
// Make newform 9555.2.a.bq in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_9555_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_9555_2_a_bq();".
// 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_9555_2_a_bq();".
// 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 := [2, -4, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [-3, 0, 1], [2, 2, -1]];
    Rf_basisdens := [1, 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_9555_a();"
function MakeCharacter_9555_a()
    N := 9555;
    order := 1;
    char_gens := [6371, 1912, 9166, 1471];
    v := [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_9555_a_Hecke(Kf)
    return MakeCharacter_9555_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, -1, 0], [1, 0, 0], [1, 0, 0], [0, 0, 0], [0, 0, -1], [-1, 0, 0], [0, -2, -1], [-2, -2, 0], [-2, 2, 1], [6, 0, 0], [-2, 2, 0], [4, 2, -1], [0, 2, 1], [0, -4, 0], [6, -2, 0], [4, 2, 1], [2, -2, -2], [-4, -2, -3], [2, 2, 2], [-4, 0, -1], [2, 4, 0], [2, -2, 1], [-2, 2, 2], [-4, -2, -1], [8, -2, -1], [-2, 4, 2], [-4, -4, -4], [-10, -2, -3], [6, 0, 2], [-2, 4, 0], [-8, 0, -2], [4, 0, 2], [2, 4, 4], [-2, -2, 1], [4, 6, -3], [-6, -2, 2], [2, 0, -4], [0, 4, -1], [10, 2, 2], [-2, 4, 0], [-8, -4, 2], [-12, -2, 1], [-4, 4, 4], [-4, 6, -1], [-2, -4, 2], [4, -4, 0], [4, 8, 0], [-10, 2, 2], [-6, -2, -4], [6, -4, -4], [4, -2, -1], [4, 0, -1], [6, 0, 0], [4, 8, 0], [10, 4, 0], [8, 0, 0], [-10, 4, 2], [-6, -2, 0], [6, 0, -4], [2, 4, 0], [0, 12, 0], [2, -4, 6], [8, -4, 1], [8, 8, -4], [14, 8, 0], [22, -4, -2], [-2, 6, 2], [-10, -4, -2], [-2, -2, 1], [10, 8, -2], [14, 4, 4], [-10, -2, 2], [8, 8, 0], [10, -4, -2], [10, 2, 8], [-6, 2, -4], [26, -4, -2], [-4, 6, 1], [2, -12, -8], [14, 0, 0], [4, 0, -6], [6, -8, -2], [2, 2, -2], [26, 4, 2], [6, -6, 1], [14, 6, 5], [0, -2, -1], [0, 2, -3], [0, -10, -3], [8, 0, 7], [-10, -2, -9], [4, 0, 3], [8, 0, -3], [-20, 8, 0], [2, -6, -4], [-16, 8, 4], [20, 2, 7], [6, -8, -4], [4, 0, 0], [-22, -4, -8], [12, -8, 0], [18, 8, 4], [-10, 6, -5], [14, 4, -6], [-22, 2, -3], [0, -2, 3], [-14, 6, 0], [18, -8, 2], [4, 12, -4], [-20, 2, 3], [-8, -8, 0], [-16, 6, -3], [-6, -4, 0], [-6, 2, 0], [2, -10, 2], [6, 12, 2], [12, 8, 1], [6, 2, 5], [-14, -4, 4], [8, 12, 0], [-22, 0, -2], [-6, -8, 4], [12, -2, 7], [-2, 10, 6], [-2, 6, -4], [-6, -12, 6], [-10, 8, 2], [24, 0, 4], [-32, 0, -2], [28, -2, 5], [-26, 6, 4], [22, -2, -4], [2, -10, -5], [18, 4, 2], [22, 4, -4], [26, 4, 2], [18, 12, 2], [-14, 10, 2], [-16, 