
// Make newform 966.2.q.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_966_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_966_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_966_2_q_a();".
// 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_966_2_q_a();".
// 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 := [1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_966_q();"
function MakeCharacter_966_q()
    N := 966;
    order := 11;
    char_gens := [323, 829, 925];
    v := [11, 11, 4];
    // 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_966_q_Hecke();"
function MakeCharacter_966_q_Hecke(Kf)
    N := 966;
    order := 11;
    char_gens := [323, 829, 925];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 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, 1, 0, 0, 0, 0, 0], [-1, 1, -1, 1, -1, 1, -1, 1, -1, 1], [-1, 0, 1, 0, -1, 0, 1, -1, 1, -1], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [-1, 0, 0, 1, 0, 0, -1, 3, -1, 0], [1, -1, 0, 1, 1, 1, -3, 1, 1, 1], [3, -3, 1, -4, 1, -3, 3, 0, 0, 0], [-2, 1, -3, 3, -1, 2, 0, 0, 1, 0], [-2, 0, -2, 2, -2, 3, 0, 4, -4, 2], [1, 0, -2, 6, -2, 0, 1, 0, 0, 0], [1, 0, 1, 0, 4, -3, 0, 0, 3, -4], [-2, 0, -1, 5, -5, 1, 0, 2, 0, 4], [0, -2, -8, -2, 0, 0, 1, 0, 0, -1], [1, -1, -5, 4, 0, 0, -4, 5, 1, -1], [-1, 0, -3, -2, -2, 0, 0, 2, 2, 3], [0, 1, -2, 4, -1, -1, -1, 4, -2, 1], [1, -1, 0, -6, 2, -5, 0, -5, 2, -6], [-1, 2, -1, 0, -5, -1, 0, 0, 1, 5], [0, 3, 4, -1, 0, -1, 4, 3, 0, 0], [4, -1, 7, -6, 2, -6, 7, -1, 4, 0], [-3, 5, 3, -3, -5, 3, 0, -1, 0, -1], [0, 0, 0, -2, 0, 3, -1, 3, 0, -2], [5, -7, 3, -6, 6, -3, 7, -5, 0, -6], [-1, -5, 1, -3, 7, -7, 3, -1, 5, 1], [5, -7, 9, -7, 5, 0, 10, -3, 3, -10], [0, 2, -6, 6, -6, 6, -2, 0, 0, 8], [4, -1, 1, -4, 0, -1, 3, 2, 3, -1], [1, 4, 1, 0, 1, 4, -3, 3, -4, -1], [5, 2, 5, -5, 5, 2, 5, 0, 2, -2], [-1, -6, 5, -2, -4, -2, 5, -6, -1, 0], [-2, -1, 1, 2, 0, -3, -2, 4, -2, -3], [0, 2, -2, 7, -2, 13, -2, 7, -2, 2], [5, 0, -7, -1, -6, -2, 2, 6, 1, 7], [-6, 0, -6, 10, -1, 3, -3, 1, -10, 6], [2, 5, -5, -2, 0, 2, -12, 5, -12, 2], [-2, 2, 0, 7, 1, 8, -8, 8, 1, 7], [5, 0, -4, 4, 0, -5, 0, 5, 0, 5], [2, 6, -1, 5, 3, 5, -1, 6, 2, 0], [-7, 3, -4, 6, -4, 3, -7, 0, -4, 4], [4, 4, -4, -4, 0, -7, 0, 9, 0, -7], [-9, 4, -1, 4, -9, 0, -1, -2, 2, 1], [0, 7, 5, 1, 2, -2, -1, -5, -7, 0], [0, -3, -3, 11, 0, 12, 0, 11, -3, -3], [-6, 4, -4, 1, -1, 4, -4, 6, 0, 9], [11, -11, 0, -6, 13, -8, 11, -8, 13, -6], [10, -6, 10, 0, 4, 2, 10, -10, -2, -4], [2, -6, 2, -2, 6, -2, 0, -8, -4, -8], [0, -1, 2, -7, 2, 5, 2, -7, 2, -1], [1, 3, -2, 10, 3, -3, -10, 2, -3, -1], [-12, 0, -5, 7, -3, 6, -6, 3, -7, 5], [-8, -3, 1, 4, 