
// Make newform 1134.2.g.k in Magma, downloaded from the LMFDB on 18 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_k();".
// 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_k();".
// 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 := [3, -12, 19, -15, 10, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [2, -1, 1, 0, 0, 0], [6, -18, 5, -8, 1, -1], [3, -5, 6, -2, 1, 0], [9, -21, 19, -16, 5, -2], [-9, 30, -22, 19, -5, 2]];
    Rf_basisdens := [1, 1, 3, 1, 3, 3];
    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, 0], [-1, 0, 0, 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 := 2;
    raw_aps := [[-1, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 2, -1], [-1, 1, 0, -1, 1, -1], [0, -1, 2, 2, 0, 1], [1, 1, 0, 0, 0, 0], [0, 2, 2, 2, 0, -2], [-1, 0, 1, 0, 1, 1], [-3, 0, 1, 0, 3, -1], [1, -3, 0, 1, 0, 0], [0, -2, 1, 1, -5, 2], [-3, 0, 2, 0, 3, 2], [4, 1, 0, -1, 0, 0], [-5, 2, 0, 1, 0, 0], [2, 0, -1, 0, -2, 2], [0, -4, 1, 1, 2, 4], [0, -5, 1, 1, 0, 5], [2, 0, 3, 0, -2, 5], [0, 1, 3, 3, 3, -1], [1, -1, 0, -3, 0, 0], [0, 1, -3, -3, -9, -1], [0, 0, 4, 0, 0, -3], [-3, 1, 0, -2, 0, 0], [-4, 0, 2, 0, 4, -1], [10, 0, 0, 2, 0, 0], [0, 3, -4, -4, -4, -3], [-2, 0, 3, 0, 2, 3], [-6, 0, 3, 0, 6, 0], [0, 2, -1, -1, -2, -2], [2, 4, 0, 3, 0, 0], [-4, -4, 0, 5, 0, 0], [-4, 0, 4, 0, 4, 6], [0, 0, -1, -1, 5, 0], [10, 3, 0, -1, 0, 0], [0, 0, -3, 0, 0, -3], [0, 5, -1, -1, 13, -5], [0, 6, 3, 3, -8, -6], [-7, 0, -2, 0, 7, 7], [-6, -7, 0, -1, 0, 0], [-11, 0, 3, 0, 11, -4], [0, 3, -3, -3, 0, -3], [17, -1, 0, 5, 0, 0], [11, 0, 2, 0, -11, 5], [0, 5, -2, -2, 9, -5], [9, -2, 0, -2, 0, 0], [0, -1, 1, 1, -6, 1], [-11, -4, 0, 4, 0, 0], [7, 2, 0, -2, 0, 0], [0, -8, 5, 5, -8, 8], [-4, 0, 2, 0, 4, -6], [9, 0, -5, 0, -9, -1], [1, -8, 0, -4, 0, 0], [0, 4, 8, 8, 5, -4], [-17, 1, 0, 2, 0, 0], [4, 0, -8, 0, -4, -5], [0, 1, -4, -4, 10, -1], [0, -2, 10, 10, 6, 2], [0, 0, -5, 0, 0, -9], [0, 3, 0, 0, 4, -3], [0, -8, 0, 1, 0, 0], [0, 3, -5, -5, -7, -3], [-11, -8, 0, 2, 0, 0], [1, -3, 0, -1, 0, 0], [0, 7, -2, -2, 3, -7], [10, 0, -2, 0, -10, 8], [-6, 0, -9, 0, 6, -6], [-11, 0, 3, 0, 11, 6], [17, 8, 0, 2, 0, 0], [0, 1, 8, 8, 16, -1], [0, -1, 0, 9, 0, 0], [0, 9, -3, -3, -3, -9], [4, 0, 9, 0, -4, 2], [0, 1, 2, 2, -28, -1], [-21, 0, 3, 0, 21, -2], [-3, 5, 0, -12, 0, 0], [11, 0, 3, 0, -11, -2], [0, -3, 8, 8, 11, 3], [-14, 0, 2, 0, 14, -2], [-4, 0, 5, 0, 4, 11], [0, -5, 2, 2, -10, 5], [-2, 10, 0, -4, 0, 0], [-14, -8, 0, -6, 0, 0], [0, -8, 2, 2, 1, 8], [4, 5, 0, -8, 0, 0], [11, 0, 6, 0, -11, 5], [14, 0, 0, 0, -14, 1], [2, 13, 0, 6, 0, 0], [-19, 0, -1, 0, 19, -9], [3, -4, 0, 5, 0, 0], [-7, 13, 0, 1, 0, 0], [12, 0, -7, 0, -12, 4], [0, -3, -5, -5, 16, 3], [0, -9, -3, -3, -8, 9], [-5, -5, 0, -13, 0, 0], [10, 0, 7, 0, -10, 