
// Make newform 1134.2.g.j in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1134_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1134_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_1134_2_g_j();".
// 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_1134_2_g_j();".
// 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 := [4, 0, 2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 1, 0], [0, 4, 0, 1], [0, 2, 0, -1]];
    Rf_basisdens := [1, 2, 2, 2];
    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_1134_g();"
function MakeCharacter_1134_g()
    N := 1134;
    order := 3;
    char_gens := [407, 325];
    v := [3, 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_1134_g_Hecke();"
function MakeCharacter_1134_g_Hecke(Kf)
    N := 1134;
    order := 3;
    char_gens := [407, 325];
    char_values := [[1, 0, 0, 0], [0, 1, 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 := 2;
    raw_aps := [[0, -1, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-1, -1, -1, 1], [0, 0, -1, -1], [2, 0, 1, -2], [0, 0, 0, 0], [0, 2, -2, 1], [0, -3, 2, -1], [0, 0, 1, -2], [-5, -5, -1, -1], [0, -4, 0, 0], [-3, 0, 2, -4], [2, 0, -2, 4], [0, -9, 2, -1], [0, 0, -1, -1], [0, 0, 0, 0], [0, 2, -2, 1], [4, 4, -1, -1], [3, 0, 1, -2], [7, 7, 0, 0], [0, 5, -2, 1], [-12, 0, 1, -2], [0, -3, -4, 2], [-4, 0, -2, 4], [12, 12, 1, 1], [0, 5, 2, -1], [0, -6, -4, 2], [-2, -2, -1, -1], [12, 0, 2, -4], [11, 0, -1, 2], [0, 12, 2, -1], [-3, -3, 4, 4], [8, 0, -3, 6], [0, -12, 2, -1], [7, 7, 1, 1], [-14, -14, 0, 0], [0, -16, -2, 1], [-3, 0, -5, 10], [0, -6, -8, 4], [-12, -12, 1, 1], [-16, 0, -1, 2], [0, 0, 4, -2], [13, 13, 2, 2], [0, 0, -4, 8], [19, 19, 1, 1], [2, 0, -2, 4], [-1, 0, 3, -6], [18, 18, -2, -2], [0, 8, -8, 4], [0, -12, 4, -2], [9, 0, -3, 6], [-17, -17, -2, -2], [-6, 0, -3, 6], [0, 3, 4, -2], [6, 6, -4, -4], [6, 6, 4, 4], [0, -4, 4, -2], [-2, -2, -5, -5], [3, 0, -6, 12], [-8, -8, 4, 4], [-6, 0, 4, -8], [-28, 0, 0, 0], [6, 6, 4, 4], [0, -1, 8, -4], [0, 0, -8, 4], [0, -10, 4, -2], [-4, 0, 2, -4], [6, 6, 3, 3], [8, 0, 4, -8], [21, 21, 0, 0], [0, -15, -2, 1], [1, 1, 3, 3], [0, 14, 10, -5], [14, 0, -3, 6], [0, 3, -10, 5], [6, 6, -1, -1], [0, -16, 2, -1], [0, -21, 4, -2], [-35, -35, 0, 0], [12, 0, -1, 2], [-10, 0, -1, 2], [-33, -33, 1, 1], [-13, 0, 4, -8], [0, -1, -6, 3], [0, -18, -4, 2], [-12, 0, 4, -8], [0, 20, 4, -2], [-30, 0, -1, 2], [-13, 0, 1, -2], [0, 0, 0, 0], [-9, -9, 1, 1], [7, 7, 1, 1], [6, 0, -3, 6], [0, 2, 6, -3], [3, 0, 5, -10], [0, -12, -2, 1], [-9, -9, -6, -6], [0, -16, 2, -1], [0, 2, 8, -4], [2, 0, -2, 4], [-24, -24, 4, 4], [6, 6, -3, -3], [0, 9, 16, -8], [4, 4, -4, -4], [-2, -2, 8, 8], [-18, 0, 5, -10], [0, -3, -4, 2], [6, 6, -4, -4], [-1, 0, -4, 8], [0, -22, 8, -4], [4, 4, -8, -8], [3, 0, 6, -12], [10, 10, 2, 2], [-16, 0, -2, 4], [21, 21, 6, 6], [-34, 0, -3, 6], [15, 15, 3, 3], [0, -24, 0, 0], [18, 0, 5, -10], [4, 4, -9, -9], [-19, 0, -2, 4], [0, -30, 2, -1], [12, 12, -7, -7], [0, 2, -2, 1], [-18, 0, 1, -2], [0, -4, -14, 7], [0, 15, 6, -3], [-13, 0, -3, 6], [0, -22, 8, -4], [-2, -2, 2, 2], [-39, 0, 1, -2], [0, -13, 2, -1], [-4, 0, 4, -8], [0, -21, 0, 0], [-4, 0, -2, 4], [-24, -24, 5, 5], [10, 10, -5, -5], [0, 0, 0, 0], [-9, -9, 0, 0], [2, 0, -6, 12], [0, -6, -18, 9], [-11, -11, 7, 7], [0, 0, -4, 