
// Make newform 8281.2.a.z in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8281_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_8281_2_a_z();".
// 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_8281_2_a_z();".
// 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, 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_8281_a();"
function MakeCharacter_8281_a()
    N := 8281;
    order := 1;
    char_gens := [1522, 3382];
    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_8281_a_Hecke(Kf)
    return MakeCharacter_8281_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, -2], [1, -2], [0, 0], [3, 0], [0, 0], [-5, 4], [-3, 0], [-5, -2], [-2, 4], [-5, 0], [5, -6], [-2, 4], [-8, 0], [1, 4], [-1, -4], [-5, 4], [3, 0], [3, 0], [-4, 8], [-7, 6], [7, -6], [0, 0], [1, -2], [10, -12], [-9, 0], [7, -6], [-13, 2], [-7, 6], [-10, 8], [8, -12], [-5, -2], [5, 2], [-4, -12], [9, -6], [-1, -12], [7, 0], [-3, 12], [-16, -4], [3, -12], [9, -18], [-6, -12], [-5, 10], [9, -6], [22, -8], [-11, -6], [4, 0], [4, 0], [7, -8], [5, -18], [-1, 8], [0, -12], [5, -6], [8, -4], [-13, 8], [-3, 18], [7, 4], [1, 12], [19, -12], [-14, -8], [-3, 18], [2, 8], [0, -12], [-31, 2], [-25, 12], [5, 2], [21, -12], [18, 0], [-19, 26], [10, 12], [-27, 6], [1, -8], [-3, -6], [-21, 12], [-4, 12], [-15, 0], [-5, 4], [-3, 18], [-11, -2], [1, -6], [20, 8], [6, -12], [-1, 20], [10, 12], [-5, 18], [-1, 2], [2, 20], [-23, 18], [-2, -20], [24, 0], [-13, 14], [15, -24], [-17, 24], [-32, 4], [-7, -12], [8, 8], [17, 2], [-5, -8], [23, 6], [-11, 6], [-24, 12], [9, 6], [-23, 22], [19, 4], [-7, -18], [29, -6], [32, -16], [-31, 2], [25, -26], [-22, 0], [-5, 18], [-11, 18], [-2, 4], [-17, 0], [16, -24], [-1, 8], [4, -24], [17, 2], [11, -4], [4, -8], [-3, 6], [10, -12], [23, -16], [7, 4], [13, 24], [10, -20], [21, 18], [15, 6], [0, -24], [-19, -6], [-21, 24], [-20, -8], [19, -18], [-14, -24], [-15, 30], [-46, 0], [5, 26], [-1, -12], [22, -8], [3, 12], [-24, 24], [-35, -2], [5, 18], [12, -24], [-25, 0], [-20, 16], [-6, 36], [31, -20], [11, 6], [-11, -8], [-3, -18], [-2, 28], [20, 0], [-17, -2], [9, 6], [24, -36], [-9, 18], [41, -10], [6, 12], [-47, -2], [29, -4], [10, -32], [-36, 12], [-21, 6], [25, -14], [-1, -4], [21, 6], [7, 12], [30, -12], [35, -28], [35, -4], [38, -36], [27, -30], [17, -30], [1, 30], [-10, 8], [16, 0], [17, 26], [-20, -24], [-5, 24], [29, -12], [-12, 0], [-14, 36], [41, 2], [43, -32], [-5, 16], [-25, 0], [35, -36], [-23, 42], [-53, 4], [11, 0], [8, 20], [31, 18], [-11, -14], [-35, -2], [15, 12], [-23, -6], [-45, 12], [-30, 12], [7, -2], [17, 2], [-4, 24], [-19, -6], [23, -36], [16, -8], [-33, 12], [1, -12], [-25, -22], [10, -20], [-31, 6], [43, 12], [38, -40], [-4, 0], [-15, 24], [-53, 4], [17, 6], [-7, -18], [-13, -16], [37, -44], [-54, 24], [7, -24], [24, -12], [1, 