
// Make newform 9576.2.a.bk in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_9576_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_9576_2_a_bk();".
// 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_9576_2_a_bk();".
// 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, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 2]];
    Rf_basisdens := [1, 1];
    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_9576_a();"
function MakeCharacter_9576_a()
    N := 9576;
    order := 1;
    char_gens := [7183, 4789, 5321, 4105, 1009];
    v := [1, 1, 1, 1, 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;

function MakeCharacter_9576_a_Hecke(Kf)
    return MakeCharacter_9576_a();
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], [-1, 0], [0, 0], [-2, 0], [0, -1], [-1, 0], [-4, 0], [4, 1], [4, -2], [-2, 0], [-6, 0], [0, -2], [-6, 1], [0, -1], [-4, 0], [6, 0], [8, -2], [-10, 1], [6, 2], [0, 0], [2, -1], [2, -4], [10, 0], [0, -3], [8, 0], [2, -3], [-6, 2], [-8, 1], [-12, 2], [-2, -3], [-2, 0], [0, 2], [-2, -2], [4, 2], [-2, -4], [-8, 2], [-4, -2], [-2, 0], [-10, -1], [-2, 0], [4, -4], [-14, 0], [-6, 4], [-12, -2], [12, -4], [-4, 6], [0, -2], [2, 6], [2, 2], [0, -2], [-14, 4], [-2, -3], [-2, 6], [16, 2], [10, 2], [-16, 4], [-10, -4], [0, 5], [-16, -2], [22, 0], [8, 2], [-14, 1], [10, 0], [-4, 1], [8, -2], [14, -2], [-12, 6], [-2, -4], [16, -1], [-12, 0], [-4, -2], [6, 4], [-8, 6], [-12, 2], [14, 2], [14, 4], [4, 3], [10, 0], [-6, -1], [-22, -2], [-34, 1], [2, 0], [-20, -6], [-4, -6], [4, -1], [10, 4], [16, 1], [8, 8], [-34, 1], [-26, -1], [0, 8], [-12, -2], [-12, 0], [-6, 9], [-22, -6], [6, -6], [16, -2], [-18, 0], [12, -8], [-2, -6], [8, 2], [-24, 1], [-12, 8], [34, 0], [-2, -3], [-12, 1], [26, 3], [10, -8], [8, 8], [14, 0], [30, 0], [-32, 2], [20, 6], [12, -9], [24, 6], [18, -7], [18, -8], [6, 11], [-34, 0], [-14, 0], [-22, -6], [-26, 3], [-4, -4], [-6, -4], [-2, 0], [-2, 7], [-8, -4], [2, -6], [-32, 6], [-18, -3], [12, 2], [-2, 12], [20, 9], [10, -12], [-2, -4], [16, -6], [14, 8], [-2, -4], [-4, 0], [38, 2], [4, -2], [2, 5], [-34, -4], [24, 4], [6, 12], [18, -4], [4, 8], [10, -5], [38, 0], [-24, -5], [-20, -4], [-12, 10], [-20, 0], [-54, -1], [-24, -8], [-24, -1], [26, -8], [-30, 6], [-16, -4], [-12, 3], [-4, 2], [-20, -4], [12, -9], [40, 0], [-16, 12], [14, 0], [6, 2], [24, 9], [-30, 5], [-26, -8], [40, 2], [-30, -8], [-16, 4], [36, 5], [16, -6], [-14, 4], [4, 10], [22, 0], [-8, -8], [-20, 0], [-6, 6], [10, 12], [-12, -8], [0, -11], [2, 6], [-24, -2], [-2, 2], [-10, -13], [10, -4], [8, -12], [32, -2], [18, -6], [0, 4], [-26, 10], [18, -12], [6, -4], [-40, 7], [-10, -13], [56, -1], [16, 4], [-34, 0], [42, 4], [30, 9], [8, -3], [20, 2], [-22, -3], [18, -2], [40, -2], [-14, 4], [-16, -7], [-28, 2], [16, -10], [-8, -4], [50, 0], [-4, 2], [-26, -2], [-6, 7], [-18, -2], [2, 18], [16, -4], [-24, 9], [0, -16], [12, -8], [-42, -8], [-2, 10], [-22, -13], [20, 10], [20, 14], [-10, -4], [0, 6], [8, 0], [24, 5], [20, 0], [14, 7], [62, 2], [2, -16], [-12, 14], [-44, 6], [30, 11], [20, 4], [-68, 2], [-34, 12], [-58, 2], [6, 11], [44, 10], [16, 14], [-44, 8], [-28, 0], [14, 0], [-4, -11], [8, -14], [18, -4], [-20, 7], [16, 4], [-10, 16], [8, -2], [-6, -6], [-6, 16], [8, 12], [18, 13], [-14, -10], [-42, -8], [-28, 1], [16, 2], [-24, -1], [-12, 5], [-20, -4], [14, 