
// Make newform 8005.2.a.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8005_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_8005_2_a_a();".
// 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_8005_2_a_a();".
// 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_8005_a();"
function MakeCharacter_8005_a()
    N := 8005;
    order := 1;
    char_gens := [1602, 4806];
    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_8005_a_Hecke(Kf)
    return MakeCharacter_8005_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], [-2], [1], [2], [-6], [-2], [6], [-4], [2], [2], [-8], [2], [-6], [8], [12], [-6], [10], [-2], [-8], [10], [-2], [-8], [6], [-2], [10], [-14], [-4], [-6], [-14], [6], [-6], [-10], [14], [18], [10], [-8], [18], [4], [0], [6], [-16], [10], [-10], [22], [26], [4], [-14], [-2], [-24], [10], [18], [22], [10], [-2], [-18], [-12], [14], [6], [-10], [30], [14], [-26], [-32], [-10], [6], [-2], [6], [2], [-8], [22], [10], [-36], [34], [2], [-34], [-16], [-2], [-18], [-30], [2], [-6], [18], [-18], [-26], [8], [16], [22], [22], [34], [18], [-30], [-14], [-4], [14], [-20], [4], [-30], [-30], [-6], [-30], [-38], [-2], [12], [-30], [-32], [2], [-30], [6], [-6], [-26], [-40], [22], [-18], [-42], [-12], [34], [14], [-12], [-34], [-14], [-34], [18], [14], [-24], [-28], [-46], [-30], [-40], [26], [6], [14], [-6], [38], [38], [-22], [10], [6], [-38], [30], [2], [36], [-6], [-4], [12], [26], [0], [42], [6], [22], [-24], [-58], [6], [16], [58], [-8], [-40], [-46], [18], [14], [-30], [-38], [6], [-40], [-38], [-2], [-6], [0], [-38], [34], [30], [6], [-34], [26], [30], [36], [26], [2], [-54], [-16], [-46], [24], [-12], [-58], [30], [36], [42], [26], [-14], [34], [-52], [14], [60], [16], [-62], [-26], [-6], [2], [-18], [-42], [-12], [18], [28], [22], [42], [-48], [2], [-44], [-26], [-18], [28], [2], [-30], [18], [50], [6], [18], [6], [-30], [2], [30], [-6], [2], [-70], [60], [-18], [-50], [-26], [-32], [-8], [10], [-34], [-46], [10], [18], [-58], [22], [-6], [46], [-4], [4], [-24], [38], [40], [10], [-42], [40], [-38], [-54], [-46], [-60], [-78], [1], [6], [42], [30], [24], [-10], [-26], [22], [-34], [-8], [-16], [26], [-26], [-42], [16], [42], [18], [-4], [-26], [50], [42], [-10], [8], [78], [-8], [14], [-58], [-70], [-60], [66], [-58], [82], [-38], [-46], [-42], [14], [-42], [-14], [-78], [66], [-74], [38], [-72], [70], [10], [62], [-18], [56], [-2], [38], [-2], [-52], [-22], [56], [-2], [-64], [18], [-40], [22], [54], [18], [-62], [-30], [30], [-74], [8], [80], [50], [-10], [68], [82], [-30], [90], [38], [2], [32], [-42], [4], [14], [-34], [30], [-38], [-52], [26], [-60], [-50], [14], [66], [78], [-10], [-42], [30], [-70], [78], [36], [74], [-8], [-38], [-78], [76], [-78], [-42], [-52], [42], [-26], [0], [-28], [-18], [-76], [-66], [-10], [72], [-10], [34], [-26], [66], [18], [-62], [-54], [10], [94], [-6], [-48], [-18], [66], [-26], [66], [50], [-2], [-18], [-74], [74], [78], [48], [10], [62], [-70], [-60], [-68], [-62], [-18], [-24], [42], [-66], [86], [94], [-94], [66], [-30], [58], [-18], [46], [30], [-102], [52], [-14], [-102], [-92], [92], [50], [54], [-16], [-100], [58], [-78], [-26], [78], [-30], [-56], [50], [-74], [-42], [-56], [-10], [78], [26], [-54], [18], [-18]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8005_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8005_2_a_a();".
// 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_8005_2_a_a(:prec:=1)
    chi := MakeCharacter_8005_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_8005_2_a_a();".
// 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_8005_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8005_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 1]>,<3,R![2, 1]>],Snew);
    return Vf;
end function;