
// Make newform 2304.3.g.l 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_l();".
// 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_l();".
// 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 := [6, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 4]];
    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_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], [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], [2, 0], [0, -1], [0, -2], [0, 0], [0, 0], [0, 0], [0, 0], [-50, 0], [0, -5], [0, 0], [0, 0], [0, 0], [0, 0], [94, 0], [0, 12], [0, 0], [0, 0], [0, 0], [-50, 0], [0, 15], [0, 10], [0, 0], [-190, 0], [190, 0], [0, 21], [0, -20], [0, 0], [0, 0], [0, -11], [0, -8], [0, 0], [0, 0], [290, 0], [0, -5], [0, 0], [0, 0], [0, 0], [-46, 0], [0, -28], [0, 0], [0, 0], [-290, 0], [-194, 0], [0, -35], [0, 0], [0, -39], [0, 30], [0, 0], [0, 0], [0, 0], [-382, 0], [0, -18], [0, 0], [0, 0], [430, 0], [0, -55], [0, 0], [0, 0], [0, 0], [-386, 0], [0, 0], [0, 0], [-530, 0], [334, 0], [0, 0], [190, 0], [0, -70], [0, 0], [0, 0], [0, 0], [0, 19], [0, 0], [0, 0], [0, 0], [-190, 0], [0, 0], [0, 0], [-718, 0], [0, -42], [0, 0], [0, 0], [670, 0], [0, 85], [0, 90], [0, 0], [530, 0], [530, 0], [0, -91], [0, 70], [0, 0], [0, -9], [0, -88], [0, 0], [0, 0], [-1010, 0], [0, 0], [0, 0], [0, 0], [0, 0], [914, 0], [0, 110], [0, 0], [0, 0], [290, 0], [0, -80], [0, 0], [0, 0], [-1198, 0], [0, -99], [0, 0], [0, 0], [0, 0], [0, -25], [0, 0], [0, 0], [0, 0], [-1006, 0], [0, -112], [0, 0], [190, 0], [1346, 0], [0, 30], [0, 0], [-50, 0], [0, 0], [0, 0], [0, 11], [0, 0], [0, 0], [0, 0], [0, 55], [0, 0], [0, 0], [862, 0], [-1154, 0], [0, 0], [-1294, 0], [0, 0], [0, 0], [670, 0], [0, -161], [0, -60], [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, 175], [0, 0], [1490, 0], [-430, 0], [0, 140], [0, 0], [0, -31], [0, 22], [0, 0], [0, 0], [0, -145], [0, 0], [482, 0], [674, 0], [0, -208], [0, 0], [0, 0], [1390, 0], [0, -65], [0, 0], [0, 0], [-2110, 0], [0, -91], [0, 0], [0, -209], [0, -38], [0, 0], [0, 0], [0, 0], [1250, 0], [0, 0], [0, 0], [-142, 0], [0, 0], [1150, 0], [0, -220], [0, 0], [1970, 0], [0, -240], [0, 0], [-2302, 0], [0, 0], [0, 0], [0, 0], [-430, 0], [0, -35], [0, 0], [-2402, 0], [0, -238], [2446, 0], [0, 105], [0, 190], [0, 0], [0, 0], [2110, 0], [-1630, 0], [0, 221], [0, 260], [0, 0], [-2258, 0], [0, -261], [0, 0], [0, 0], [2546, 0], [0, 0], [0, 105], [0, 0], [0, -111], [0, 130], [0, 0], [0, 0], [0, 0], [0, -279], [0, 168], [0, 0], [0, 0], [0, 165], [0, 0], [0, 0], [0, 0], [-478, 0], [-2014, 0], [0, 182], [0, 0], [0, -210], [0, 0], [0, 299], [0, 0], [0, 0], [0, 0], [0, 319], [0, 28], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-1682, 0], [1774, 0], [0, -92], [0, 0], [0, 0], [-3074, 0], [2830, 0], [0, -259], [0, -340], [0, 0], [0, 0], [0, 0], [0, 0], [-3310, 0], [0, 0], [0, 0], [-766, 0], [0, 0], [0, 0], [-1970, 0], [0, -85], [-3170, 0], [0, -119], [0, -230], [0, 0], [-1102, 0], [0, 252], [0, 0], [0, 205], [0, 0], [0, 0], [0, 0], [0, 0], [3650, 0], [3746, 0], [0, -345], [0, 0], [1490, 0], [0, 100], [0, 0], [0, 308], [0, 0], [-3790, 0], [0, 385], [-3554, 0], [0, 322], [0, 0], [3790, 0], [-3694, 0], [0, 215]];
    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_l();".
// 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_l(:prec:=2)
    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_l();".
// 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_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2304_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<7,R![96, 0, 1]>,<11,R![384, 0, 1]>,<13,R![0, 1]>],Snew);
    return Vf;
end function;