
// Make newform 3640.2.a.t in Magma, downloaded from the LMFDB on 23 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_t();".
// 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_t();".
// 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, -1, -6, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, 0, 1, 0], [-1, -5, 0, 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], [0, 1, 0, 0], [-1, 0, 0, 0], [1, 0, 0, 0], [0, 0, -1, 1], [1, 0, 0, 0], [-3, 0, 0, -1], [-1, -2, 0, -1], [-2, -2, -1, 1], [-4, -1, 1, -1], [-2, 1, -2, -2], [0, -1, 2, -2], [-1, -2, -1, 2], [-2, 2, 0, -2], [0, 0, 1, 3], [0, -2, -2, 2], [0, -3, 1, 3], [6, -2, 1, 3], [1, 2, 1, -2], [-4, 4, 2, 0], [4, 2, 1, 1], [1, 0, 1, -2], [-4, 2, 0, 0], [0, -1, 3, -1], [-2, -2, 3, 1], [4, 0, 5, -3], [-1, 2, -2, 3], [-2, 6, 0, -2], [0, 0, 2, 2], [-6, 0, -1, -3], [-8, 2, 3, -1], [0, -4, 0, 0], [-6, 5, -1, -1], [-6, -6, -2, 4], [-6, 0, -3, 1], [-6, 0, 4, 0], [4, 5, -4, 4], [-2, -1, 1, 3], [-6, 6, -2, -4], [-6, 3, 1, -7], [-7, -4, 2, -3], [4, -2, -1, -5], [-4, 5, -1, -5], [-3, 2, -4, -3], [-8, 5, 4, 0], [-4, 4, -2, 6], [-5, -4, 0, 5], [-10, -4, 7, -1], [-8, 6, 0, -4], [1, 4, -2, -5], [-4, -6, -1, 3], [-10, 0, -4, 4], [7, -6, -2, -5], [-10, 0, 1, 1], [-14, -6, 2, 0], [-8, -6, -4, 6], [8, 10, -3, 1], [6, 4, -7, 1], [0, 8, 0, 8], [0, 10, -2, -4], [3, 0, 3, -4], [12, -6, 0, 2], [-6, 8, 0, 2], [6, 12, 0, -6], [-2, -5, -1, 9], [-8, -8, 1, -3], [-4, 2, 1, -1], [-8, 8, 3, -1], [2, -4, 0, 10], [4, -3, -1, -3], [4, 4, 3, -1], [-4, -8, -4, 6], [4, -8, 0, 0], [6, 6, 4, -2], [2, -2, -6, 0], [0, 4, -1, -3], [13, 8, -6, 3], [14, 2, -2, 2], [8, 2, 8, -2], [6, 2, 4, -10], [-18, -8, 1, -3], [12, 6, -7, -1], [2, 0, 2, -2], [-11, 4, 0, -9], [-10, -14, 4, 2], [-8, -10, 4, -8], [4, -4, -2, 4], [-2, 1, -5, -5], [20, 3, -1, 1], [-17, -2, 4, -3], [-3, -4, -8, 1], [16, -3, 3, -1], [-14, 7, 1, 3], [4, 0, -4, -4], [-4, -2, -5, -3], [0, 4, 8, 0], [13, -8, 0, 5], [6, -6, 2, 2], [-2, 2, -5, 9], [8, -12, -2, 6], [2, 2, -6, 4], [-9, 6, 1, -6], [-11, -4, 3, 2], [-6, -7, -8, 6], [20, -9, 1, 5], [10, 6, 2, 6], [-16, -6, 0, -4], [-20, 8, 0, -2], [-14, -8, 7, -5], [8, -18, -4, 2], [10, -5, -11, -1], [12, -8, -1, -5], [-21, -6, -2, 9], [-4, -3, 1, -13], [4, 0, 8, -6], [-14, 5, 2, -8], [-22, -2, -4, -2], [2, -5, -5, -3], [0, -4, -6, -2], [3, -4, 2, 3], [7, 12, -12, 3], [-8, -10, 7, 3], [-16, 8, -5, -9], [9, 2, 9, -2], [16, 11, -5, 5], [-1, 4, -8, 7], [10, -8, -5, -5], [0, -8, 8, 0], [6, -7, 5, 3], [20, 6, -4, -2], [4, 8, 0, 8], [-13, 6, 2, -9], [6, 3, -10, -2], [-22, 12, 4, -6], [10, -5, -8, 0], [4, -14, 1, 7], [12, 12, -8, 2], [12, 0, -2, 10], [-20, -11, -2, 2], [-1, 20, 2, -3], [-4, 10, 0, 6], [18, -6, 4, 10], [-24, 2, -9, -1], [-12, -4, 4, -4], [16, -2, -2, -4], [-34, 