
// Make newform 1134.2.g.m in Magma, downloaded from the LMFDB on 16 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_m();".
// 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_m();".
// 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], [0, 0, 1, 0, 0, 0], [0, 1, 1, 0, 1, 0], [0, 0, 1, 1, 0, 0], [-3, -2, 0, 1, 0, 0], [0, -2, 0, 0, -2, 2], [-2, 0, 0, 0, 2, -3], [-2, 0, 0, 0, 2, 1], [0, -3, 0, -2, 0, 0], [0, -1, -1, -1, 6, 1], [1, 0, 0, 0, -1, 0], [-1, -1, 0, -2, 0, 0], [2, -3, 0, 3, 0, 0], [-3, 0, 3, 0, 3, 3], [0, -1, 1, 1, 5, 1], [0, 1, -2, -2, -5, -1], [3, 0, -2, 0, -3, -1], [0, 1, 4, 4, 2, -1], [4, 7, 0, -2, 0, 0], [0, 1, 4, 4, 8, -1], [0, 0, 1, 0, 0, -4], [-1, 2, 0, -3, 0, 0], [-2, 0, 1, 0, 2, 4], [-8, 2, 0, 2, 0, 0], [0, -4, 3, 3, -10, 4], [1, 0, 0, 0, -1, 3], [7, 0, 1, 0, -7, -5], [0, 1, 1, 1, -1, -1], [13, 1, 0, 3, 0, 0], [3, -5, 0, -5, 0, 0], [0, 0, -2, 0, 0, 0], [0, 5, 3, 3, 8, -5], [-5, -1, 0, -4, 0, 0], [-1, 0, 8, 0, 1, 5], [0, 1, -2, -2, 2, -1], [0, -3, -3, -3, 1, 3], [7, 0, -3, 0, -7, 6], [-5, 1, 0, 0, 0, 0], [-2, 0, -1, 0, 2, -11], [0, -7, -2, -2, -1, 7], [0, 1, 0, 4, 0, 0], [9, 0, -3, 0, -9, 6], [0, 0, -3, -3, -5, 0], [-11, -8, 0, 4, 0, 0], [0, 3, 0, 0, 13, -3], [-1, 6, 0, -6, 0, 0], [-7, 6, 0, -6, 0, 0], [0, 7, -1, -1, 13, -7], [2, 0, -4, 0, -2, -2], [-6, 0, 2, 0, 6, -3], [5, -4, 0, -2, 0, 0], [0, 0, 0, 0, 13, 0], [-3, 6, 0, -3, 0, 0], [-4, 0, -1, 0, 4, 2], [0, -8, -1, -1, -2, 8], [0, 6, 8, 8, 0, -6], [3, 0, -8, 0, -3, -1], [0, 0, -9, -9, -14, 0], [5, -1, 0, 9, 0, 0], [0, 1, -2, -2, 20, -1], [5, -4, 0, 10, 0, 0], [-8, 5, 0, -4, 0, 0], [0, -2, 1, 1, 1, 2], [-4, 0, 2, 0, 4, -2], [-3, 0, 9, 0, 3, 3], [16, 0, -3, 0, -16, -3], [15, -2, 0, -2, 0, 0], [0, 6, -3, -3, -6, -6], [-11, -1, 0, 8, 0, 0], [0, 1, -4, -4, -2, -1], [5, 0, 9, 0, -5, 5], [0, 6, -3, -3, 10, -6], [-4, 0, 5, 0, 4, 1], [13, 8, 0, -1, 0, 0], [10, 0, -3, 0, -10, -5], [0, 2, -3, -3, 23, -2], [-12, 0, -8, 0, 12, 2], [-1, 0, -4, 0, 1, -13], [0, 0, 3, 3, 4, 0], [-10, 8, 0, -2, 0, 0], [0, -2, 0, -2, 0, 0], [0, 14, -8, -8, 11, -14], [-4, -6, 0, 3, 0, 0], [9, 0, 1, 0, -9, 14], [-4, 0, -5, 0, 4, -10], [8, -4, 0, -7, 0, 0], [26, 0, 2, 0, -26, 1], [0, -3, 0, 1, 0, 0], [-8, -13, 0, 8, 0, 0], [11, 0, -9, 0, -11, -7], [0, -3, -8, -8, -3, 3], [0, 15, 0, 0, 7, -15], [14, 5, 0, -2, 0, 0], [-1, 0, -13, 0, 1, -5], [-36, 0, 0, -3, 0, 0], [-1, 0, -4, 0, 1, -4], [0, 11, 1, 1, -1, -11], [-2, 0, 9, 0, 2, -12], [15, 0, 4, 0, -15, -7], [-12, 4, 0, 7, 0, 0], [0, -3, 8, 8, 18, 3], [0, 11, 5, 5, 20, -11], [-21, 0, 0, 0, 21, -6], [0, 14, -7, -7, 3, -14], [0, -8, 1, 1, 26, 8], [8, 