
// Make newform 3640.2.a.x in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3640_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_3640_2_a_x();".
// 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_3640_2_a_x();".
// 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, -6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, -1, 1, 0], [1, -6, -1, 1]];
    Rf_basisdens := [1, 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_3640_a();"
function MakeCharacter_3640_a()
    N := 3640;
    order := 1;
    char_gens := [911, 1821, 1457, 521, 561];
    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_3640_a_Hecke(Kf)
    return MakeCharacter_3640_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], [1, -1, 0, 0], [-1, 0, 0, 0], [-1, 0, 0, 0], [1, 1, 1, 1], [1, 0, 0, 0], [2, 1, 1, 0], [1, 0, 0, 1], [1, -1, -1, -1], [0, -2, -1, -1], [1, -1, 0, -2], [-4, 2, -1, 3], [-2, -1, -1, 4], [3, -1, 1, 1], [0, 2, 0, 2], [4, -2, 0, -2], [3, 1, -2, -2], [2, -4, 0, -4], [4, 3, 1, 0], [1, -7, 1, -5], [2, -2, 0, 2], [2, 3, -3, 2], [0, 2, 0, 6], [6, 4, -1, -3], [6, 0, 2, -6], [4, 4, -2, 4], [5, 2, 0, -3], [3, 1, -3, -1], [-4, -2, 0, 6], [10, 0, 2, 2], [3, 1, 1, 5], [4, 2, 0, 2], [2, 0, -1, -1], [0, -2, 2, 4], [-4, 2, 0, 2], [-1, 7, -1, 1], [4, -6, -1, 3], [4, 2, -1, -7], [2, 2, 4, -4], [0, -6, 1, 1], [14, -7, -1, -6], [-6, 0, 2, -2], [-4, -2, 1, -1], [-6, -1, -3, 0], [0, -6, -3, -7], [10, 6, -4, 0], [-4, -3, -1, 8], [2, 0, 4, -6], [0, 6, -4, 6], [-16, 5, 3, 2], [-6, 2, 0, 6], [7, -3, 1, 1], [-6, -7, 1, 0], [-2, 6, 0, -8], [4, 10, 0, 10], [3, 1, -7, 1], [8, 0, -2, 0], [-7, 5, 5, 1], [-4, 8, 0, -4], [-10, 2, -2, -8], [7, -2, -6, -1], [0, 4, 8, 2], [2, 4, -2, 0], [-4, 10, -2, 12], [-2, -4, -5, -3], [-6, 0, -4, -12], [7, 3, -1, 3], [-6, 4, 4, -4], [-5, -7, -1, -3], [-24, 2, -1, -1], [14, -4, 0, 0], [21, -1, 1, -3], [15, -1, 3, -13], [-8, 6, 2, 4], [13, -1, -3, 3], [6, 8, 0, -6], [-4, -1, 9, -2], [0, 12, 0, 2], [-10, 6, 6, 0], [8, 6, -4, -6], [12, -2, 2, -8], [-16, 14, 0, 10], [-3, 7, 5, 5], [-6, 5, -7, 4], [6, 2, 2, 6], [-7, 11, 1, 7], [-14, 4, -2, 6], [2, -8, -5, -1], [0, 2, -3, -3], [2, -3, -1, 6], [-7, 4, 0, 1], [1, -5, -4, 2], [-6, 0, 1, -1], [8, -8, 0, -4], [5, -11, 1, 1], [5, -3, -3, 1], [4, 5, -1, -6], [26, 2, 0, -6], [11, -1, -3, 3], [-20, 8, 6, -2], [-13, 1, -1, 5], [8, 1, 7, 10], [7, 0, 12, -5], [2, 0, -1, 3], [-6, -4, 5, -5], [6, 6, -4, 10], [-8, -6, -6, 8], [2, 6, -4, -2], [12, 8, -4, 0], [-14, -4, -2, 2], [9, -13, -2, -12], [-18, -4, -4, 0], [6, 13, 3, 12], [7, -15, -2, 6], [11, -3, 3, -17], [4, 10, 5, 9], [8, -2, 8, 2], [15, -7, 0, 2], [16, 0, 2, -6], [16, -7, 3, -4], [-4, 15, -3, -10], [2, -10, -4, -14], [14, -12, 4, -16], [-14, 1, 3, 10], [-5, -11, -4, -8], [-24, -3, -1, 10], [-20, 0, 8, 0], [-8, 2, -2, 0], [-9, 1, -6, 4], [-6, -12, 4, 0], [-9, 1, -11, -9], [-4, -9, -3, 8], [-2, 12, -3, 3], [4, -2, 8, -6], [-2, -8, -5, 1], [-12, 6, 0, -6], [8, -2, 2, 