2], [22, 4], [28, -12], [30, 6], [-8, 12], [10, 0], [20, -14], [36, 2], [-50, 4], [50, -4], [-2, -7], [-28, 2], [28, -6], [-6, 17], [14, 12], [20, 4], [-24, 6], [-14, -16], [-12, 9], [-52, 6], [78, -2], [6, -2], [-6, 11], [-24, -7], [-28, -4], [-2, -12], [36, -5], [28, -10], [16, 1], [8, 14], [-32, 14], [-14, 12], [12, -11], [-8, -8], [2, 17], [-24, 14], [-10, -6], [-14, -7], [62, 0], [8, -6], [46, 0], [-6, -11], [-18, 10], [68, -5], [0, 14], [4, 20], [-14, 0], [6, -15], [30, 5], [-26, 6], [-42, 0], [44, 8], [-22, 12], [66, 2], [12, 2], [-6, -14], [-30, -12], [52, -4], [-4, 12], [38, 5], [-14, -4], [-2, 0], [4, -15], [-4, -6], [26, -1], [4, 16], [54, -7], [-26, -8], [42, -4], [-18, -2], [48, -4], [22, 12], [-2, 0], [-18, 8], [32, 0], [-36, -11], [78, -1], [10, -14], [52, 12], [-32, 0], [-16, 13], [-20, 16], [-34, 18], [58, 8], [24, 0], [6, -4], [-18, -10], [-22, -19], [-6, 7], [-6, -2], [-66, 5], [-58, 8], [52, 1], [38, -11], [34, -11], [-32, 14], [2, 12], [20, -9], [-44, 6], [-62, -4], [24, -20], [-24, 14], [-8, 6], [24, 3], [8, -4], [-42, -4], [36, 4], [-84, -4], [26, 20], [-12, 9], [42, 12], [2, 10], [-2, -20], [12, -18], [30, -8], [64, -6], [-36, -6], [-36, 2], [18, -18], [24, 14], [-20, -14], [-74, -6], [-18, 8], [0, 0], [-20, 8], [-18, 5], [62, -6], [-40, 8], [58, -12], [-80, -2], [-58, 6], [-42, -16], [6, 20], [64, 0], [30, -6], [-46, 10], [-36, 18], [22, -12], [-28, 7], [36, 0], [18, -13], [34, -8], [-42, -2], [-6, 13], [-16, -6], [-22, -16], [60, 3], [30, -11], [-60, 10], [-48, -9], [0, -16], [12, -11], [-18, -4], [24, -18], [52, 8], [18, -16], [54, -4], [-90, -1], [2, -6], [-8, -26], [10, 21], [10, 24], [0, 4], [-8, -10], [-24, -4], [-10, -24], [-72, -1], [-22, 12], [-58, -4], [-32, 6], [-24, 0], [54, 9], [-50, 4], [-26, 20], [-42, -11], [-18, 10], [36, -1], [16, -6], [24, 4], [-22, 0], [-18, -8], [52, 12], [24, 4], [-68, 2], [28, 17], [18, -8], [-4, -19], [34, -18], [12, -12], [-38, 10], [-4, 15], [76, 0], [-28, 22], [42, 17], [14, 0], [-28, -20], [50, -8], [-16, -16], [2, -13], [22, 0], [48, 6], [-24, -12], [-70, -3], [-28, 14], [-78, -4], [-58, 11], [14, 20], [42, -12], [28, 2], [-10, 19], [-20, 5], [-46, 16], [50, 8], [18, -16], [-58, -8], [-8, 4], [-12, 10], [-18, 20], [-8, 30], [-12, 16], [-4, -14], [38, -12], [-24, -20], [-46, -12], [34, -2], [24, -8], [30, -16], [4, -32], [6, -22], [-20, -18], [-12, 12], [22, 2], [-48, -12], [-40, 21], [-40, 0], [22, -8], [36, -15], [-34, -15], [52, -2], [50, 2], [-4, 20], [90, 7], [6, 11], [-46, 4], [-28, 5], [-28, 8], [-62, -4], [26, 6], [-74, 8], [30, -15], [44, -10], [-2, -24], [8, 14], [28, 19], [60, 12], [-22, 24], [-34, 5], [10, -20], [12, -1], [-6, 17], [-14, 18], [0, -16], [-36, -13]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9576_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9576_2_a_bk();".
// 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_9576_2_a_bk(:prec:=2)
    chi := MakeCharacter_9576_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(3833) - 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_9576_2_a_bk();".
// 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_9576_2_a_bk( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9576_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-12, 0, 1]>,<11,R![0, 1]>,<13,R![2, 1]>,<17,R![-12, 0, 1]>],Snew);
    return Vf;
end function;