2, 0, -9, 0, 0], [7, 0, -8, 0, -7, -14], [0, 0, 3, 3, 15, 0], [17, 3, 0, 3, 0, 0], [9, 0, -5, 0, -9, -1], [0, -7, 2, 2, 6, 7], [-15, 15, 0, 5, 0, 0], [0, -11, 1, 1, 17, 11], [17, -3, 0, 15, 0, 0], [0, 3, 1, 1, -24, -3], [-13, -15, 0, 6, 0, 0], [0, 5, -5, -5, 2, -5], [-19, 0, -7, 0, 19, -13], [2, 5, 0, 3, 0, 0], [0, 16, -5, -5, 8, -16], [-7, -12, 0, -12, 0, 0], [0, 0, 0, 0, 0, -3], [0, -7, 4, 4, -10, 7], [-8, 0, 0, 0, 8, -9], [-7, -19, 0, -4, 0, 0], [-17, 0, -9, 0, 17, -9], [31, 0, 7, 0, -31, 10], [-17, -7, 0, -1, 0, 0], [6, 0, 13, 0, -6, 5], [0, -16, -1, -1, -12, 16], [5, -4, 0, -8, 0, 0], [-6, 0, 1, 0, 6, 2], [-6, 4, 0, 1, 0, 0], [-5, 0, -3, 0, 5, -2], [-2, 20, 0, 5, 0, 0], [0, 6, 5, 5, 18, -6], [0, 11, 5, 5, 18, -11], [-13, -13, 0, -11, 0, 0], [0, 3, 8, 8, -3, -3], [-5, 17, 0, 2, 0, 0], [-31, 0, 7, 0, 31, -1], [0, -11, 4, 4, -19, 11], [-9, -9, 0, 6, 0, 0], [0, 1, -8, -8, -19, -1], [-34, -4, 0, 3, 0, 0], [14, 9, 0, -3, 0, 0], [0, -2, -13, -13, -5, 2], [-11, 0, 9, 0, 11, 0], [-1, 0, 0, 0, 1, -1], [17, 0, 8, 0, -17, 4], [-21, -3, 0, -9, 0, 0], [-4, -6, 0, -12, 0, 0], [-1, 0, 11, 0, 1, -13], [0, -1, -16, -16, -3, 1], [-29, -2, 0, 2, 0, 0], [-13, 0, -13, 0, 13, -11], [42, 0, 3, 0, -42, 3], [-7, 0, 0, -3, 0, 0], [0, -2, -2, -2, -8, 2], [1, 0, -4, 0, -1, -8], [4, 16, 0, -2, 0, 0], [-37, 6, 0, -3, 0, 0], [-5, 0, 4, 0, 5, 10], [0, -6, -9, -9, 18, 6], [0, -11, 17, 17, 22, 11], [0, 3, -3, -3, 22, -3], [0, 0, 3, 3, 4, 0], [-7, 9, 0, 3, 0, 0], [17, 0, 2, 0, -17, 14], [0, -1, -12, -12, 2, 1], [16, -7, 0, 2, 0, 0], [29, 0, 8, 0, -29, 17], [0, -1, 5, 5, -3, 1], [0, -7, -13, -13, -18, 7], [22, 10, 0, -6, 0, 0], [39, 10, 0, -2, 0, 0], [0, -5, 0, 0, -41, 5], [-28, -3, 0, -3, 0, 0], [13, 0, 6, 0, -13, 21], [-15, 0, 13, 0, 15, 8], [-1, 11, 0, -3, 0, 0], [-8, 14, 0, 11, 0, 0], [20, 0, -18, 0, -20, -7], [0, 10, 10, 10, -14, -10], [0, 13, 11, 11, -26, -13], [0, 26, 8, 8, 3, -26], [0, -9, -6, -6, -32, 9], [-39, 0, 10, 0, 39, 5], [0, -4, -8, -8, 29, 4], [-16, 0, 5, 0, 16, -14], [-2, -2, 0, -10, 0, 0], [17, 0, -13, 0, -17, 1], [-25, 0, -2, 0, 25, -4], [0, -10, 4, 4, -13, 10], [0, 17, -22, -22, -10, -17], [0, 15, -15, -15, 1, -15], [2, 0, 17, 0, -2, 22], [1, 10, 0, 11, 0, 0], [10, 0, -2, 0, -10, 7], [0, 9, 15, 15, 6, -9], [15, 4, 0, -11, 0, 0], [34, 0, 0, 0, -34, -9], [0, 20, -4, -4, -9, -20], [-12, 15, 0, 0, 0, 0], [0, -16, 4, 4, -19, 16], [4, 0, 0, 0, -4, -6], [-13, 0, -7, 0, 13, -16], [-26, -17, 0, 5, 0, 0], [0, -12, 6, 6, -26, 12], [-2, 0, -6, 0, 2, 24], [-20, 13, 0, -3, 0, 0], [-3, -14, 0, 16, 0, 0], [4, 0, 8, 0, -4, 13], [0, -9, -6, -6, -3, 9], [-2, 0, -12, 