2, 1], [10, -8, 0], [34, 2, -2], [-8, -14, 7], [28, -4, -4], [18, 6, -4], [-6, -8, 8], [24, 4, -3], [24, 2, 11], [-4, 2, 1], [-10, -2, -3], [14, 6, 4], [14, 8, 8], [-2, -8, -4], [0, 12, -8], [2, -10, -11], [-20, 8, -6], [28, -12, -4], [26, -18, -7], [12, -2, 7], [14, -8, 4], [12, 2, 11], [22, -14, -6], [0, -14, -7], [26, 6, 14], [0, 4, 8], [-18, -8, -2], [-30, 2, -6], [-2, 10, 3], [-18, -4, -6], [-30, 8, -4], [12, -10, -5], [-30, -2, 2], [-18, 20, 4], [-8, 8, 4], [16, 2, 5], [-34, -6, 5], [-6, -4, -2], [-10, -18, 4], [-4, -2, -3], [-12, -12, 0], [2, 0, 8], [16, 8, 3], [20, 8, 8], [-18, 8, 0], [-2, -12, 8], [2, 10, -4], [-34, -4, -14], [18, -4, -6], [0, -20, -5], [-10, -4, 2], [28, 0, 7], [-22, 4, -10], [6, -14, 0], [22, 6, 5], [-48, -2, 1], [8, 20, -4], [8, 6, 11], [10, 8, 8], [-18, 16, 8], [2, 8, 2], [-34, 10, 3], [10, -8, -4], [22, 2, -5], [-8, -6, -9], [-16, 6, 9], [-4, 12, 3], [8, -6, 13], [30, 2, 4], [-28, -16, 0], [36, 2, 9], [34, 2, 9], [6, -4, 14], [18, -8, -4], [0, 0, -4], [6, 2, -4], [14, 6, 6], [10, -4, -6], [0, -8, -8], [-26, 8, -4], [6, -2, -4], [-6, 0, 0], [56, 6, 1], [-30, -2, -8], [10, 12, 4], [-30, -10, 2], [-18, 6, 11], [4, -6, 3], [-4, 10, 1], [-32, -8, 2], [16, 16, 6], [-8, 16, 11], [34, -4, 2], [6, 14, 17], [-10, -14, -10], [-2, -4, -6], [0, 4, 14], [10, -6, -8], [-38, -4, 6], [-22, -8, -8], [-32, 12, 6], [12, -4, 0], [-6, -10, 6], [-22, -6, -3], [16, -16, -12], [-2, 0, -6], [2, -8, -6], [16, -16, 0], [26, 6, -2], [-32, 0, -11], [-10, 22, 14], [-28, -4, -4], [6, -8, -4], [-10, -20, -12], [-6, -14, -12], [-48, -2, 5], [16, -6, -3], [36, -12, -3], [22, 0, 0], [4, -16, -7], [14, 4, 0], [-8, 14, 7], [-16, 8, -14], [26, 10, -9], [-22, 24, 10], [22, -4, -6], [-2, 12, -8], [18, 10, -1], [-12, -10, 5], [28, 6, -1], [-36, -8, 5], [10, -4, 4], [42, 8, 0], [-40, 4, 1], [0, -18, 3], [-60, -4, 4], [22, 8, 20], [-32, 0, -9], [-6, -10, 6], [14, 16, -10], [30, -20, -6], [8, -20, -10], [-26, -6, -5], [-22, 6, -14], [40, -8, -8], [14, -4, 8], [-6, 26, 14], [-8, -8, -4], [6, 28, -2], [6, -20, -14], [-54, -2, 0], [-30, 4, -14], [-6, 4, 6], [4, -16, -8], [6, 24, 2], [10, 6, 2], [22, -8, 0], [-6, -24, 0], [-40, -8, 0], [8, -14, 9], [-8, -4, 8], [32, -12, -3], [22, 16, 12], [22, 12, 10], [-34, 10, 5], [18, 2, -9], [-12, 8, 4], [12, -2, -1], [-18, -2, -5], [40, 18, -5], [38, 6, -6], [2, 4, -14], [38, 2, 19], [-30, -8, -4], [-14, -24, 12], [-36, -16, 8], [-34, 6, 2], [-8, 6, 1], [-64, 0, -1], [20, -24, -13], [-22, -12, -4], [46, 