
// Make newform 2304.3.g.e in Magma, downloaded from the LMFDB on 24 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_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_g_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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
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], [1], [1]];
    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], [6], [0], [0], [24], [16], [0], [0], [42], [0], [-24], [-80], [0], [0], [90], [0], [-120], [0], [0], [110], [0], [0], [-160], [130], [-198], [0], [0], [120], [224], [0], [0], [-176], [0], [102], [0], [264], [0], [0], [-330], [0], [-360], [0], [190], [-390], [0], [0], [0], [0], [120], [416], [0], [418], [0], [-64], [0], [138], [0], [504], [320], [0], [570], [0], [0], [-50], [-150], [0], [350], [0], [360], [-544], [0], [0], [-504], [0], [0], [-378], [456], [-80], [-782], [0], [-840], [0], [-290], [0], [0], [-560], [850], [-522], [0], [0], [0], [0], [0], [0], [0], [-918], [880], [0], [-840], [0], [-330], [0], [-1040], [0], [-1150], [0], [-736], [0], [-1102], [0], [1224], [-1216], [0], [0], [400], [0], [0], [630], [0], [-600], [-770], [1350], [0], [0], [-1302], [-1320], [0], [0], [216], [0], [0], [0], [-936], [-1520], [-962], [-390], [0], [1110], [-560], [0], [858], [0], [0], [1080], [0], [-1656], [-464], [0], [0], [-696], [-1600], [0], [0], [0], [0], [0], [1840], [-430], [-1482], [0], [1456], [0], [0], [496], [0], [0], [744], [-1118], [-90], [0], [1320], [0], [2030], [0], [-640], [0], [-1722], [0], [-1560], [0], [0], [-264], [1856], [0], [-282], [-2184], [0], [658], [0], [-2050], [0], [0], [2262], [0], [1664], [98], [-2376], [-1984], [0], [-2442], [0], [-1224], [1598], [0], [-2070], [0], [0], [-1120], [0], [2590], [102], [0], [0], [0], [2542], [0], [-2480], [0], [-2730], [2040], [0], [2800], [0], [0], [2760], [1184], [0], [0], [0], [-456], [0], [0], [2240], [0], [0], [1378], [2790], [0], [0], [0], [0], [0], [-2520], [-2944], [0], [0], [0], [0], [0], [-2856], [-160], [0], [3182], [-2550], [0], [1560], [0], [570], [1870], [0], [0], [-2280], [2664], [-656], [0], [-1482], [1760], [0], [2310], [-3480], [0], [590], [0], [-2530], [0], [0], [840], [1298], [0], [0], [0], [0], [3720], [0], [0], [610], [-2970], [0], [2720], [-1098], [0], [1376], [0], [2184], [-3498], [0], [-1830], [0], [0], [-3410], [630], [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_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_g_e(:prec:=1)
    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_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_g_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2304_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![-6, 1]>,<7,R![0, 1]>,<11,R![0, 1]>,<13,R![-24, 1]>],Snew);
    return Vf;
end function;