
// Make newform 1088.2.a.k in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1088_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_1088_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_1088_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_1088_a();"
function MakeCharacter_1088_a()
    N := 1088;
    order := 1;
    char_gens := [511, 69, 513];
    v := [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_1088_a_Hecke(Kf)
    return MakeCharacter_1088_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 := [[0], [2], [-2], [-2], [-2], [-2], [1], [-4], [-2], [-2], [10], [-10], [2], [-4], [0], [-6], [-4], [-2], [16], [10], [-6], [6], [4], [6], [-14], [6], [-16], [18], [6], [-6], [20], [-22], [22], [2], [2], [-4], [10], [-18], [-6], [-18], [4], [-10], [16], [-22], [-18], [-6], [-10], [-12], [22], [6], [18], [-24], [10], [8], [6], [-12], [-10], [-8], [-26], [26], [14], [18], [24], [18], [-14], [-10], [-4], [18], [22], [18], [-30], [-12], [-26], [6], [2], [20], [-26], [-34], [10], [10], [-2], [22], [6], [-26], [-22], [-32], [-30], [-18], [-30], [-8], [-12], [10], [-10], [12], [-2], [-34], [42], [-14], [-24], [-34], [-14], [-18], [36], [-6], [-2], [14], [-28], [34], [-40], [-38], [-14], [2], [-30], [-10], [-28], [34], [34], [24], [30], [-48], [10], [-14], [14], [-6], [-10], [-2], [38], [-6], [32], [-14], [52], [26], [2], [-10], [22], [30], [6], [2], [18], [10], [26], [14], [-26], [-10], [-22], [-42], [14], [18], [-20], [24], [-42], [18], [-48], [-30], [-2], [-26], [24], [-6], [-38], [6], [18], [-10], [36], [-12], [2], [-54], [-54], [14], [-46], [22], [24], [46], [14], [-10], [44], [42], [-14], [6], [12], [46], [-64], [-54], [38], [-38], [60], [6], [-2], [24], [58], [-14], [34], [-34], [-4], [62], [-54], [-54], [26], [-26], [-46], [0], [-2], [30], [2], [-26], [56], [-2], [8], [4], [-38], [8], [34], [54], [-18], [60], [54], [2], [16], [-46], [-10], [-34], [34], [-30], [2], [-38], [-16], [22], [-54], [30], [28], [54], [-46], [34], [28], [30], [-40], [60], [42], [6], [-22], [52], [-42], [56], [32], [-18], [58], [66], [-38], [6], [52], [36], [-70], [-54], [-12], [-22], [-18], [24], [-50], [50], [30], [46], [66], [56], [46], [-34], [-30], [0], [-30], [-22], [46], [-46], [30], [-16], [-38], [-52], [18], [-28], [-12], [46], [38], [-36], [-64], [50], [-50], [74], [-54], [-32], [-30], [-26], [-28], [-26], [-2], [18], [30], [-42], [30], [-10], [72], [-26], [6], [4], [54], [-18], [-50], [10], [34], [2], [-48], [62], [48], [2], [38], [-66], [-78], [36], [-88], [-70], [36], [66], [26], [18], [38], [18], [42], [64], [-14], [66], [-54], [-26], [34], [-82], [62], [46], [90], [-56], [14], [-42], [50], [42], [-62], [12], [90], [-54], [-58], [64], [-82], [-42], [30], [-32], [66], [6], [-4], [10], [74], [-14], [-10], [86], [-92], [-14], [-22], [36], [-26], [-30], [-56], [86], [12], [-54], [-2], [-16], [-22], [36], [14], [-66], [-42], [40], [-50], [-30], [38], [86], [-30], [26], [-74], [26], [-22], [-54], [-58], [-50], [-100], [-74], [18], [48], [26], [-50], [-48], [0], [-28], [98], [88], [-70], [70], [-42], [-2], [46], [-24], [82], [-94], [50], [-50], [78], [-76], [42], [66], [22], [-32], [-38], [-38], [-66], [46], [-62], [34], [64], [2], [-74], [2], [-84], [2], [42], [-30], [90], [-8], [-66]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1088_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1088_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_1088_2_a_k(:prec:=1)
    chi := MakeCharacter_1088_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_1088_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_1088_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1088_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 1]>,<5,R![2, 1]>,<7,R![2, 1]>],Snew);
    return Vf;
end function;