2], [-14, 14, -2, 10], [4, 18, -5, 7], [26, 3, 1, 4], [-11, -11, -5, -15], [-4, -4, -2, -6], [-7, 17, 3, -3], [-4, -4, -8, -12], [14, -22, 0, -14], [-1, 3, 3, -9], [-16, 0, 6, 8], [6, 4, -1, -7], [-22, -4, -8, -14], [2, 19, -3, -6], [16, -1, -3, 2], [18, -12, 6, -18], [11, -9, 3, -1], [17, -1, 2, 16], [1, -15, 5, 5], [-14, 5, 7, 6], [16, 6, -3, -1], [-22, -4, -1, 13], [30, 1, -3, 4], [-4, -7, 7, -6], [2, -9, 13, -6], [10, -2, 4, -10], [-18, 0, 3, 5], [-6, -2, -6, 14], [32, -7, 3, -6], [34, -8, 6, -20], [-6, -1, 5, 6], [4, 15, -3, -2], [-14, 16, 6, 2], [-12, 1, -1, -2], [-11, -3, -5, 11], [12, 6, 11, 19], [-2, 10, -6, -6], [-10, 0, -4, -4], [12, -6, 8, -10], [14, -2, -2, 4], [-35, 13, 3, 7], [-12, 12, -2, 6], [9, -1, 6, 12], [-16, 12, 0, -6], [30, -4, -1, -3], [-34, 0, -6, -4], [-20, 0, -2, 2], [18, -20, -10, -10], [-4, -12, -4, -20], [26, 10, -8, 2], [-24, 20, 0, 12], [0, 14, 6, 16], [-10, 2, 0, -10], [7, 5, 0, 6], [26, 3, 3, 4], [18, 0, -3, -9], [-4, -14, -9, -15], [-8, -3, 3, 2], [-15, 19, 3, 5], [36, 2, 0, 6], [-14, 2, 6, -4], [-8, -8, -8, -20], [14, 2, -4, 10], [3, -19, 4, 6], [16, -16, -6, -10], [-22, -4, 6, 0], [2, -4, -10, 14], [2, 20, -8, 0], [-37, 6, -2, 7], [4, 11, -11, 10], [-9, -14, 8, -5], [-11, -1, 5, 15], [10, 4, 4, -16], [8, -12, 0, -8], [14, -8, -6, -2], [10, -2, -2, 16], [19, -15, -3, -3], [6, 10, 8, 4], [25, 9, -11, 5], [-2, -12, 13, -5], [-23, 9, 5, -5], [-10, -2, -8, -14], [40, -6, 5, -5], [-8, -17, 9, -2], [0, 25, -1, 18], [9, -14, -4, 1], [-18, 4, -12, -8], [36, 9, 3, 8], [21, 0, 0, 5], [-20, 6, 5, -11], [30, -20, -1, -11], [0, 2, 5, 3], [27, -6, 0, -1], [1, -3, 7, 19], [-8, -12, 0, -8], [-22, 20, 4, 22], [-7, 13, -1, 13], [-8, -6, 15, -9], [-3, -16, -8, -19], [8, -16, -2, -2], [-20, 2, 0, 6], [-4, 10, -13, -5], [0, 15, 1, 0], [14, -12, -2, -4], [-16, 2, 7, 5], [34, -2, 8, 4], [27, 13, -6, 0], [-32, -4, 0, 20], [30, 10, -12, 2], [2, -16, 2, -8], [-4, -14, 4, -6], [1, 8, 4, -3], [5, -23, -5, -21], [1, -13, -13, -9], [12, -23, -1, -18], [34, -5, 3, -12], [14, -18, -10, 2], [-2, -12, -6, 18], [2, 0, -12, 12], [3, 3, 13, -3], [-8, -3, -13, 10], [-4, 12, 0, -20], [4, 13, -5, -22], [54, -2, -2, -4], [-19, 23, -1, 27], [-37, 25, 5, 3], [8, -6, 15, 11], [-4, -2, -3, -15], [-10, -12, -8, -32], [-16, 14, -6, -12], [32, -8, 8, -12], [-2, -13, -5, 8], [-10, 0, -9, 5], [-36, 2, -8, 6], [0, -8, 6, -8], [-22, -13, -3, -2], [-6, -6, -4, 18], [28, -15, -13, -8], [26, 3, -1, -8], [50, -22, -4, -4], [-2, 7, -7, 10], [-12, 26, -4, 18], [14, -14, 0, -18], [12, 0, 6, -2], [27, -4, 6, 11], [-33, -5, -9, -3], [9, -13, 4, -2], [-26, 16, 0, -12], [12, -4, 8, 12], [28, -5, 13, -28], [14, -2, 0, -2], [-38, -16, 0, 0], [29, -12, 10, -7], [18, -10, 4, -14], [34, -12, -4, -12], [21, -3, 7, -7], [38, -12, -1, 11], [17, -16, -8, 5], [36, 20, -6, 2], [4, 12, 2, 12], [-6, -8, -2, 12], [12, 5, 7, 10], [-16, 14, 0, 18], [-22, -10, 8, -12], [22, -25, -5, 8], [-28, -3, 7, 2], [-26, -12, -3, 