2], [8, 11, 0, -8, 0, 0], [-5, 0, -2, 0, 5, -14], [0, 0, -15, -15, -18, 0], [10, 0, 6, 0, -10, 9], [-16, 0, 9, 0, 16, -7], [-12, -15, 0, -2, 0, 0], [0, 15, -7, -7, -1, -15], [0, 10, 3, 3, 11, -10], [-3, 0, -6, 0, 3, 6], [0, 9, 2, 2, 9, -9], [0, -11, 6, 6, 12, 11], [-14, 11, 0, 0, 0, 0], [21, 0, 6, 0, -21, -6], [0, -5, -4, -4, -29, 5], [8, 6, 0, -3, 0, 0], [-14, 0, -1, 0, 14, 10], [0, -11, 1, 1, -5, 11], [0, -8, 0, 1, 0, 0], [0, 4, 3, 3, 6, -4], [-20, -10, 0, -5, 0, 0], [0, 12, 1, 1, 7, -12], [8, -9, 0, -3, 0, 0], [0, 0, 3, 3, 33, 0], [-22, 0, -1, 0, 22, -10], [-9, -2, 0, -11, 0, 0], [0, 3, -8, -8, 2, -3], [9, 4, 0, -4, 0, 0], [9, 0, 3, 0, -9, -9], [0, -7, -5, -5, 11, 7], [-5, 0, 3, 0, 5, -9], [-30, -7, 0, -1, 0, 0], [8, 0, -5, 0, -8, 4], [17, 0, 2, 0, -17, -1], [-14, -18, 0, -1, 0, 0], [17, 0, 3, 0, -17, 14], [0, 17, 6, 6, 8, -17], [9, -6, 0, 0, 0, 0], [-36, 0, -2, 0, 36, -9], [8, 17, 0, 8, 0, 0], [15, 0, 4, 0, -15, 17], [8, 13, 0, -2, 0, 0], [0, 11, 6, 6, 22, -11], [0, -2, -5, -5, 13, 2], [16, 4, 0, 5, 0, 0], [0, -9, -1, -1, 14, 9], [-4, 7, 0, 13, 0, 0], [-10, 0, 3, 0, 10, 4], [0, -7, -13, -13, -8, 7], [2, -7, 0, 7, 0, 0], [0, 5, -13, -13, -2, -5], [-10, -3, 0, 8, 0, 0], [21, 4, 0, -7, 0, 0], [0, 5, 4, 4, -1, -5], [-23, 0, 0, 0, 23, -15], [-8, 0, 3, 0, 8, 11], [9, 0, 6, 0, -9, -2], [32, 8, 0, -5, 0, 0], [-14, -10, 0, -14, 0, 0], [32, 0, -1, 0, -32, 2], [0, 7, 7, 7, 16, -7], [5, 14, 0, -2, 0, 0], [10, 0, -11, 0, -10, -4], [-11, 0, -5, 0, 11, 4], [7, -7, 0, -14, 0, 0], [0, -4, -6, -6, 10, 4], [5, 0, 10, 0, -5, -12], [0, 10, 0, -8, 0, 0], [1, -1, 0, 10, 0, 0], [13, 0, 4, 0, -13, -2], [0, -7, 4, 4, -10, 7], [0, -14, -1, -1, 13, 14], [0, -8, 5, 5, -1, 8], [0, -13, -8, -8, 30, 13], [10, -4, 0, 19, 0, 0], [7, 0, -8, 0, -7, 10], [0, -13, 9, 9, 13, 13], [17, -11, 0, 13, 0, 0], [0, 0, 9, 0, 0, -15], [0, 16, -11, -11, -8, -16], [0, -4, 15, 15, 17, 4], [-24, 8, 0, -22, 0, 0], [1, -20, 0, 12, 0, 0], [0, -5, 21, 21, 2, 5], [7, 14, 0, 7, 0, 0], [-14, 0, -3, 0, 14, 15], [-7, 0, 0, 0, 7, -13], [26, -6, 0, -7, 0, 0], [28, 9, 0, -10, 0, 0], [-29, 0, 3, 0, 29, -1], [0, 4, -2, -2, 48, -4], [0, 2, 13, 13, 5, -2], [0, 0, 8, 8, 21, 0], [0, -3, -15, -15, -17, 3], [-8, 0, -13, 0, 8, -5], [0, 6, 0, 0, -3, -6], [4, 0, -20, 0, -4, 5], [0, -12, 0, 18, 0, 0], [10, 0, 1, 0, -10, -16], [19, 0, 14, 0, -19, -12], [0, 30, 0, 0, 9, -30], [0, -9, -11, -11, -11, 9], [0, 8, 1, 1, -12, -8], [18, 0, 6, 0, -18, 1], [9, 9, 0, -12, 0, 0], [19, 0, -5, 0, -19, -5], [0, -10, 13, 13, 17, 10], [-1, -7, 0, 8, 0, 0], [5, 0, -11, 0, -5, -11], [0, -12, 8, 8, 27, 12], [-1, -25, 0, 1, 0, 0], [0, -22, 10, 10, -19, 22], [18, 0, -10, 0, -18, 2], [35, 0, -4, 0, -35, 5], [5, -16, 0, -11, 0, 0], [0, -10, 28, 28, 30, 10], [10, 0, 12, 0, -10, -6], [43, 0, 0, 13, 0, 