
// Make newform 2304.3.e.e in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2304_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2304_e_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_e_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_e_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, 4, 0], [0, 3, 0, 3], [0, 6, 0, -6]];
    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_e();"
function MakeCharacter_2304_e()
    N := 2304;
    order := 2;
    char_gens := [1279, 2053, 1793];
    v := [2, 2, 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_e_Hecke();"
function MakeCharacter_2304_e_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, 1, 0], [0, 0, 0, 1], [0, -1, 0, 0], [-18, 0, 0, 0], [0, 0, -1, 0], [0, 0, 0, 2], [0, -9, 0, 0], [0, 0, -3, 0], [0, 0, 0, 1], [-36, 0, 0, 0], [0, 0, -7, 0], [0, 0, 0, -8], [0, -9, 0, 0], [0, 0, -19, 0], [0, -20, 0, 0], [36, 0, 0, 0], [0, 0, 0, 14], [0, 27, 0, 0], [56, 0, 0, 0], [0, 0, 0, 3], [0, 19, 0, 0], [0, 0, -21, 0], [104, 0, 0, 0], [0, 0, 35, 0], [0, 0, 0, -21], [0, -32, 0, 0], [-126, 0, 0, 0], [0, 0, -21, 0], [0, 0, 0, -19], [0, 38, 0, 0], [0, 0, -49, 0], [0, 0, 0, -14], [0, 0, 23, 0], [0, 0, 0, 29], [252, 0, 0, 0], [0, 0, 0, 12], [0, 0, 0, 0], [0, 0, -7, 0], [0, 14, 0, 0], [-126, 0, 0, 0], [0, 0, 0, 0], [298, 0, 0, 0], [0, 0, -77, 0], [0, 0, 0, 7], [0, 0, 0, -6], [0, 0, 0, 33], [0, -71, 0, 0], [126, 0, 0, 0], [0, 0, -67, 0], [0, -90, 0, 0], [-32, 0, 0, 0], [0, 5, 0, 0], [0, 0, -91, 0], [0, 18, 0, 0], [0, 0, -93, 0], [0, 0, 0, -11], [-126, 0, 0, 0], [0, 0, -21, 0], [0, 0, 0, 24], [0, 0, 45, 0], [0, 0, 0, -34], [0, 126, 0, 0], [-58, 0, 0, 0], [0, 0, 43, 0], [0, 0, 0, 42], [-8, 0, 0, 0], [0, -7, 0, 0], [-252, 0, 0, 0], [0, 0, -133, 0], [0, -63, 0, 0], [0, 0, 0, 1], [-252, 0, 0, 0], [0, 0, 0, 70], [0, 126, 0, 0], [0, 0, -49, 0], [756, 0, 0, 0], [0, 0, 119, 0], [136, 0, 0, 0], [0, 7, 0, 0], [-270, 0, 0, 0], [0, 63, 0, 0], [-574, 0, 0, 0], [0, 0, 0, -11], [0, 179, 0, 0], [0, 0, -47, 0], [40, 0, 0, 0], [0, 0, 175, 0], [0, 0, 0, -77], [0, 13, 0, 0], [0, -171, 0, 0], [0, 0, 0, 27], [0, -110, 0, 0], [0, 0, 0, -56], [0, 225, 0, 0], [0, 0, -43, 0], [0, 0, -1, 0], [0, 0, 0, -30], [-414, 0, 0, 0], [0, 0, 0, 8], [0, 0, -65, 0], [0, 223, 0, 0], [0, 0, 199, 0], [0, 0, 0, -52], [-326, 0, 0, 0], [0, 100, 0, 0], [0, 0, 213, 0], [0, -189, 0, 0], [-778, 0, 0, 0], [0, 0, 0, 45], [180, 0, 0, 0], [0, 0, 107, 0], [0, 0, 0, -4], [0, 0, 0, -89], [0, 0, 55, 0], [0, 0, 0, 140], [0, 81, 0, 0], [0, 0, 209, 0], [0, -280, 0, 0], [-180, 0, 0, 0], [302, 0, 0, 0], [0, 0, -183, 0], [0, -265, 0, 0], [0, 0, 0, 84], [0, 0, -67, 0], [-882, 0, 0, 0], [0, 63, 0, 0], [0, 0, 0, -5], [-162, 0, 0, 0], [0, 0, 0, 28], [0, 252, 0, 0], [0, 0, 0, -59], [-1026, 0, 0, 0], [0, 0, 297, 0], [694, 0, 0, 0], [0, 0, -339, 0], [0, 0, 0, 102], [0, 0, -151, 0], [0, 0, 161, 0], [0, 0, 0, -154], [0, 0, 27, 0], [0, 0, 0, 77], [0, -194, 0, 0], [-702, 0, 0, 0], [0, 153, 0, 0], [252, 0, 0, 0], [0, 0, 153, 0], [0, 0, 0, 40], [0, 252, 0, 0], [-1260, 0, 0, 0], [0, 0, 357, 