3, -3, -4, -1, 3, 8], [-1, -9, 6, 2, -2, -6, 9, 1, 0, 3], [3, 6, -6, -3, 0, 5, -11, 0, -11, 5], [-2, -2, -5, 8, 1, 8, -5, -2, -2, 0], [-10, 8, -4, 4, -8, 10, 0, 13, -11, 13], [-17, 17, 0, 6, -6, 10, -11, 10, -6, 6], [0, -20, 13, -18, 14, -18, 14, -18, 13, -20], [-3, 1, 2, 1, -3, 0, 3, 12, -12, -3], [14, 0, 6, -9, 16, -5, 5, -16, 9, -6], [3, 9, 4, 9, 3, 0, -2, -12, 12, 2], [0, -9, 2, -4, 15, -11, 15, -4, 2, -9], [9, -13, 6, -11, 6, -13, 9, 0, 8, -8], [-5, 11, -5, 0, -10, 0, 6, -6, 0, 10], [9, -12, 12, -9, 0, -7, 4, -7, 4, -7], [-6, 7, -8, -1, 1, 8, -7, 6, 0, 3], [-1, -5, 6, -5, -1, 0, -10, 0, 0, 10], [-5, 9, -5, -1, 1, 5, -9, 5, 0, 11], [-4, 2, -4, 0, -10, 2, -11, 11, -2, 10], [-7, -2, 2, 3, 2, 3, 2, -2, -7, 0], [3, -15, 6, -6, 15, -3, 0, -7, 11, -7], [-3, 6, -3, 0, 5, 8, 0, 0, -8, -5], [8, -16, 16, -12, 12, -16, 16, -8, 0, -5], [11, 0, -6, -7, -7, 6, -6, 7, 7, 6], [9, -4, 3, -4, 9, 0, 6, -7, 7, -6], [8, -13, 13, -8, 0, -15, 2, 2, 2, -15], [8, -8, 8, 0, 8, -9, 4, -4, 9, -8], [16, -13, 2, -13, 7, -13, 2, -13, 16, 0], [11, -14, 0, 2, 0, -14, 11, 0, 4, -4], [4, -4, 0, 0, 0, -9, 1, -9, 0, 0], [-4, -1, -8, -1, -4, 0, -7, -11, 11, 7], [0, 10, -6, 10, 0, 0, -4, -4, 4, 4], [0, 2, -7, 13, 1, 12, 1, 13, -7, 2], [4, -5, 18, -10, 18, -5, 4, 0, 17, -17], [-5, 4, 5, -5, -4, 5, 0, -14, 3, -14], [-4, -1, -1, -1, -26, -1, -1, -1, -4, 0], [-16, 18, -23, 11, -23, 18, -16, 0, -13, 13], [-4, 12, -12, 4, 0, -8, -7, 6, -7, -8], [-5, 5, -8, -11, 8, -8, 11, 8, -5, 5], [-6, 0, 0, -14, 5, -6, 6, -5, 14, 0], [3, 2, 5, -11, 6, -6, 11, -5, -2, -3], [0, 16, -13, 23, -5, 16, -5, 23, -13, 16], [7, -17, 4, -4, 17, -7, 0, -9, 11, -9], [10, 0, 0, 0, 0, -10, 0, 15, 1, 15], [7, 0, 7, 0, -4, 12, 3, -3, -12, 4], [0, -18, 10, -9, 9, -19, 9, -9, 10, -18], [-11, 2, -3, -7, -6, 6, 7, 3, -2, 11], [18, -16, 21, -1, 10, -10, 1, -21, 16, -18], [4, -7, 4, 0, -9, 3, -4, 4, -3, 9], [10, -11, 6, -4, 6, -11, 10, 0, -8, 8], [22, -28, 28, -22, 0, -23, 14, -7, 14, -23], [9, -14, -7, -14, 9, 0, 9, -21, 21, -9], [-13, 5, -10, 5, -13, 0, -5, 15, -15, 5], [-5, -1, 1, 5, 0, 13, -16, 7, -16, 13], [3, 10, -1, -7, -1, 10, 3, 0, -13, 13], [-12, 17, 2, -2, -17, 12, 0, 24, -6, 24], [-3, 18, -6, 22, -9, 22, -6, 18, -3, 0], [12, 9, -9, -12, 0, -1, 2, 2, 2, -1], [5, -9, -3, -9, 5, 0, 8, -16, 16, -8], [-22, 0, 3, 4, -4, 15, -15, 4, -4, -3], [-28, 10, -3, -2, -8, 8, 2, 3, -10, 28], [-8, -1, -15, 0, 0, 15, 1, 8, 0, 14], [-2, 30, -2, 0, -1, 1, -7, 7, -1, 1], [1, -2, 5, -5, 2, -1, 0, -9, 4, -9], [-7, -3, 8, -8, 0, -8, 8, -3, -7, 0], [10, -10, 0, 7, 1, -13, -7, -13, 1, 7], [3, -5, 14, -7, 17, -17, 7, -14, 5, -3], [-10, 0, -18, 4, -7, 23, -23, 7, -4, 18], [-2, 6, -13, 31, -4, 