8], [16, 16, -8, -8], [-24, 0, 2, -4], [-10, 0, 9, -18], [3, 3, 2, 2], [0, 14, -14, 7], [0, -21, -10, 5], [0, -4, -14, 7], [45, 0, -2, 4], [20, 0, 7, -14], [0, -27, 6, -3], [4, 4, -8, -8], [27, 0, -3, 6], [0, 8, -12, 6], [0, 27, 8, -4], [5, 0, 6, -12], [6, 6, 11, 11], [0, 30, 4, -2], [15, 0, -2, 4], [23, 0, -7, 14], [0, 18, -4, 2], [3, 3, 4, 4], [-24, -24, -2, -2], [-29, -29, 5, 5], [-14, -14, 0, 0], [-16, 0, -4, 8], [0, 0, 0, 0], [24, 24, 4, 4], [-4, 0, -9, 18], [0, 36, 12, -6], [19, 19, 2, 2], [34, 34, 4, 4], [-36, 0, -4, 8], [26, 0, 5, -10], [30, 30, 5, 5], [8, 0, 4, -8], [0, -34, -8, 4], [0, -28, -4, 2], [12, 0, -8, 16], [2, 0, -9, 18], [0, 33, -12, 6], [9, 9, 5, 5], [-12, -12, -1, -1], [22, 22, 4, 4], [10, 10, 2, 2], [0, 11, -8, 4], [3, 3, -5, -5], [0, -25, 4, -2], [-6, 0, -7, 14], [0, 20, 16, -8], [0, 6, 8, -4], [-6, -6, -2, -2], [15, 15, -4, -4], [13, 13, -2, -2], [0, 32, -8, 4], [-36, 0, -4, 8], [0, -39, -10, 5], [-18, -18, 4, 4], [-4, 0, -2, 4], [0, 2, -2, 1], [25, 25, -2, -2], [18, 0, -5, 10], [-30, -30, 8, 8], [0, -1, 22, -11], [0, -6, -4, 2], [-39, 0, -6, 12], [40, 40, 1, 1], [0, 59, 0, 0], [-36, 0, -4, 8], [11, 0, 1, -2], [0, -6, 6, -3], [-33, -33, -1, -1], [0, 53, -4, 2], [-41, -41, -3, -3], [-27, -27, 10, 10], [0, 0, -24, 12], [42, 0, 3, -6], [0, -16, -2, 1], [-16, 0, 6, -12], [0, -12, 4, -2], [0, -7, -18, 9], [0, 0, 0, 0], [14, 0, -8, 16], [0, -21, 0, 0], [-6, -6, 12, 12], [0, -37, 2, -1], [0, 42, 4, -2], [4, 4, -4, -4], [-56, -56, 0, 0], [8, 0, 10, -20], [51, 51, -2, -2], [2, 0, -6, 12], [12, 12, -6, -6], [0, -4, 4, -2], [0, 54, 8, -4], [-12, 0, 4, -8], [39, 0, 9, -18], [-48, -48, 5, 5], [0, 2, 12, -6], [40, 40, -6, -6], [0, 32, -8, 4], [51, 0, -6, 12], [0, 3, -10, 5], [35, 0, 7, -14], [-66, -66, -2, -2], [-14, -14, -6, -6], [3, 0, -1, 2], [2, 0, 12, -24], [0, -3, 24, -12], [-27, -27, -9, -9], [-1, 0, -4, 8], [-12, -12, -15, -15], [0, 6, 14, -7], [34, 34, 5, 5], [28, 28, -7, -7], [48, 0, -4, 8], [0, 56, 0, 0], [-8, -8, -2, -2], [-48, 0, 7, -14], [-14, -14, -7, -7], [14, 0, 0, 0], [12, 12, -6, -6], [0, 26, 2, -1], [-48, 0, 0, 0], [0, 0, 0, 0], [50, 0, 3, -6], [-30, -30, 0, 0], [0, -4, -10, 5], [-20, -20, 4, 4], [-44, -44, 8, 8], [0, -40, 8, -4], [5, 0, 6, -12], [0, 5, -26, 13], [0, 24, 22, -11], [10, 10, -4, -4], [0, 41, 4, -2], [0, -12, -16, 8], [-36, -36, -10, -10], [-32, -32, -4, -4], [33, 0, -1, 2], [62, 0, -4, 8], [0, 8, -8, 4], [0, 3, 22, -11], [28, 28, -2, -2], [-18, 0, -13, 26], [64, 64, -4, -4], [39, 0, 2, -4], [-42, -42, -1, -1], [30, 30, 6, 6], [0, 6, -16, 8], [54, 0, 6, -12], [-22, 0, 0, 0], [6, 6, 11, 11], [0, 41, 22, -11], [6, 0, 3, -6], [0, -30, -12, 6], [14, 0, 7, -14], [0, 47, 16, -8], [0, 12, 22, -11], [16, 16, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1134_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1134_2_g_j();".
// 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_1134_2_g_j(:prec:=4)
    chi := MakeCharacter_1134_g();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_1134_2_g_j();".
// 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_1134_2_g_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1134_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<11,R![324, 0, 18, 0, 1]>,<17,R![0, 1]>,<23,R![81, 54, 45, -6, 1]>],Snew);
    return Vf;
end function;