
// Make newform 8381.2.a.e in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8381_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_8381_2_a_e();".
// 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_8381_2_a_e();".
// 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, 0, 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_8381_a();"
function MakeCharacter_8381_a()
    N := 8381;
    order := 1;
    char_gens := [581, 8093];
    v := [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_8381_a_Hecke(Kf)
    return MakeCharacter_8381_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 := [[-1, 1], [-1, 1], [1, 0], [0, -2], [-1, -1], [-1, 2], [0, 0], [6, 0], [2, 4], [-1, 0], [-3, 5], [4, 0], [-4, -6], [5, 1], [1, 3], [1, -6], [2, 4], [2, -2], [0, -4], [6, -2], [-4, 0], [1, -1], [2, -4], [-4, 6], [4, 6], [-8, -4], [-2, 2], [12, -2], [-7, 4], [2, -8], [-10, -4], [-10, 8], [12, 0], [-14, 0], [-5, 2], [0, 10], [0, -6], [-11, -5], [6, 2], [-18, -4], [2, 6], [3, 8], [14, -8], [8, 2], [-2, 0], [-8, -6], [1, -13], [-6, -2], [-6, -10], [-12, -6], [-7, 8], [-14, 4], [7, -8], [13, -5], [-21, -2], [7, 3], [-6, 18], [1, -11], [6, -8], [15, -12], [-6, 4], [2, -4], [-7, -7], [-14, -8], [-7, 2], [6, -18], [-1, 1], [-2, 14], [6, 6], [9, 10], [10, -12], [11, 5], [-18, 0], [-15, 8], [-10, -12], [-12, -6], [20, 12], [-25, -4], [13, -4], [-2, 12], [18, 6], [6, -22], [-14, 4], [8, -16], [6, 4], [-30, -4], [18, -12], [18, 12], [14, 0], [-26, 0], [-3, 25], [3, 7], [20, 6], [-17, 3], [-2, 12], [-13, -9], [-19, -6], [15, -10], [16, 8], [16, 4], [-16, 14], [-6, 8], [-5, -3], [-34, 4], [8, 16], [-10, 14], [2, -4], [-11, 6], [-17, -19], [20, 2], [-5, -9], [-9, 0], [-12, -8], [-35, -1], [-34, 2], [2, 14], [-24, 6], [34, -4], [16, -10], [13, -1], [22, -8], [-1, -16], [22, 0], [-4, 12], [-48, 0], [-9, 22], [-15, 10], [6, -10], [-10, -8], [4, 32], [-3, -5], [-18, 4], [-14, 8], [34, 6], [6, -28], [-18, 22], [-28, -6], [6, -34], [-2, -38], [-8, 20], [-8, 2], [-7, -6], [26, -20], [-23, 7], [-10, 14], [15, 5], [6, -12], [9, 2], [7, -9], [-14, -22], [23, -10], [-14, 0], [-4, -30], [-27, -7], [6, 20], [31, 11], [-6, -10], [-30, -8], [6, 16], [17, -28], [-21, -13], [-13, 16], [-31, 3], [-10, 4], [39, 2], [-5, -19], [10, -2], [0, 20], [-19, 6], [4, 8], [-20, 22], [-17, -8], [2, -10], [2, -8], [-26, 4], [22, 12], [0, -32], [14, -10], [-33, 17], [-26, -20], [43, -3], [-7, 7], [2, -8], [19, 30], [-28, 24], [21, 8], [-26, -14], [2, 4], [-18, 2], [-28, 6], [3, 24], [-5, 13], [21, 13], [-28, -16], [34, 20], [-2, -4], [-44, 8], [34, -4], [43, -16], [-48, -4], [-44, -6], [-26, 24], [32, 18], [4, -14], [-10, 20], [-27, -8], [7, 33], [58, 2], [-1, 26], [31, 23], [-24, -14], [22, -16], [-49, 3], [-39, 13], [-38, -16], [13, 18], [0, -4], [20, -4], [-48, -8], [-38, -4], [-26, 2], [12, 18], [-2, -4], [-3, -21], [-52, -8], [18, -4], [0, 16], [-34, 16], [17, 27], [24, 30], [-12, -8], [0, 20], [-2, -20], [-16, -28], [10, -6], [18, 28], [-22, 18], [-4, -26], [-28, 10], [7, 7], [-23, -27], [26, 30], [46, 14], [-36, 8], [18, -8], [50, -14], [18, -8], [10, 28], [-34, 22], [75, 1], [50, 2], [-17, 22], [-3, -19], [0, -28], [48, -18], [-24, -16], [16, -14], [27, 29], [54, 4], [-31, 4], [6, -8], [-55, 9], [-22, 36], [22, 26], [-18, 14], [21, 33], [-32, 12], [-6, -2], [-18, 4], [-9, 28], [7, -26], [-18, -38], [-51, 12], [-18, -28], [36, -16], [-9, -33], [-10, 12], [-55, -20], [4, -16], [-6, -32], [2, -10], [-26, -14], [20, -40], [18, 48], [-14, 16], [10, 28], [22, -26], [-32, 18], [0, -34], [57, 12], [3, 12], [-62, 4], [-19, -12], [-29, 9], [18, -40], [-15, -34], [45, 27], [21, -14], [0, -26], [-22, -12], [-2, -26], [46, -16], [-42, -24], [-41, 15], [-2, 12], [-47, 6], [7, -39], [-19, 10], [36, -10], [5, 24], [34, 32], [16, 14], [10, 16], [-34, 8], [32, 10], [-57, -4], [6, 44], [22, 2], [-3, 6], [26, -34], [15, -3], [49, -6], [14, 0], [21, -31], [-55, -12], [24, 6], [10, 30], [-48, 0], [-15, 31], [53, 16], [36, 20], [-27, 2], [-56, 0], [-22, 24], [37, 21], [-20, 12], [-10, -32], [26, -8], [6, -40], [-60, -8], [-62, -6], [-22, -36], [-2, 46], [-36, 20], [10, 28], [-33, -29], [-46, -14], [-54, 16], [37, 31], [-34, 0], [30, -20], [22, 44], [-6, 16], [10, -28], [-6, -38], [-42, 14], [-81, -1], [12, -18], [18, -14], [2, 42], [-33, 20], [7, -40], [35, 45], [-14, 50], [-27, 13], [-60, -14], [-36, -40], [86, 6], [-18, -24], [-25, -3], [-42, 2], [-24, 42], [2, 0], [-26, -52], [21, 36], [-18, 28], [-41, 15], [-46, 30], [-15, -42], [43, -2], [-20, 4], [31, -30], [-37, 21], [2, 14], [-28, -42], [20, 50], [22, -48], [-41, 6], [1, -29], [-58, 4], [54, 0], [-5, -2], [-47, -1], [19, 51], [-29, 43], [1, -54], [-40, 32], [64, -16], [-26, 6], [18, -12], [50, -28], [-26, -12], [10, 28], [27, 32], [51, 18], [50, -28], [15, 8], [-82, -14], [1, -15], [60, -20], [-30, 48], [-7, 6], [-32, 26], [22, 48], [32, -18], [-6, 0], [43, -1], [18, 54], [40, 4], [-25, -15], [-6, 64], [-54, 12], [15, 15], [51, -25], [-15, -12], [31, 16], [-14, -20], [36, 36], [-18, 16], [15, -17]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8381_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8381_2_a_e();".
// 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_8381_2_a_e(:prec:=2)
    chi := MakeCharacter_8381_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_8381_2_a_e();".
// 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_8381_2_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8381_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 2, 1]>,<3,R![-1, 2, 1]>],Snew);
    return Vf;
end function;