0], [0, 0, 0, 112], [0, 54, 0, 0], [0, 0, 0, -88], [0, -36, 0, 0], [0, 0, 0, 77], [0, 0, -297, 0], [-1154, 0, 0, 0], [0, 0, -257, 0], [0, -280, 0, 0], [0, 0, 79, 0], [0, 0, 0, -37], [0, -193, 0, 0], [0, 0, 281, 0], [0, 126, 0, 0], [0, 0, 0, -43], [756, 0, 0, 0], [-1240, 0, 0, 0], [0, 0, -111, 0], [0, 370, 0, 0], [1404, 0, 0, 0], [0, 198, 0, 0], [976, 0, 0, 0], [0, 0, 0, 223], [0, 0, -21, 0], [0, 0, 0, 20], [0, 0, -233, 0], [0, 0, 0, -7], [972, 0, 0, 0], [0, 0, 0, 31], [0, -215, 0, 0], [-882, 0, 0, 0], [0, 0, -245, 0], [0, 126, 0, 0], [0, 0, -271, 0], [558, 0, 0, 0], [0, 0, 0, 84], [656, 0, 0, 0], [0, -18, 0, 0], [128, 0, 0, 0], [0, 334, 0, 0], [0, 0, 0, -68], [0, 0, -217, 0], [0, 50, 0, 0], [0, 0, -91, 0], [442, 0, 0, 0], [1854, 0, 0, 0], [0, 0, 19, 0], [0, -180, 0, 0], [0, 0, -47, 0], [0, 0, 0, -235], [1890, 0, 0, 0], [616, 0, 0, 0], [0, 77, 0, 0], [0, 0, 51, 0], [0, 0, 0, -237], [0, -11, 0, 0], [0, 0, -467, 0], [0, 0, 0, 56], [-938, 0, 0, 0], [0, 0, -423, 0], [0, 0, 0, -187], [0, 550, 0, 0], [0, -207, 0, 0], [-14, 0, 0, 0], [0, 0, 0, -225], [0, 0, 41, 0], [0, 630, 0, 0], [0, 0, -111, 0], [-126, 0, 0, 0], [0, 0, 0, -231], [0, 0, 31, 0], [0, 0, 0, 87], [0, 61, 0, 0], [-2772, 0, 0, 0], [0, 0, -651, 0], [0, 315, 0, 0], [0, 0, 0, 81], [0, 280, 0, 0], [-558, 0, 0, 0], [0, 0, 0, -252], [0, 0, 0, 327], [0, 0, -203, 0], [0, 0, 0, -252], [0, -81, 0, 0], [2168, 0, 0, 0], [0, 0, 279, 0], [0, 497, 0, 0], [0, 405, 0, 0], [0, 91, 0, 0], [0, 0, 0, 42], [0, 0, 0, -209], [1260, 0, 0, 0], [0, 0, -611, 0], [0, 567, 0, 0], [0, 0, 0, -257], [0, 434, 0, 0], [0, 0, 0, 126], [0, 378, 0, 0], [-2142, 0, 0, 0], [0, 0, 277, 0], [0, -567, 0, 0], [1552, 0, 0, 0], [0, 0, -373, 0], [0, -346, 0, 0], [-558, 0, 0, 0], [0, 0, 0, 0], [0, 0, 101, 0], [-254, 0, 0, 0], [0, 0, 0, 103], [0, 170, 0, 0], [-126, 0, 0, 0], [1260, 0, 0, 0], [0, 0, -351, 0], [0, 0, 0, -154], [0, 0, -375, 0], [0, 0, 61, 0], [0, 0, 0, -154], [0, 0, -345, 0], [756, 0, 0, 0], [0, 0, 0, 252], [256, 0, 0, 0], [0, 0, 0, -235], [1520, 0, 0, 0], [0, 0, 0, 217], [0, -277, 0, 0], [-2700, 0, 0, 0], [-1078, 0, 0, 0], [0, 224, 0, 0], [0, 171, 0, 0], [0, 0, 0, 97], [0, -783, 0, 0], [2772, 0, 0, 0], [0, 0, 0, -14], [0, 324, 0, 0], [592, 0, 0, 0], [0, 0, 319, 0], [0, 0, 0, 141], [0, 0, -245, 0], [0, 0, 609, 0], [0, 428, 0, 0], [0, 0, 91, 0], [0, -578, 0, 0], [-900, 0, 0, 0], [0, 0, -33, 0], [0, 0, 0, 113], [0, 0, 419, 0], [0, 707, 0, 0], [0, 0, 0, 414], [2618, 0, 0, 0], [0, 0, -169, 0], [0, 0, 0, -169]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2304_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2304_3_e_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_e_e(:prec:=4)
    chi := MakeCharacter_2304_e();
    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_e_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_e_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2304_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![18, 0, 1]>,<7,R![-72, 0, 1]>,<13,R![18, 1]>,<19,R![-288, 0, 1]>,<31,R![-72, 0, 1]>],Snew);
    return Vf;
end function;