0], [-5, 16, 0, -6, 0, 0], [-7, 0, 1, 0, 7, -3], [0, 1, 5, 5, 34, -1], [7, 0, 9, 0, -7, 3], [0, 15, -8, -8, 8, -15], [0, 4, -18, -18, -43, -4], [18, 0, -11, 0, -18, 8], [-12, 10, 0, -5, 0, 0], [-8, 0, 10, 0, 8, 14], [11, -2, 0, 10, 0, 0], [33, 0, 0, 0, -33, 12], [-18, 0, 15, 0, 18, -2], [-14, -8, 0, 17, 0, 0], [35, -3, 0, 6, 0, 0], [-14, 0, -16, 0, 14, 0], [0, 11, -9, -9, 28, -11], [9, 0, 4, 0, -9, 18], [-3, 0, 12, 0, 3, 9], [0, -1, -15, -15, 23, 1], [0, 2, -19, -19, 7, -2], [22, 18, 0, -13, 0, 0], [0, -5, -22, -22, 4, 5], [-2, 20, 0, -2, 0, 0], [0, 28, 2, 2, 25, -28], [0, 0, 15, 0, 0, 1], [-24, 0, -12, 0, 24, 9], [21, -3, 0, 6, 0, 0], [-18, 8, 0, -7, 0, 0], [0, 24, -2, -2, 10, -24], [-19, 0, -8, 0, 19, 4], [0, -3, -18, -18, 7, 3], [30, 0, 7, 0, -30, -6], [-29, -7, 0, 9, 0, 0], [34, 0, 1, 0, -34, 13], [-5, 13, 0, -18, 0, 0], [0, 2, -4, -4, -6, -2], [0, -15, 0, 0, -8, 15], [-21, -30, 0, 15, 0, 0], [-7, -32, 0, -2, 0, 0], [-5, 0, 25, 0, 5, -8], [0, 1, 20, 20, 22, -1], [-1, -5, 0, -11, 0, 0], [0, -24, 4, 4, -32, 24], [38, 0, -21, 0, -38, -8], [0, -12, 21, 21, -5, 12], [0, 4, -2, -2, 28, -4], [47, -11, 0, 6, 0, 0], [8, 0, 19, 0, -8, -14], [0, 1, -24, -24, -21, -1], [27, 2, 0, 23, 0, 0], [0, 24, 4, 4, 29, -24], [4, 30, 0, -4, 0, 0], [0, 19, -10, -10, -14, -19], [-8, 0, 19, 0, 8, 2], [11, 2, 0, 13, 0, 0], [-31, -5, 0, -15, 0, 0], [-26, 12, 0, -10, 0, 0], [0, 6, -3, -3, -6, -6], [25, 0, 8, 0, -25, 7], [0, -17, -4, -4, 20, 17], [0, -15, 26, 26, 6, 15], [29, 0, 1, 0, -29, -2], [-38, 12, 0, -4, 0, 0], [33, 0, 10, 0, -33, 15], [9, 0, -17, 0, -9, -7], [0, 23, -18, -18, -1, -23], [-16, 0, 2, 0, 16, 0], [-38, 0, -8, 0, 38, 19], [0, -9, -9, -9, 39, 9], [0, -1, 15, 15, -4, 1], [57, -6, 0, 6, 0, 0], [-48, -8, 0, 11, 0, 0], [-4, 0, -9, 0, 4, 26], [18, 0, 13, 0, -18, -16], [0, 0, -1, -1, -57, 0], [-28, -21, 0, 2, 0, 0], [0, -19, -20, -20, -15, 19], [20, 8, 0, -8, 0, 0], [0, -12, 0, 0, -27, 12], [0, 10, -10, -10, 4, -10], [-12, 0, -9, 0, 12, 6], [-3, -19, 0, -7, 0, 0], [18, -2, 0, -13, 0, 0], [0, 29, -2, -2, -1, -29], [43, 0, 3, 0, -43, 0], [44, 14, 0, -14, 0, 0], [3, 0, -9, 0, -3, 3], [43, 10, 0, -6, 0, 0], [32, 0, -13, 0, -32, 21], [-18, 0, -9, 0, 18, -6], [0, 26, 11, 11, 13, -26]];
    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_k();".
// 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_k(:prec:=6)
    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_k();".
// 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_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1134_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![9, -12, 31, 26, 21, 5, 1]>,<11,R![1089, -858, 643, -92, 27, 1, 1]>,<17,R![28224, 7392, 2608, 160, 60, 4, 1]>,<23,R![9, -12, 37, 34, 45, 7, 1]>],Snew);
    return Vf;
end function;