4, 10], [-16, -20, -4], [6, -20, 0], [10, -24, -12], [0, 0, 11], [50, -4, 0], [-20, 26, 15], [30, 22, 4], [28, 0, 5], [-34, 2, -3], [-70, -4, 0], [-14, 20, 16], [-16, -26, -5], [-24, 0, 0], [-54, -10, -2], [-10, -10, -6], [-38, 14, 8], [-38, 4, 8], [42, 8, -6], [-10, -24, 8], [-32, 0, -10], [16, 6, 5], [-4, 6, 7], [-12, 18, 3], [4, -12, 8], [50, -16, -4], [12, 8, 16], [28, 22, 1], [-16, -4, -17], [-54, -6, -10], [-8, -14, 9], [-2, 6, 0], [-20, -10, -9], [8, -22, -17], [8, 16, 14], [-16, 30, 9], [-6, 12, 12], [16, -20, -11], [-34, 2, -2], [-10, 12, 0], [22, -10, 12], [-2, 16, -12], [-54, 12, 6], [8, -8, 12], [-6, 22, 2], [-28, 8, 2], [38, -12, 2], [-14, 12, -6], [-8, -16, -21], [-56, 10, 7], [-28, -8, -10], [-16, 4, 12], [-26, 6, 6], [-34, 8, -8], [26, 14, 13], [18, 12, 14], [16, -8, 3], [44, 4, 18], [-48, 14, 3], [-22, 4, 6], [46, -8, 0], [14, -8, 8], [50, 4, 20], [70, 2, 2], [-14, 4, -8], [14, -2, 18], [26, 26, 2], [-22, 14, -12], [-2, -20, 14], [-22, 10, -10], [6, 2, 11], [-50, 12, -2], [30, -16, -8], [0, -16, 11], [-20, -8, 0], [0, -4, 11], [34, 24, -8], [-2, -6, -4], [6, 0, 8], [42, 2, 3], [18, -8, 0], [22, -20, 4], [4, -2, -1], [12, 12, 17], [-22, -8, -18], [56, 6, 13], [-14, 6, -3], [14, -32, -16], [-6, -4, 4], [-44, -16, -1], [28, -4, 7], [6, -8, -8], [-64, 6, -5], [-26, -10, 13], [70, -10, -1], [2, -8, -12], [10, 8, 8], [32, 20, 9], [-20, -28, -2], [50, -8, 2], [-58, -6, -3], [-2, -24, 0], [-20, -18, -19], [-14, 26, 18], [8, 20, 8], [-38, 4, -16], [-6, -4, -18], [-30, -6, -11], [8, -18, -13], [-14, 2, 16], [32, -8, -6], [-10, 8, -4], [16, -16, -13], [-26, 22, 5], [54, 6, 12], [-8, -6, 11], [30, -8, 12], [58, -8, 8], [10, -4, 0], [8, 4, 8], [6, 2, -6], [-2, 2, 14], [12, 2, 5], [-66, 8, 6], [-28, 20, 14], [38, -24, -4], [-34, -4, -8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9555_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9555_2_a_bq();".
// 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_9555_2_a_bq(:prec:=3)
    chi := MakeCharacter_9555_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(3137) - 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_9555_2_a_bq();".
// 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_9555_2_a_bq( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9555_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-2, -7, 0, 1]>,<11,R![-16, -16, -1, 1]>,<17,R![76, -32, -1, 1]>,<19,R![-64, -16, 6, 1]>,<23,R![-128, -16, 7, 1]>],Snew);
    return Vf;
end function;