
// Make newform 8281.2.a.bd in Magma, downloaded from the LMFDB on 18 May 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_bd();".
// 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_bd();".
// 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, -2, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-2, 0, 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_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], [0, 0, 0], [0, 0, 0], [0, 0, 0], [4, -2, 1], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-3, 7, -5], [-4, 1, 4], [0, 0, 0], [-1, 9, -5], [0, 0, 0], [-2, 7, 2], [0, 0, 0], [7, 1, 3], [0, 0, 0], [0, 0, 0], [5, 7, -1], [7, -11, 9], [0, 0, 0], [12, -6, 13], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-4, 2, -13], [-8, 4, -15], [16, -8, 13], [-6, -5, 6], [0, 0, 0], [16, -8, 17], [0, 0, 0], [4, 9, -4], [-2, -11, 2], [0, 0, 0], [-6, -7, 6], [0, 0, 0], [0, 0, 0], [-20, 10, -13], [0, 0, 0], [10, -9, -10], [8, 5, -8], [26, 0, 0], [0, 0, 0], [11, 9, 1], [0, 0, 0], [0, 0, 0], [0, 0, 0], [8, -15, -8], [10, 3, -10], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-4, 2, 13], [0, 0, 0], [0, 0, 0], [24, -12, 13], [26, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-8, 4, 11], [19, -15, 17], [-8, -11, 8], [5, 23, -9], [0, 0, 0], [0, 0, 0], [-3, -25, 11], [0, 0, 0], [-5, 29, -17], [-14, 1, 14], [0, 0, 0], [8, -4, -13], [0, 0, 0], [8, -21, -8], [0, 0, 0], [0, 0, 0], [26, 0, 0], [-20, 10, -31], [0, 0, 0], [0, 0, 0], [14, 3, -14], [9, 23, -7], [-32, 16, -21], [0, 0, 0], [12, -6, 29], [0, 0, 0], [0, 0, 0], [28, -14, 33], [-23, 19, -21], [-10, 23, 10], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-23, -1, -11], [-12, 6, 13], [25, -17, 21], [0, 0, 0], [-32, 16, -13], [18, -11, -18], [0, 0, 0], [0, 0, 0], [0, 0, 0], [14, -23, -14], [0, 0, 0], [0, 0, 0], [12, -25, -12], [26, 0, 0], [0, 0, 0], [-36, 18, -35], [-27, 11, -19], [0, 0, 0], [0, 0, 0], [4, -27, -4], [-10, -17, 10], [0, 0, 0], [32, -16, 39], [0, 0, 0], [52, 0, 0], [0, 0, 0], [9, 31, -11], [-20, 7, 20], [0, 0, 0], [0, 0, 0], [0, 0, 0], [52, 0, 0], [14, -27, -14], [-20, 10, -39], [4, 25, -4], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [11, -43, 27], [0, 0, 0], [21, 19, 1], [-25, -11, -7], [29, -1, 15], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-44, 22, -37], [24, -12, -7], [0, 0, 0], [-44, 22, -39], [0, 0, 0], [-29, 33, -31], [-19, -25, 3], [31, -3, 17], [0, 0, 0], [0, 0, 0], [0, 0, 0], [1, 43, -21], [-25, 41, -33], [-11, 47, -29], [0, 0, 0], [-32, 16, -47], [0, 0, 0], [22, 1, -22], [0, 0, 0], [24, -11, -24], [0, 0, 0], [52, 0, 0], [0, 0, 0], [2, -33, -2], [7, 41, -17], [0, 0, 0], [0, 0, 0], [-18, 31, 18], [40, -20, 49], [0, 0, 0], [0, 0, 0], [12, -6, -23], [0, 0, 0], [-12, -23, 12], [0, 0, 0], [27, 17, 5], [0, 0, 0], [-8, 4, -39], [0, 0, 0], [48, -24, 25], [0, 0, 0], [0, 0, 0], [-33, 37, -35], [-7, 51, -29], [0, 0, 0], [4, 0, 0], [0, 0, 0], [-37, 21, -29], [24, -12, -13], [0, 0, 0], [0, 0, 0], [-20, 33, 20], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-20, 0, 0], [32, -16, -5], [0, 0, 0], [29, 19, 5], [0, 0, 0], [4, -2, 39], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-24, 0, 0], [0, 0, 0], [14, 25, -14], [-28, 15, 28], [56, -28, 39], [0, 0, 0], [-16, -23, 