11], [9, -3, -7, -7], [18, -5, 17, -14], [-16, -11, 7, 10], [1, -5, -5, -31], [-16, -4, -8, -14], [17, -15, 3, 25], [-20, -4, -6, -6], [31, 16, -6, -5], [36, -14, -10, 0], [2, -5, -15, 4], [33, 15, -1, -15], [-14, -16, 3, -7], [6, 18, -10, 8], [27, -29, 1, -25], [9, 17, 1, -1], [2, -10, -8, 2], [8, 11, -3, 2], [-8, 10, 8, 30], [12, 13, -5, -2], [6, 18, -14, 8], [16, -18, 13, -17], [28, 14, -1, 3], [-14, -8, -10, -2], [15, 13, -11, 7], [28, -26, -6, 8], [-9, 19, -5, 29], [8, 24, -12, -8], [-4, -16, -10, -10], [24, 6, 4, 18], [-42, 4, 0, -2], [-16, 20, -6, 2], [15, 11, -5, -9], [10, -2, 6, 10], [0, -20, -4, -20], [6, -26, -6, 0], [38, -16, 5, -13], [-13, 1, -8, 18], [-32, -14, 15, -9], [-34, 22, 10, 4], [-12, -26, 9, -7], [-10, 8, 5, -9], [-6, -8, 20, -8], [18, -8, -17, 1], [12, -10, 6, -26], [2, 10, 10, -10], [-21, 6, 8, 7], [2, -12, -4, -16], [-5, 25, -8, 8], [8, 18, -2, 6], [4, 6, 4, 26], [5, 3, 8, 10], [-44, 2, -11, 9], [38, -32, -3, -13], [-15, -4, 10, -11], [47, -3, 6, 18], [-14, 8, 14, 10], [2, -32, 5, 19], [-8, 2, -9, 19], [-22, 36, 4, 28], [-3, -7, 21, 3], [-19, 15, -7, -17], [-11, -5, -2, -18], [14, 16, 6, 30], [6, 12, -4, 16], [14, 5, 15, 2], [-16, -14, -13, -5], [-22, 9, -5, 2], [2, 14, 0, 4], [4, -2, -5, 1], [-10, 28, 4, 4], [24, -18, -2, 12], [-4, 0, 20, 4], [-7, -20, 8, -11], [-20, -7, -9, 12], [38, -20, 6, -34], [-10, -14, -2, -24], [-26, 16, -3, 15], [-20, -22, -5, -17], [-50, 0, -4, -12], [-22, 1, -1, -6], [12, 11, 9, 14], [-53, -3, -8, 12], [-12, 12, -6, 26], [-7, 27, -5, 11], [-48, 19, 5, 14], [-32, 16, -4, 0], [33, 19, 4, -10], [10, -28, -6, -22], [-16, -4, 2, -20], [71, 1, 5, -9], [-5, -7, 0, -32], [-5, 35, -5, 29], [34, 6, 10, -8], [-1, 5, -4, 6], [-6, 8, 14, 22], [4, 14, 3, -7], [40, 10, 0, -6], [0, 19, 1, -30], [22, -30, -6, -28], [8, -14, -15, -17], [-6, -10, -10, -20], [4, -13, 1, 26], [8, 28, -2, 2], [-12, -9, 5, -2], [-26, -2, -12, -14], [30, -16, -20, 6], [-67, 1, -5, -3], [4, -6, 8, -6], [-4, 18, 0, -18], [33, -25, 1, -1], [38, -9, -3, 14], [-18, 4, -4, -24], [28, 16, 14, 36], [15, -1, 9, -1], [-13, 23, 15, 11], [-30, -8, -8, 24], [-7, -9, 6, 12], [30, -28, 0, 4], [10, -4, 16, -4], [0, -33, 5, -20], [16, 16, 4, 8], [-10, 2, 22, -12], [32, 2, 12, -16], [-29, 3, -5, -5], [-34, -8, -10, 22], [31, -29, -11, 3], [-8, -8, -6, 26]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3640_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3640_2_a_x();".
// 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_3640_2_a_x(:prec:=4)
    chi := MakeCharacter_3640_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(2999) - 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_3640_2_a_x();".
// 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_3640_2_a_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3640_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-4, 10, -3, -3, 1]>,<11,R![-64, 80, -12, -6, 1]>],Snew);
    return Vf;
end function;