
// Make newform 1134.2.e.s in Magma, downloaded from the LMFDB on 23 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1134_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1134_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_1134_2_e_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_1134_2_e_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 := [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_e();"
function MakeCharacter_1134_e()
    N := 1134;
    order := 3;
    char_gens := [407, 325];
    v := [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_1134_e_Hecke();"
function MakeCharacter_1134_e_Hecke( : Kf := false)
    N := 1134;
    order := 3;
    char_gens := [407, 325];
    char_values := [[-1, -1, 0, 0], [-1, -1, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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 := [[1, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-1, -1, 1, -1], [0, 0, 1, 1], [-2, -2, 1, 1], [0, 0, 0, 0], [-2, -2, 1, 1], [0, 3, -2, 1], [0, 0, 2, -1], [5, 0, -1, 2], [4, 4, 0, 0], [-3, -3, -2, -2], [0, 2, 4, -2], [9, 0, 1, -2], [0, 0, -2, 1], [0, 0, 0, 0], [2, 0, 1, -2], [-4, 0, -1, 2], [-3, 0, -1, 2], [0, -7, 0, 0], [5, 0, 1, -2], [0, 12, 2, -1], [-3, -3, -2, -2], [0, -4, 4, -2], [-12, -12, -1, -1], [0, 5, 2, -1], [-6, -6, -2, -2], [0, 2, 2, -1], [12, 12, -2, -2], [11, 0, -1, 2], [0, -12, -2, 1], [3, 3, -4, -4], [-8, -8, -3, -3], [0, 12, -2, 1], [7, 7, 1, 1], [14, 0, 0, 0], [16, 16, 1, 1], [-3, -3, 5, 5], [6, 0, -4, 8], [0, -12, 2, -1], [-16, 0, -1, 2], [0, 0, 2, -4], [-13, 0, 2, -4], [0, 0, 4, -8], [0, -19, -2, 1], [-2, -2, -2, -2], [0, -1, -6, 3], [-18, -18, 2, 2], [0, 8, -8, 4], [-12, -12, 2, 2], [9, 9, 3, 3], [-17, -17, -2, -2], [6, 0, 3, -6], [0, -3, -4, 2], [-6, -6, 4, 4], [0, 6, 8, -4], [4, 4, -2, -2], [-2, -2, -5, -5], [0, -3, -12, 6], [8, 0, 4, -8], [-6, -6, -4, -4], [-28, 0, 0, 0], [6, 0, -4, 8], [-1, 0, -4, 8], [0, 0, -4, 8], [-10, 0, -2, 4], [4, 4, 2, 2], [6, 0, -3, 6], [0, 8, -8, 4], [-21, -21, 0, 0], [-15, -15, -1, -1], [1, 1, 3, 3], [0, 14, 10, -5], [14, 0, -3, 6], [0, -3, 10, -5], [-6, -6, 1, 1], [16, 16, -1, -1], [0, 21, -4, 2], [35, 0, 0, 0], [12, 12, 1, 1], [0, -10, 2, -1], [0, -33, 2, -1], [-13, 0, 4, -8], [-1, 0, 3, -6], [18, 0, -2, 4], [12, 0, -4, 8], [20, 0, -2, 4], [0, 30, -2, 1], [13, 13, 1, 1], [0, 0, 0, 0], [9, 9, -1, -1], [0, -7, -2, 1], [6, 6, 3, 3], [0, 2, 6, -3], [-3, 0, -5, 10], [0, 12, 2, -1], [0, -9, -12, 6], [16, 16, -1, -1], [-2, -2, -4, -4], [0, 2, 4, -2], [0, -24, 8, -4], [6, 0, 3, -6], [-9, 0, 8, -16], [-4, 0, -4, 8], [0, 2, -16, 8], [0, 18, 10, -5], [-3, -3, -2, -2], [6, 0, 4, -8], [0, -1, 8, -4], [0, -22, 8, -4], [0, -4, 16, -8], [3, 3, -6, -6], [10, 10, 2, 2], [-16, 0, -2, 4], [-21, -21, -6, -6], [34, 34, -3, -3], [0, 15, 6, -3], [0, 24, 0, 0], [0, -18, 10, -5], [-4, 0, -9, 18], [0, -19, 4, -2], [30, 0, 1, -2], [0, 12, -14, 7], [2, 0, 1, -2], [18, 0, -1, 2], [-4, 0, 7, -14], [15, 15, 3, 3], [0, -13, 6, -3], [0, -22, 8, -4], [0, 2, -4, 2], [-39, -39, -1, -1], [0, -13, 2, -1], [-4, 0, 4, -8], [0, 21, 0, 0], [4, 4, -2, -2], [0, -24, 10, -5], [-10, 0, -5, 10], [0, 0, 0, 0], [0, -9, 0, 0], [2, 0, -6, 12], [6, 0, -9, 18], [11, 0, 