
// Make newform 1064.2.q.k in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1064_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1064_q_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_1064_2_q_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_1064_2_q_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 := [4, 0, 2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 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_1064_q();"
function MakeCharacter_1064_q()
    N := 1064;
    order := 3;
    char_gens := [799, 533, 913, 1009];
    v := [3, 3, 2, 3];
    // 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_1064_q_Hecke();"
function MakeCharacter_1064_q_Hecke(Kf)
    N := 1064;
    order := 3;
    char_gens := [799, 533, 913, 1009];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [-1, 0, -1, 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 := 2;
    raw_aps := [[0, 0, 0, 0], [0, -1, 2, -1], [1, 0, 1, 0], [1, -1, 1, -2], [0, 3, 1, 3], [-2, 0, 0, -1], [0, 2, -4, 2], [1, 0, 1, 0], [-3, 1, -3, 0], [-8, 0, 0, -1], [0, 1, 6, 1], [8, 1, 8, 0], [0, 0, 0, -4], [1, 0, 0, -3], [-3, -3, -3, 0], [0, 1, -4, 1], [0, 2, 2, 2], [5, 0, 5, 0], [0, 2, 6, 2], [-12, 0, 0, 3], [0, 0, -7, 0], [6, 7, 6, 0], [5, 0, 0, 7], [2, 11, 2, 0], [0, 0, 0, -4], [0, -6, -7, -6], [-8, -4, -8, 0], [-10, 5, -10, 0], [0, 8, -8, 8], [4, 0, 0, 5], [-4, 0, 0, -11], [4, 4, 4, 0], [0, -4, -5, -4], [-3, 0, 0, -11], [1, -4, 1, 0], [0, -4, 0, -4], [0, -8, -13, -8], [-7, 5, -7, 0], [20, 0, 0, -1], [4, -4, 4, 0], [0, -6, -2, -6], [-2, 0, 0, -9], [-5, 3, -5, 0], [0, -3, -2, -3], [-9, 0, 0, 6], [0, -5, -5, -5], [-2, 0, 0, 10], [-12, 0, 0, -3], [0, -1, 0, -1], [-16, 0, -16, 0], [12, 4, 12, 0], [-10, 0, 0, 4], [0, -9, -4, -9], [-1, 0, 0, -15], [-14, 0, -14, 0], [0, -2, -24, -2], [0, 0, -16, 0], [-15, -9, -15, 0], [0, 12, -11, 12], [-8, 0, 0, -7], [0, -11, -1, -11], [4, 0, 0, 4], [-4, 0, 0, 16], [0, 0, 8, 0], [13, 10, 13, 0], [6, -3, 6, 0], [10, -6, 10, 0], [-2, 0, 0, 5], [0, -1, 11, -1], [0, 0, 0, 4], [0, 0, 24, 0], [-7, 21, -7, 0], [0, -16, -6, -16], [12, 4, 12, 0], [14, 0, 0, -1], [8, 3, 8, 0], [0, 8, -4, 8], [10, 0, 10, 0], [-14, -16, -14, 0], [0, 8, -18, 8], [-1, 0, 0, -21], [-26, 0, 0, 1], [0, 7, -4, 7], [12, 0, 0, 0], [-18, 6, -18, 0], [28, 8, 28, 0], [8, 0, 0, 16], [-3, 2, -3, 0], [-9, 0, 0, 14], [-15, 0, 0, -9], [-5, -9, -5, 0], [0, -5, -17, -5], [0, -10, -26, -10], [11, 0, 0, -21], [5, 3, 5, 0], [9, 0, 0, 9], [2, 0, 2, 0], [0, 7, -12, 7], [4, 8, 4, 0], [15, -10, 15, 0], [-6, 0, 0, -11], [0, -6, 27, -6], [0, -19, 16, -19], [-6, -8, -6, 0], [0, 21, -13, 21], [0, 2, 31, 2], [-34, 0, 0, 0], [7, 8, 7, 0], [0, -15, 0, -15], [16, 0, 0, -23], [-6, 7, -6, 0], [0, 6, -24, 6], [-1, 0, 0, 20], [0, 7, 19, 7], [1, 0, 0, -13], [0, 8, 22, 8], [36, 0, 0, -6], [0, 5, 35, 5], [16, -10, 16, 0], [6, 0, 0, 13], [0, -19, -18, -19], [-18, 0, 0, -16], [0, 8, 0, 0], [0, 10, -10, 10], [-6, -16, -6, 0], [-1, 0, 0, -14], [-13, 28, -13, 0], [8, 8, 8, 0], [-1, 0, 0, 3], [40, -8, 40, 0], [0, -16, 22, -16], [34, 0, 0, -9], [-14, -6, -14, 0], [-43, 0, 0, 8], [-1, -10, -1, 0], [-29, 0, 0, 18], [0, -5, -8, -5], [0, 3, 16, 3], [4, 0, 0, -35], [0, 12, -11, 12], [-10, 0, 0, 30], [-25, -12, -25, 0], [0, -9, -3, -9], [-12, 0, 0, 8], [0, -8, 28, -8], [22, 