4, -3, -11], [16, 12, -6, 4], [2, -9, -3, -9], [10, -4, 7, 7], [13, -6, -10, -3], [-3, -22, 7, 6], [6, 10, -2, -6], [14, 16, 2, -4], [2, -1, 15, -7], [-4, -14, -6, 10], [29, 4, 2, 5], [2, 5, 6, 2], [-10, 7, 4, 8], [-7, 12, -2, -15], [21, 12, -6, -1], [25, -2, -4, -1], [-16, -8, 0, 6], [2, -15, -5, 5], [4, -4, -5, -7], [-19, -10, -12, 17], [0, -10, -6, 10], [1, -4, -7, -10], [-5, 14, -3, 6], [16, 2, 4, -2], [-1, 18, 4, -13], [-14, 4, -8, -6], [16, -15, -1, 9], [-6, 8, -7, -3], [-22, -4, -4, -8], [34, 4, 2, 4], [6, 10, 14, -10], [2, 8, 5, -11], [30, 10, -12, 2], [-10, 3, -13, 9], [16, -4, -3, 9], [10, 1, 7, -15], [0, -8, 7, -7], [-2, 2, -2, 0], [8, 4, 5, 9], [28, 0, -10, 6], [18, 10, 2, -6], [6, 4, 16, -6], [6, -12, 4, -2], [6, 10, 6, -6], [22, -1, 0, 8], [-25, 12, 2, 3], [12, 7, 2, 0], [20, -9, -11, 1], [-7, 8, 2, 15], [-16, 0, 2, -6], [-14, -2, 2, -4], [4, 8, 4, 14], [8, -22, 0, 8], [18, 12, -8, 8], [6, 7, 1, -5], [-10, 6, -11, 1], [12, 2, 0, 8], [12, -12, 0, 10], [2, 12, -2, 10], [-9, 14, 4, 7], [-35, -10, -4, 9], [-13, -16, 14, -5], [-4, -2, 16, -4], [-16, -4, 6, -12], [-6, 24, 9, -7], [-16, 2, -7, 1], [-22, -12, -5, 13], [-34, 0, -1, -11], [-12, 10, -8, -12], [-14, 0, -19, 1], [-2, 17, -2, -2], [-20, 0, -15, 1], [4, -2, -10, 12], [10, -7, -5, 17], [-7, 6, 7, -6], [11, 8, 10, -7], [-21, -4, 1, 8], [-38, 12, 2, -6], [1, -6, 14, 5], [19, -12, 1, 2], [24, -11, -7, 11], [-22, 15, 11, -3], [-10, 13, -6, 2], [25, -6, 8, 15], [10, -2, 3, 7], [-14, -8, -8, 2], [0, 20, 0, 4], [14, -2, 8, 12], [26, -5, 4, 18], [-45, 12, -3, -2], [-4, -16, 9, -1], [28, -10, 3, 5], [-12, -1, 2, -18], [-11, 4, 8, -5], [-24, -14, 10, 2], [-8, -13, 0, -10], [-20, 4, -1, 13], [-10, -23, 13, -1], [-4, 14, 8, -16], [-28, -2, -3, 5], [-28, 6, 0, -8], [-4, -2, 7, 5], [11, 18, 6, -19], [-64, -2, 1, -1], [0, -16, 3, -9], [-17, 2, 11, 0], [-35, 10, -10, -11], [-12, -32, 6, 2], [-2, -12, 12, 0], [16, 4, 11, -13], [4, 14, 6, -2], [-21, 12, -4, -1], [4, 14, 0, -8], [45, -14, 5, 12], [-16, -4, -7, 1], [-38, -8, 2, -6], [-16, 16, 2, -6], [48, -3, 5, 7], [4, -25, 2, 6], [-14, 6, 5, -5], [-8, 10, -2, -10], [-8, -24, -5, 13], [7, 14, 11, -22], [24, 1, -6, 0], [24, -12, 14, 18], [-8, -2, 9, -19], [15, 2, 11, -10], [-2, 0, -1, 9], [-9, 4, -22, -1], [3, 8, -8, 17], [2, 8, 4, -6], [-3, -2, 4, 19], [38, 14, -9, 3], [-2, -34, 2, 6], [6, 2, -10, 16], [-1, -22, 0, 5], [-8, -14, 2, -8], [6, 17, 5, -5], [-2, 16, 14, -10], [16, -6, 2, 10], [7, 8, -17, 6], [-18, 2, 9, 7], [-34, -4, -4, 4], [5, 20, -13, -8], [-22, 16, -12, -8], [2, -20, 0, -8], [26, 14, -9, 9], [-38, -7, -11, 5], [-25, 6, 0, -13], [-4, -14, 2, -2], [44, 4, 8, -6], [-2, -2, -16, 6], [21, -6, -5, 10], [-38, 16, -5, -1], [40, 0, 6, 14], [-37, 24, 4, 1], [35, 6, 5, -10], [-20, 11, 