7, -14], [0, 0, 4, -8], [0, -16, 16, -8], [0, 24, 4, -2], [0, -10, -18, 9], [-3, -3, -2, -2], [0, 14, -14, 7], [-21, -21, -5, -5], [0, -4, -14, 7], [-45, 0, 2, -4], [20, 0, 7, -14], [0, 27, -6, 3], [4, 4, -8, -8], [0, -27, -6, 3], [-8, -8, 6, 6], [-27, 0, 4, -8], [5, 0, 6, -12], [6, 0, -11, 22], [-30, 0, 2, -4], [-15, 0, 2, -4], [-23, -23, -7, -7], [18, 18, -2, -2], [3, 0, -4, 8], [24, 24, 2, 2], [0, 29, -10, 5], [-14, -14, 0, 0], [-16, 0, -4, 8], [0, 0, 0, 0], [-24, -24, -4, -4], [4, 4, -9, -9], [0, -36, -12, 6], [19, 19, 2, 2], [-34, 0, 4, -8], [-36, -36, 4, 4], [0, 26, -10, 5], [0, 30, 10, -5], [8, 0, 4, -8], [-34, 0, 4, -8], [-28, 0, 2, -4], [0, -12, -16, 8], [-2, -2, -9, -9], [33, 33, -6, -6], [9, 0, -5, 10], [12, 12, 1, 1], [0, -22, -8, 4], [10, 10, 2, 2], [0, 11, -8, 4], [0, 3, -10, 5], [25, 25, -2, -2], [0, 6, -14, 7], [-20, -20, -8, -8], [-6, 0, 4, -8], [0, -6, -4, 2], [15, 0, 4, -8], [-13, 0, -2, 4], [32, 0, 4, -8], [0, 36, -8, 4], [-39, -39, -5, -5], [-18, 0, -4, 8], [0, -4, 4, -2], [0, 2, -2, 1], [25, 25, -2, -2], [-18, 0, 5, -10], [0, -30, 16, -8], [1, 1, -11, -11], [0, 6, 4, -2], [0, 39, -12, 6], [-40, 0, 1, -2], [-59, -59, 0, 0], [-36, -36, 4, 4], [0, 11, -2, 1], [6, 0, 3, -6], [-33, 0, 1, -2], [53, 0, 2, -4], [41, 0, -3, 6], [27, 27, -10, -10], [0, 0, -12, -12], [42, 42, -3, -3], [0, -16, -2, 1], [16, 16, 6, 6], [0, 12, -4, 2], [7, 7, 9, 9], [0, 0, 0, 0], [0, 14, 16, -8], [21, 0, 0, 0], [0, -6, 24, -12], [-37, 0, -1, 2], [-42, 0, 2, -4], [-4, 0, -4, 8], [0, 56, 0, 0], [-8, -8, 10, 10], [51, 0, 2, -4], [0, 2, 12, -6], [-12, -12, 6, 6], [0, -4, 4, -2], [54, 54, 4, 4], [-12, -12, -4, -4], [-39, 0, -9, 18], [48, 48, -5, -5], [-2, -2, -6, -6], [-40, 0, -6, 12], [-32, -32, 4, 4], [51, 51, 6, 6], [-3, 0, -5, 10], [35, 0, 7, -14], [-66, 0, 2, -4], [14, 0, -6, 12], [-3, 0, 1, -2], [-2, -2, 12, 12], [-3, -3, 12, 12], [-27, 0, 9, -18], [0, -1, 8, -4], [12, 12, 15, 15], [6, 6, 7, 7], [0, -34, -10, 5], [28, 28, -7, -7], [-48, 0, 4, -8], [-56, -56, 0, 0], [-8, -8, -2, -2], [0, 48, 14, -7], [14, 0, -7, 14], [14, 0, 0, 0], [12, 0, 6, -12], [26, 0, -1, 2], [48, 0, 0, 0], [0, 0, 0, 0], [-50, -50, 3, 3], [-30, 0, 0, 0], [0, -4, -10, 5], [0, 20, -8, 4], [-44, -44, 8, 8], [0, -40, 8, -4], [-5, -5, 6, 6], [-5, -5, 13, 13], [0, -24, -22, 11], [10, 10, -4, -4], [-41, -41, -2, -2], [12, 0, -8, 16], [-36, 0, 10, -20], [32, 0, -4, 8], [0, -33, -2, 1], [0, 62, 8, -4], [0, 8, -8, 4], [3, 3, 11, 11], [0, -28, 4, -2], [-18, -18, 13, 13], [64, 64, -4, -4], [-39, 0, -2, 4], [42, 42, 1, 1], [0, 30, 12, -6], [0, -6, 16, -8], [54, 54, -6, -6], [0, -22, 0, 0], [6, 0, -11, 22], [41, 0, -11, 22], [0, -6, 6, -3], [-30, -30, -6, -6], [0, 14, -14, 7], [0, 47, 16, -8], [12, 12, 11, 11], [0, -16, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1134_e_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1134_2_e_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_1134_2_e_s(:prec:=4)
    chi := MakeCharacter_1134_e();
    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_e_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_1134_2_e_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1134_e();
    // 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;