0, 0, 6], [-27, 0, 0, -18], [0, 12, 24, 12], [-11, -27, -11, 0], [-2, -19, -2, 0], [-10, 19, -10, 0], [-4, 0, 0, 8], [-28, 0, 0, 2], [0, -20, 0, 0], [0, 11, -34, 11], [-10, 0, 0, -11], [-7, -1, -7, 0], [13, -16, 13, 0], [39, 0, 0, -6], [0, -1, 26, -1], [26, 0, 26, 0], [-4, 0, 0, -39], [0, 0, 0, 12], [-22, 19, -22, 0], [0, -8, 6, -8], [0, 24, -16, 24], [0, 18, -18, 18], [0, -28, -20, -28], [6, 0, 0, -36], [-3, -38, -3, 0], [0, -11, -20, -11], [-12, 0, 0, 17], [-12, 22, -12, 0], [0, 28, -3, 28], [0, 9, 22, 9], [21, 0, 0, 2], [45, 0, 0, -3], [0, -12, 9, -12], [-20, 0, 0, -19], [-7, 16, -7, 0], [19, 7, 19, 0], [-6, 0, 0, 30], [-8, 0, 0, 13], [8, -3, 8, 0], [0, 6, 36, 6], [0, -32, 10, -32], [0, 0, 10, 0], [0, 6, 18, 6], [8, 23, 8, 0], [0, 15, -13, 15], [38, 4, 38, 0], [4, 0, 0, 8], [0, -8, 0, 0], [8, -33, 8, 0], [0, 24, -12, 24], [0, -25, 8, -25], [0, 8, 9, 8], [-11, -24, -11, 0], [-23, 0, 0, -8], [39, -9, 39, 0], [0, -8, 54, -8], [-40, 0, 0, -12], [-28, -4, -28, 0], [0, 28, -18, 28], [20, 0, 0, -24], [0, -6, -41, -6], [3, 21, 3, 0], [-32, 5, -32, 0], [20, 0, 0, -30], [0, -17, 40, -17], [19, -26, 19, 0], [8, 0, 0, 30], [15, 0, 0, 15], [-14, -14, -14, 0], [0, -3, -28, -3], [20, -3, 20, 0], [0, -1, -9, -1], [0, 17, 16, 17], [-44, 4, -44, 0], [25, 0, 0, -12], [-16, -4, -16, 0], [-24, 0, 0, 16], [-24, -23, -24, 0], [-28, -14, -28, 0], [24, 0, 0, 1], [11, 0, 0, 12], [-6, 27, -6, 0], [0, 42, 2, 42], [-20, -9, -20, 0], [-23, 11, -23, 0], [0, 16, 9, 16], [0, 33, -24, 33], [12, 0, 0, 43], [0, 35, -12, 35], [-24, 0, 0, -22], [0, 6, 4, 6], [-33, 6, -33, 0], [4, 16, 4, 0], [-17, 0, 0, -11], [42, 0, 0, 5], [0, -7, -32, -7], [1, -35, 1, 0], [0, 7, -3, 7], [24, 3, 24, 0], [-6, 0, 0, 32], [-29, 15, -29, 0], [32, 0, 0, -8], [0, 1, 0, 1], [0, 21, 28, 21], [23, 0, 0, -19], [-65, 0, 0, 0], [-4, 30, -4, 0], [0, 16, 0, 16], [-6, 0, 0, -9], [0, -10, -3, -10], [11, 33, 11, 0], [0, -14, 17, -14], [0, -26, -26, -26], [-6, 0, 0, 25], [0, 48, 0, 0], [0, 19, 2, 19], [0, 0, 0, -29], [0, -12, 64, -12], [4, 0, 0, 39], [0, -2, -9, -2], [-16, 12, -16, 0], [-10, 0, 0, -16], [23, 0, 0, 8], [-6, 0, 0, 6], [0, -12, -39, -12], [20, 12, 20, 0], [0, -20, -36, -20], [0, 26, 31, 26], [-17, 11, -17, 0], [32, 0, 0, -9], [16, 12, 16, 0], [-49, -5, -49, 0], [0, -28, -36, -28], [20, -4, 20, 0], [-9, -1, -9, 0], [0, 30, 18, 30], [0, -28, 40, -28], [21, 0, 0, 43], [-2, 0, 0, 20], [-52, -18, -52, 0], [25, -5, 25, 0], [0, -8, -8, -8], [-26, 0, 0, -12], [0, -3, 15, -3], [-24, 0, 0, 0], [0, 4, -1, 4], [0, 15, 11, 15], [-8, 20, -8, 0], [-56, 0, 0, -4], [-8, 0, 0, 24], [0, 28, 8, 28], [-32, -15, -32, 0], [-24, 0, 0, 12], [14, 6, 14, 0], [-55, 0, 0, 21], [69, -2, 69, 0], [2, 56, 2, 0], [0, 9, -11, 9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1064_q_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1064_2_q_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_1064_2_q_k(:prec:=4)
    chi := MakeCharacter_1064_q();
    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_1064_2_q_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_1064_2_q_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1064_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, 8, 14, 4, 1]>,<5,R![1, -1, 1]>,<11,R![289, -34, 21, 2, 1]>],Snew);
    return Vf;
end function;