16], [22, -35, -22], [0, 0, 0], [17, -57, 37], [0, 0, 0], [32, 0, 0], [0, 0, 0], [-2, -37, 2], [-23, -33, 5], [0, 0, 0], [20, -10, -21], [8, -41, -8], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-56, 28, -53], [28, -14, -13], [0, 0, 0], [52, 0, 0], [0, 0, 0], [0, 0, 0], [-78, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-30, 17, 30], [-40, 0, 0], [56, -28, 27], [0, 0, 0], [2, -41, -2], [0, 0, 0], [0, 0, 0], [-43, 23, -33], [-31, 55, -43], [0, 0, 0], [0, 0, 0], [0, 0, 0], [43, -31, 37], [44, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [4, -43, -4], [0, 0, 0], [36, -18, 61], [-28, 33, 28], [0, 0, 0], [-44, 22, -63], [0, 0, 0], [-48, 0, 0], [0, 0, 0], [0, 0, 0], [28, -14, 59], [-40, 20, 3], [32, -19, -32], [0, 0, 0], [0, 0, 0], [1, -61, 31], [0, 0, 0], [0, 0, 0], [0, 0, 0], [44, -22, 65], [-64, 32, -39], [8, -4, -37], [0, 0, 0], [0, 0, 0], [56, -28, 65], [0, 0, 0], [-5, -59, 27], [0, 0, 0], [47, -23, 35], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [28, 11, -28], [6, -47, -6], [45, -1, 23], [-36, 18, -65], [31, 33, -1], [-34, 15, 34], [0, 0, 0], [-34, 21, 34], [8, -4, -39], [0, 0, 0], [47, -7, 27], [-64, 32, -65], [60, 0, 0], [68, -34, 43], [0, 0, 0], [0, 0, 0], [-10, 49, 10], [0, 0, 0], [-32, 16, -65], [0, 0, 0], [-41, 57, -49], [0, 0, 0], [25, -69, 47], [-64, 32, -29], [0, 0, 0], [-12, 6, -55], [0, 0, 0], [64, 0, 0], [-56, 28, -13], [-78, 0, 0], [45, -53, 49], [0, 0, 0], [0, 0, 0], [52, 0, 0], [0, 0, 0], [-17, -55, 19], [0, 0, 0], [0, 0, 0], [0, 0, 0], [36, -23, -36], [-16, -35, 16], [0, 0, 0], [49, -45, 47], [-45, -11, -17], [-68, 0, 0], [0, 0, 0], [-68, 34, -69], [0, 0, 0], [0, 0, 0], [0, 0, 0], [48, -24, -1], [-31, 71, -51], [0, 0, 0], [72, -36, 65], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-9, 73, -41], [-37, -31, -3], [26, 0, 0], [0, 0, 0], [-22, 51, 22], [-14, 53, 14], [0, 0, 0], [37, -69, 53], [0, 0, 0], [36, -18, -17], [43, 21, 11], [-21, -55, 17], [0, 0, 0], [23, 53, -15], [-40, 20, 13], [0, 0, 0], [-36, 35, 36], [0, 0, 0], [76, -38, 65], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-32, 45, 32], [-55, 27, -41], [16, -55, -16], [0, 0, 0], [0, 0, 0], [-76, 38, -45], [0, 0, 0], [37, -73, 55], [0, 0, 0], [48, -24, 77], [0, 0, 0], [52, 0, 0], [0, 0, 0], [104, 0, 0], [-16, 8, 39], [0, 0, 0], [0, 0, 0], [19, -79, 49], [-56, 28, -79], [0, 0, 0], [-32, 47, 32], [104, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [72, -36, 31], [80, -40, 59], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-4, -51, 4], [-57, 29, -43], [-76, 38, -39], [-55, 7, -31], [0, 0, 0], [104, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-20, 57, 20], [0, 0, 0], [-38, -1, 38], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-44, 22, 13], [26, 0, 0], [0, 0, 0], [0, 0, 0]];
    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_bd();".
// 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_bd(:prec:=3)
    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_bd();".
// 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_bd( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8281_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-13, -4, 3, 1]>,<3,R![0, 1]>,<5,R![0, 1]>,<11,R![-13, 20, -9, 1]>,<17,R![0, 1]>],Snew);
    return Vf;
end function;