
// Make newform 2005.2.a.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2005_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_2005_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_2005_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_2005_a();"
function MakeCharacter_2005_a()
    N := 2005;
    order := 1;
    char_gens := [402, 1206];
    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_2005_a_Hecke(Kf)
    return MakeCharacter_2005_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], [0], [1], [0], [-4], [4], [4], [-6], [8], [-6], [-2], [4], [2], [-4], [0], [-4], [-6], [14], [-12], [-10], [-6], [6], [4], [6], [0], [-14], [0], [0], [-14], [-2], [-12], [-18], [12], [-10], [-22], [-8], [-12], [-4], [-16], [2], [8], [-22], [-6], [24], [-6], [2], [-2], [16], [0], [-10], [12], [-24], [-2], [-30], [-2], [-16], [10], [26], [16], [-2], [12], [0], [20], [-16], [14], [12], [8], [-14], [-12], [22], [24], [2], [20], [-10], [-8], [16], [-30], [-22], [1], [6], [-16], [38], [38], [-26], [-6], [4], [-22], [14], [-22], [24], [-16], [-20], [32], [20], [16], [-16], [30], [-6], [12], [-26], [36], [38], [12], [10], [-10], [-18], [4], [36], [-8], [38], [-24], [28], [-20], [8], [26], [-26], [-4], [-36], [-30], [-26], [14], [-36], [42], [24], [-40], [22], [42], [-36], [28], [34], [24], [-36], [40], [26], [-18], [-2], [-42], [-32], [-38], [50], [0], [-18], [32], [-12], [-30], [-18], [-26], [-18], [16], [-36], [-32], [-18], [4], [-36], [32], [-36], [34], [-6], [32], [-46], [-36], [14], [16], [24], [-2], [-32], [-42], [22], [-26], [-28], [58], [34], [-48], [-8], [-16], [58], [26], [18], [-56], [-22], [56], [-12], [-42], [24], [-24], [2], [6], [-52], [2], [-10], [-46], [-52], [20], [14], [-52], [38], [2], [54], [58], [-24], [30], [-64], [44], [-10], [0], [52], [-54], [60], [6], [-24], [42], [10], [8], [12], [-40], [46], [-52], [-38], [0], [44], [-42], [-4], [46], [24], [52], [-34], [-16], [-30], [0], [10], [26], [-28], [46], [-18], [-44], [56], [50], [-30], [54], [0], [12], [-20], [-4], [-50], [-66], [12], [-16], [-2], [-32], [56], [-2], [-6], [-28], [-22], [2], [26], [58], [-64], [44], [64], [52], [4], [-58], [66], [82], [-10], [14], [-30], [-20], [-4], [50], [-72], [-18], [22], [-2], [-40], [52], [-58], [74], [-34], [36], [58], [48], [-38], [-4], [22], [-36], [50], [-64], [2], [-18], [28], [48], [46], [-2], [-62], [62], [6], [6], [52], [-28], [2], [58], [-36], [-70], [16], [-36], [62], [-38], [4], [-64], [-42], [-22], [44], [-8], [22], [-84], [66], [-64], [26], [-24], [64], [6], [24], [-88], [46], [-10], [20], [20], [-4], [-26], [-74], [-58], [68], [30], [72], [34], [36], [-26], [44], [-14], [-46], [42], [30], [42], [-46], [46], [-12], [52], [-42], [-44], [10], [2], [4], [-58], [8], [-64], [-72], [-14], [-24], [68], [78], [-56], [-18], [88], [40], [-28], [-44], [38], [20], [26], [-96], [-6], [-4], [70], [36], [18], [8], [-58], [-4], [30], [-68], [16], [-12], [62], [-32], [-2], [-92], [48], [88], [-14], [90], [-82], [-60], [58], [-22], [-68], [58], [-54], [-18], [-94], [12], [-40], [92], [-74], [-20], [-102], [66], [-60], [-6], [-28], [-12], [-60], [-60], [50], [-46], [64], [56], [-6], [-84], [30], [-54], [24], [-10], [86], [88], [100], [-22], [-92], [94]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2005_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2005_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_2005_2_a_a(:prec:=1)
    chi := MakeCharacter_2005_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_2005_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_2005_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2005_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 1]>,<11,R![4, 1]>],Snew);
    return Vf;
end function;