10], [-27, 48], [20, -4], [-42, 24], [-13, -28], [25, 10], [-9, -18], [17, -16], [3, 12], [9, -6], [4, -24], [-1, 32], [-36, 24], [-37, 20], [21, -18], [-7, -10], [-44, 28], [-52, 20], [39, -48], [-37, -18], [13, -54], [13, 18], [-10, 8], [-41, 46], [-4, 12], [-25, -4], [31, -12], [-8, -32], [34, -12], [37, -14], [-25, 20], [74, 0], [-13, -28], [21, -24], [-1, 48], [3, 12], [-18, 48], [-7, 18], [73, -6], [28, -32], [-31, 30], [22, -12], [-31, 50], [55, -18], [46, -32], [-30, 36], [16, 12], [-5, -8], [-25, 24], [23, -48], [45, 6], [-13, 18], [-62, 12], [-43, -12], [-39, 72], [65, -6], [-39, 6], [3, 6], [-27, 30], [27, -48], [-24, -12], [14, 24], [-23, 48], [19, -26], [35, 12], [30, -12], [55, 0], [26, -4], [31, -8], [17, 26], [-7, 50], [-52, 20], [-50, 24], [11, -40], [-61, 6], [70, -8], [-41, 22], [44, -12], [51, -24], [5, 26], [-3, -42], [-48, 0], [15, 6], [2, -36], [-71, 10], [-18, 36], [61, -32], [19, -48], [3, 42], [57, -30], [-25, -40], [-27, -6], [-20, 4], [-53, 0], [-16, 44], [-53, 28], [-38, 60], [10, -32], [37, -18], [13, -30], [-26, 16], [40, -36], [-55, 38], [-5, 48], [-23, 36], [1, 0], [35, -34], [74, -16], [41, -18], [-65, 28], [-16, 36], [-1, -4], [-25, 0], [-24, 24], [22, 12], [53, 14], [46, 0], [11, 18], [-11, -6], [34, -32], [6, -36], [32, 0], [-3, -6], [36, 24], [-5, 48], [21, 12], [4, -56], [-57, 48], [-7, -24], [9, -18], [-58, 44], [1, -18], [-73, 12], [30, -36], [-83, 4], [-25, -16], [43, 16], [0, -24], [34, 36], [55, -44], [-3, -6], [-79, 8], [-11, 18], [-1, 24], [-50, 4], [-37, 12], [70, -36], [49, -38], [-81, -6], [-19, -40], [-54, 60], [13, -18], [3, 48], [19, -20], [8, 8], [5, 42], [27, -48], [-14, 24], [65, 14], [94, 4], [-36, 0], [-47, -2], [44, -48], [-69, 12], [-5, -24], [-29, 12], [-43, 0], [28, -56], [-46, 60], [41, -22], [43, 16], [11, -6], [33, 36], [-17, -12], [-69, 24], [-6, 0], [12, -12], [57, -42], [57, 6], [-1, 32], [-43, -36], [-31, 38], [-71, 10], [-21, -18], [9, 18], [-46, -16], [43, -24], [-79, 20], [67, -48], [-5, -44], [-28, -40], [75, 18], [-58, 24], [47, -4], [21, -36], [-7, -30], [10, 4], [-21, 66], [27, 24], [49, -30], [-20, -8], [-16, -40], [10, 48], [41, -22], [-29, -2], [42, -24], [-53, 60], [-47, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8281_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8281_2_a_z();".
// 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_8281_2_a_z(:prec:=2)
    chi := MakeCharacter_8281_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_8281_2_a_z();".
// 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_8281_2_a_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8281_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -3, 1]>,<3,R![-5, 0, 1]>,<5,R![-5, 0, 1]>,<11,R![-3, 1]>,<17,R![-11, 6, 1]>],Snew);
    return Vf;
end function;