
// Make newform 1008.2.a.k in Magma, downloaded from the LMFDB on 17 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1008_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_1008_2_a_k();".
// 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_1008_2_a_k();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_1008_a();"
function MakeCharacter_1008_a()
    N := 1008;
    order := 1;
    char_gens := [127, 757, 785, 577];
    v := [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_1008_a_Hecke( : Kf := Kf)
    return MakeCharacter_1008_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]];
    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], [2], [1], [-2], [2], [6], [4], [-6], [0], [4], [10], [2], [4], [-4], [-12], [-12], [6], [4], [14], [-2], [8], [16], [-6], [-18], [14], [4], [6], [2], [-16], [8], [-8], [-12], [-16], [12], [-24], [6], [12], [-4], [6], [10], [-14], [-26], [-26], [-12], [0], [12], [-24], [-28], [10], [-4], [-14], [-26], [-12], [26], [18], [10], [12], [-2], [12], [-28], [6], [-4], [-20], [-14], [12], [-20], [14], [-6], [-18], [-26], [6], [32], [22], [28], [24], [-8], [22], [12], [-2], [4], [-2], [-10], [10], [0], [18], [-8], [22], [-18], [16], [-4], [36], [-8], [-6], [4], [-16], [-14], [6], [-16], [-10], [36], [20], [28], [-4], [-4], [34], [12], [-26], [-18], [-22], [40], [6], [28], [-40], [-24], [12], [44], [20], [-32], [-42], [-10], [-2], [30], [-30], [-40], [48], [-6], [-32], [-28], [26], [44], [-2], [8], [-38], [-30], [26], [-10], [-32], [30], [32], [-16], [44], [-48], [-26], [-46], [36], [-10], [-30], [4], [50], [-18], [54], [20], [-20], [28], [6], [-32], [34], [34], [-6], [14], [24], [-32], [48], [-52], [24], [-8], [22], [26], [-18], [54], [22], [30], [26], [56], [6], [-28], [40], [-12], [-34], [32], [24], [-10], [6], [34], [-42], [-30], [8], [10], [-56], [14], [-34], [-52], [-42], [30], [-18], [-34], [-18], [-22], [-12], [4], [16], [42], [6], [-56], [22], [0], [-26], [24], [-60], [2], [-30], [-8], [-48], [-36], [-70], [-24], [54], [42], [-8], [-22], [44], [16], [24], [-40], [-6], [18], [-38], [20], [-46], [50], [-60], [-8], [64], [68], [-32], [62], [68], [42], [16], [34], [-32], [16], [-70], [42], [-36], [28], [-28], [-4], [-6], [34], [-66], [62], [62], [54], [30], [22], [52], [38], [-54], [32], [18], [34], [38], [-2], [-40], [20], [-58], [36], [8], [70], [-12], [14], [-40], [-18], [-20], [-26], [10], [-2], [36], [-72], [-40], [56], [38], [0], [-26], [-34], [28], [28], [-14], [-60], [24], [-20], [60], [46], [46], [-16], [-6], [4], [40], [-54], [-80], [16], [6], [-68], [-2], [26], [58], [6], [2], [-72], [-32], [68], [-4], [42], [-50], [56], [18], [-26], [68], [-44], [-26], [22], [-88], [-32], [22], [-20], [28], [-42], [-64], [50], [-32], [-76], [4], [-52], [20], [30], [18], [-50], [-44], [86], [84], [-6], [8], [40], [-82], [-14], [36], [40], [-26], [60], [18], [-68], [0], [-10], [70], [-64], [-96], [-24], [14], [-90], [-22], [86], [46], [8], [6], [18], [56], [-18], [-30], [-20], [-34], [44], [-44], [38], [-24], [-6], [-30], [30], [-74], [-90], [96], [-24], [72], [48], [24], [-88], [66], [68], [-28], [98], [-22], [30], [-92], [14], [-42], [-56], [-2], [20], [32], [94], [24], [0], [34], [-22], [32], [6], [-48], [-88], [36], [26], [-72], [-6], [84], [26], [-70], [-46], [68], [-18], [60], [96], [-50], [-18], [-40], [-54], [-82], [-62], [-60], [72], [60]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1008_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1008_2_a_k();".
// 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_1008_2_a_k(:prec:=1)
    chi := MakeCharacter_1008_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_1008_2_a_k();".
// 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_1008_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1008_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<11,R![2, 1]>,<13,R![-2, 1]>],Snew);
    return Vf;
end function;