4, -31, 13, -6, 2], [-20, 0, 0, -4, 4, 0, 0, 20, 0, -8], [-1, 16, -1, 0, -2, -15, 1, -1, 15, 2], [21, -11, 21, -18, 21, -11, 21, 0, 7, -7], [-14, 2, 1, -2, 1, 2, -14, 0, -15, 15], [-11, 11, 0, 22, -1, 8, -2, 8, -1, 22], [0, 3, 6, 10, 4, -13, 4, 10, 6, 3], [-15, 0, -3, 7, -3, 15, -15, 3, -7, 3], [1, 2, -2, -1, 0, 7, 7, -5, 7, 7], [8, -2, -9, 9, 2, -8, 0, 8, 2, 8], [6, 11, -2, 12, -2, 11, 6, 0, 4, -4], [-9, 7, -23, -9, 9, 23, -7, 9, 0, 6], [18, -20, 22, -21, 32, -32, 21, -22, 20, -18], [-19, 19, -6, 13, -18, 18, -13, 6, -19, 19], [0, 15, -5, 6, -2, 3, -2, 6, -5, 15], [-2, -4, -2, 0, -24, -12, 3, -3, 12, 24], [8, -14, 14, -10, -4, -10, 14, -14, 8, 0], [8, 7, -9, 4, -7, 4, -9, 7, 8, 0], [-20, 20, 0, 13, 0, 3, -11, 3, 0, 13], [6, -1, 21, -4, 4, -21, 1, -6, 0, 12], [-3, 2, 21, 2, -3, 0, 9, 7, -7, -9], [-2, 8, -2, 0, 2, 5, 12, -12, -5, -2], [-2, -3, 0, 0, 3, 2, 0, -13, -11, -13], [19, -20, 11, -15, 11, -20, 19, 0, 14, -14], [0, -9, 1, -4, 10, 12, 10, -4, 1, -9], [10, -12, 13, -12, 10, 0, 2, -17, 17, -2], [31, 0, -4, -2, 12, -9, 9, -12, 2, 4], [12, 0, 22, -22, 21, -25, 25, -21, 22, -22], [14, -8, 8, -14, 0, -9, 16, -33, 16, -9], [-2, 5, -11, -2, 17, -2, -11, 5, -2, 0], [-2, -10, 2, 8, 2, -10, -2, 0, 5, -5], [-14, 1, -14, 0, -6, 22, -17, 17, -22, 6], [5, -11, 11, -5, 0, -2, -7, -25, -7, -2], [-6, 26, -5, 26, -21, 21, -26, 5, -26, 6], [0, -1, 4, -5, 6, 37, 6, -5, 4, -1], [-25, 23, -6, 12, -12, 6, -23, 25, 0, 12], [-6, 6, 0, 6, -4, 15, 5, 15, -4, 6], [-6, 6, 0, 19, -8, 29, -17, 29, -8, 19], [-3, 3, 24, -11, 11, -24, -3, 3, 0, -14], [1, 0, 20, -9, -7, -6, 6, 7, 9, -20], [-5, -7, -23, -1, 1, 23, 7, 5, 0, 6], [7, 15, 9, -21, 9, 15, 7, 0, 1, -1], [8, -1, -10, 18, 8, 18, -10, -1, 8, 0], [-5, 11, 9, -9, -11, 5, 0, -6, 15, -6], [24, -24, 0, 6, 3, 9, 10, 9, 3, 6], [-3, 0, 0, 22, 1, 3, -3, -1, -22, 0], [7, 7, 5, 7, 7, 0, -1, -14, 14, 1], [17, -15, 15, -17, 0, -25, 27, -32, 27, -25], [7, -9, 14, -21, 14, -9, 7, 0, 15, -15], [-1, -7, -29, 13, -8, 13, -29, -7, -1, 0], [16, 2, 16, 0, -1, -8, 21, -21, 8, 1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_966_q_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_966_2_q_a();".
// 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_966_2_q_a(:prec:=10)
    chi := MakeCharacter_966_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(997) - 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_966_2_q_a();".
// 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_966_2_q_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_966_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, 15, 115, 526, 1950, 2003, 1137, 412, 97, 14, 1]>],Snew);
    return Vf;
end function;