
// Make newform 1088.2.a.a in Magma, downloaded from the LMFDB on 18 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_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_1088_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_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], [-4], [-4], [-2], [-2], [-1], [-8], [-8], [-4], [-4], [8], [-2], [4], [0], [6], [-4], [-8], [-4], [-8], [-6], [0], [4], [-6], [-2], [-18], [-8], [-6], [12], [-6], [16], [2], [-10], [-14], [-14], [-8], [10], [-6], [12], [-12], [-8], [8], [-8], [-10], [0], [24], [22], [24], [-14], [-6], [-6], [-24], [-26], [-28], [6], [0], [4], [8], [4], [22], [-26], [-30], [0], [12], [22], [-32], [28], [-6], [34], [6], [6], [0], [8], [6], [-26], [8], [26], [-16], [-34], [22], [10], [-2], [0], [-14], [-32], [4], [30], [-6], [6], [-16], [-24], [16], [16], [-24], [14], [-16], [-6], [-10], [-36], [-16], [26], [-18], [12], [18], [26], [14], [-16], [2], [-16], [-38], [20], [26], [42], [10], [-32], [14], [14], [24], [36], [0], [-50], [-26], [16], [-42], [22], [-34], [-4], [24], [-32], [-26], [-16], [-52], [16], [50], [-10], [-18], [-30], [-38], [6], [38], [-2], [-32], [-40], [14], [2], [12], [20], [6], [-4], [0], [12], [-30], [36], [-48], [14], [-20], [0], [30], [22], [-24], [42], [-14], [-48], [-12], [-38], [24], [24], [-16], [-34], [-16], [0], [-2], [-4], [-22], [-56], [-30], [62], [-48], [-24], [-14], [-8], [42], [-40], [-22], [-48], [30], [-8], [24], [-2], [-20], [-38], [-46], [-32], [10], [-18], [-54], [-22], [-8], [-62], [24], [-64], [0], [-22], [-26], [-40], [26], [-56], [-56], [50], [40], [22], [18], [36], [48], [60], [38], [56], [-14], [20], [22], [-14], [-36], [46], [44], [20], [22], [-30], [24], [8], [18], [38], [38], [-40], [-6], [4], [48], [-66], [24], [50], [16], [-30], [40], [-32], [-54], [-22], [-12], [-52], [18], [-4], [24], [2], [42], [0], [-46], [18], [60], [-32], [46], [72], [70], [12], [-16], [-32], [-4], [78], [-60], [30], [-50], [2], [-38], [60], [64], [58], [-56], [78], [-8], [-36], [-50], [-70], [-60], [32], [-68], [28], [-58], [-54], [-8], [30], [8], [64], [50], [44], [-42], [78], [30], [24], [4], [-24], [-34], [18], [-52], [66], [42], [68], [-2], [-10], [-34], [-72], [56], [-24], [-46], [8], [-60], [-18], [-24], [8], [-10], [0], [12], [-70], [6], [-86], [-66], [42], [-64], [14], [-6], [-78], [-82], [-56], [40], [-76], [-16], [0], [-8], [-14], [-42], [-10], [18], [34], [72], [-42], [66], [-56], [8], [-2], [-6], [-48], [-28], [-36], [24], [76], [22], [-38], [-4], [-22], [-26], [40], [70], [-74], [-72], [-8], [90], [-32], [-46], [48], [6], [44], [-56], [14], [12], [-50], [-24], [6], [-40], [76], [6], [-52], [-10], [78], [74], [92], [70], [16], [66], [10], [28], [-80], [-38], [-42], [72], [50], [56], [-96], [-72], [-88], [-82], [56], [-26], [38], [6], [16], [62], [-24], [-82], [-86], [52], [-14], [78], [4], [54], [-6], [-22], [64], [50], [94], [0], [-32], [68], [-58], [-56], [-2], [28], [-64], [0], [86], [18], [-30], [42], [20], [-36]];
    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_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_1088_2_a_a(: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_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_1088_2_a_a( : 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![4, 1]>,<7,R![4, 1]>],Snew);
    return Vf;
end function;