21, -3], [24, 8, -12, 12], [15, 0, 12, -3], [-15, -10, 10, -17], [-30, 4, -8, -2], [36, 6, -2, 4], [12, -10, -4, 6], [18, 28, -4, 2], [19, 2, 12, 1], [12, 12, 10, -2], [-23, -16, 7, 10], [-2, -8, 6, 12], [-6, -19, 7, 9], [4, -10, -10, -4], [-4, -10, 8, 14], [2, 20, -14, 6], [32, 8, -4, -6], [-25, -16, -1, -8], [12, -14, -3, 23], [17, -22, 2, 3], [-20, 2, -11, -23], [-6, -19, 1, 11], [2, 7, -1, -1], [32, 22, 0, -6], [8, -8, -1, 5], [4, -10, 2, 6], [-6, -20, -2, -6], [-18, 4, -16, 14], [16, -16, 7, 17], [8, 10, -2, 6], [10, -18, 18, 0], [8, -2, -26, -2], [2, -6, -1, 19], [14, 16, -4, -6], [46, 0, 6, 0], [-4, 10, -1, -17], [24, 9, 1, 17], [-10, 11, -9, -7], [18, 7, -7, -11], [0, 14, -7, 17], [-24, -21, 9, -13], [-2, 19, -8, -4], [-16, 0, 19, -13], [12, 29, 11, -25], [-28, 20, 8, -10], [-4, 8, 8, -16], [-3, -24, 19, 4], [-12, 36, 1, 1], [-24, 19, 3, -19], [10, -24, 0, 24], [-26, 12, -5, 15], [22, -7, 11, -7], [20, -7, -3, 15], [-34, 25, -1, -3], [5, 12, -9, 28], [-36, 1, 11, -7], [32, 20, -11, 13], [10, -17, 1, -13], [-6, 7, 17, -11], [-12, 16, 2, 12], [32, -4, -11, 5], [-40, 10, 1, -9], [28, -9, 2, -18], [-6, 22, 7, 5], [36, 16, -2, 12], [-25, -14, -2, 19], [18, -1, -2, 0], [-19, -8, 8, -21], [-44, 12, -5, -23], [-38, -21, 13, -5], [-18, 20, -4, -20], [-4, 8, 10, 2], [-30, -6, 10, 4], [-1, 22, 12, -5], [-31, -4, -3, -2], [24, -10, 9, 13], [6, -4, -6, 18], [-2, -3, -11, 27], [8, -7, 11, -11], [6, -8, 8, 8], [-9, -14, -12, 21], [-1, 22, -6, 9], [-8, 3, -3, -5], [56, -14, -6, 2], [-62, 2, 0, 4], [7, 22, -17, -8], [-24, -20, 4, 16], [22, -11, 1, -1], [14, -18, -6, -6], [24, 14, -10, 16], [0, 18, 3, 1], [22, 19, -1, 3], [-60, -8, 6, -12], [-48, -18, 7, 3], [-6, -17, -6, -18], [66, -4, -10, -2], [24, -21, 5, 13], [22, 40, -10, -4], [3, 4, -18, -11], [-34, -16, 4, 4], [54, -7, -11, 3], [40, -24, -7, -3], [19, -8, 24, 7], [28, 14, -6, 30], [37, -10, -7, 0], [26, 4, -16, -4], [-50, -18, -2, 12], [16, -30, 12, 6], [14, 18, -4, 26], [-8, 28, -8, -16], [28, 0, -8, 24], [-37, 12, 6, 7], [-18, -12, 5, -1], [-28, -8, -16, 2], [-6, -16, 14, -14], [12, 36, 2, -10], [30, -2, 9, -21], [-14, -5, -9, -23], [-6, 0, -10, 10], [36, 10, 6, -4], [-45, -6, 6, -5], [-18, 4, 4, 6], [22, 6, -2, -10], [-30, -18, 4, 4], [-28, 2, -8, -4], [-10, -44, -4, 20], [-20, 2, -18, -12], [18, 14, -10, -4]];
    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_t();".
// 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_t(: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_t();".
// 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_t( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3640_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, -1, -6, 0, 1]>,<11,R![-16, -36, -18, 1, 1]>],Snew);
    return Vf;
end function;