0, 2, 15], [0, 4, 1, 1, 32, -4], [0, 18, -14, -14, -9, -18], [-9, 0, 11, 0, 9, -15], [-27, -15, 0, -7, 0, 0], [-10, 0, 8, 0, 10, -14], [37, 14, 0, 2, 0, 0], [-7, 0, 20, 0, 7, 8], [-31, 0, -1, 0, 31, 7], [23, -13, 0, 7, 0, 0], [-19, -24, 0, -9, 0, 0], [14, 0, -20, 0, -14, 8], [0, 21, -10, -10, -3, -21], [-3, 0, -8, 0, 3, -16], [33, 0, 9, 0, -33, 12], [0, -7, 8, 8, 18, 7], [0, 13, 7, 7, -34, -13], [31, 17, 0, -1, 0, 0], [0, 0, -21, -21, -30, 0], [8, -6, 0, 6, 0, 0], [0, -2, 20, 20, 31, 2], [5, 0, -4, 0, -5, 1], [-38, 0, 19, 0, 38, 4], [35, -10, 0, 13, 0, 0], [5, -25, 0, -13, 0, 0], [0, -18, 10, 10, 0, 18], [-29, 0, 6, 0, 29, 6], [0, -6, 3, 3, 25, 6], [-15, 0, 1, 0, 15, -1], [12, -3, 0, -4, 0, 0], [-21, 0, 12, 0, 21, -15], [21, -14, 0, 19, 0, 0], [0, -4, 14, 14, -22, 4], [0, 6, -3, -3, 10, -6], [24, -3, 0, 21, 0, 0], [-11, 20, 0, -10, 0, 0], [-14, 0, -11, 0, 14, 1], [0, -24, 3, 3, -18, 24], [28, -7, 0, 14, 0, 0], [0, 12, -8, -8, -24, -12], [3, 0, 5, 0, -3, -21], [0, -3, 9, 9, 16, 3], [0, 14, 14, 14, -6, -14], [-31, -16, 0, 11, 0, 0], [13, 0, 9, 0, -13, 3], [0, -8, 13, 13, 11, 8], [24, 3, 0, 13, 0, 0], [0, 10, 4, 4, -25, -10], [28, -4, 0, 14, 0, 0], [0, -12, 15, 15, -24, 12], [-19, 0, -13, 0, 19, -17], [-12, -3, 0, 15, 0, 0], [66, 9, 0, -2, 0, 0], [-32, -10, 0, -10, 0, 0], [0, -9, 27, 27, 33, 9], [-39, 0, -11, 0, 39, -22], [0, -18, 15, 15, -26, 18], [0, -4, 5, 5, 12, 4], [-2, 0, 3, 0, 2, 9], [22, 8, 0, 8, 0, 0], [15, 0, 7, 0, -15, 2], [-38, 0, 6, 0, 38, -17], [0, 8, 17, 17, -3, -8], [-22, 0, -10, 0, 22, -38], [-24, 0, -3, 0, 24, -18], [0, 9, 12, 12, 24, -9], [0, -7, 2, 2, -27, 7], [-53, 10, 0, 2, 0, 0], [-31, 9, 0, -9, 0, 0], [9, 0, -29, 0, -9, -1], [27, 0, -13, 0, -27, -15], [0, -19, -1, -1, -6, 19], [24, 18, 0, 5, 0, 0], [0, 0, -3, -3, -23, 0], [36, 0, 0, -12, 0, 0], [0, -14, -16, -16, 19, 14], [0, 38, -8, -8, 26, -38], [41, 0, -7, 0, -41, 11], [-2, -5, 0, 18, 0, 0], [23, 9, 0, -15, 0, 0], [0, -24, -21, -21, 9, 24], [10, 0, -9, 0, -10, -9], [-2, -2, 0, -16, 0, 0], [-44, 0, -8, 0, 44, 7], [51, 4, 0, -2, 0, 0], [-7, 0, -4, 0, 7, 25], [1, 0, 13, 0, -1, 7], [0, 1, -23, -23, -10, -1]];
    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_m();".
// 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_m(: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_m();".
// 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_m( : 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, 18, 33, 12, 7, -1, 1]>,<11,R![9, -18, 33, -12, 7, 1, 1]>,<17,R![576, 288, 240, 0, 28, 4, 1]>,<23,R![9, 36, 123, 78, 37, 7, 1]>],Snew);
    return Vf;
end function;