 0], [-1154, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 15], [0, 0, 0, 0], [0, 0, 77, 0], [0, 0, 0, 0], [0, 241, 0, 0], [0, 0, -75, 0], [0, 0, 0, 30], [0, 0, 0, 0], [0, 601, 0, 0], [0, 0, -91, 0], [0, 0, 0, 0], [-1534, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -7], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 21], [0, 0, 0, 0], [0, 529, 0, 0], [0, 0, 0, 0], [0, 0, -115, 0], [0, 0, 0, 0], [0, 0, -29, 0], [0, 0, 0, 0], [0, 0, 0, -35], [0, 0, 0, 0], [0, 0, 119, 0], [0, 0, 0, 0], [0, 0, 0, -17], [0, 0, 0, 0], [0, 0, 0, 65], [0, 0, 0, 0], [0, 433, 0, 0], [0, 0, 0, 0], [-1198, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 767, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -23, 0, 0], [0, 0, -45, 0], [1826, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 63, 0], [0, 0, 0, 0], [-1298, 0, 0, 0], [0, 313, 0, 0], [0, 0, 0, 0], [0, 0, 0, 24], [0, 0, 0, 0], [0, 671, 0, 0], [0, 0, -93, 0], [0, -1079, 0, 0], [0, 0, 0, 0], [0, 0, -99, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 161, 0], [0, 0, 0, 68], [814, 0, 0, 0], [0, 0, 0, 0], [-1538, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 56], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [2398, 0, 0, 0], [0, 0, -175, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -1081, 0, 0], [0, 0, -35, 0], [-2302, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 887, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -91], [-478, 0, 0, 0], [0, 0, 0, 0], [0, -1009, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [2158, 0, 0, 0], [0, -1319, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 37, 0], [0, -913, 0, 0], [0, 0, 0, 0], [0, 1223, 0, 0], [0, 0, 0, 0], [0, 0, 203, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -1441, 0, 0], [0, 0, 0, 0], [0, 0, -11, 0], [0, 0, 0, 105], [0, -1417, 0, 0], [0, 0, 0, 0], [0, 0, 0, -95], [0, 0, 0, 0], [-1822, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -104], [0, -191, 0, 0], [0, 0, -217, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -599, 0, 0], [0, 0, 0, 0], [0, 0, 0, 84], [0, 0, 0, 0], [0, 0, 225, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-146, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -143, 0], [0, 0, 0, -30], [0, 0, 0, 0], [-1586, 0, 0, 0], [0, -649, 0, 0], [0, 0, 0, 0], [0, 0, 185, 0], [0, 0, -41, 0], [0, 0, 0, 0], [0, 0, 0, 120], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 105], [0, 0, 0, 0], [0, 0, 187, 0], [0, 0, 0, -70], [3406, 0, 0, 0], [0, -407, 0, 0], [3358, 0, 0, 0], [0, 1391, 0, 0], [0, 0, 0, 0], [0, 0, 123, 0], [-1874, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -481, 0, 0], [0, 0, 0, 0], [0, 0, 43, 0], [0, 0, 0, 133], [0, 0, 0, 0], [-2978, 0, 0, 0], [0, 0, 0, 0], [0, -1871, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 279, 0], [0, 0, 0, 0], [0, -937, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -115], [914, 0, 0, 0], [0, 0, 0, 0], [0, -647, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2304_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2304_3_g_s();".
// 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_g_s(:prec:=4)
    chi := MakeCharacter_2304_g();
    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_g_s();".
// 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_g_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2304_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![4, 0, 1]>,<11,R![0, 1]>,<13,R![-192, 0, 1]>],Snew);
    return Vf;
end function;