
// Make newform 1088.2.a.g in Magma, downloaded from the LMFDB on 06 September 2025.
// 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_g();".
// 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_g();".
// 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 := 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]];
    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], [0], [2], [-4], [-2], [-1], [-4], [6], [-8], [-2], [-4], [-2], [4], [12], [6], [-4], [-4], [-4], [-6], [-6], [-10], [12], [-10], [-10], [-2], [-16], [4], [0], [10], [4], [-12], [2], [20], [18], [-4], [-14], [24], [-2], [16], [-20], [20], [12], [14], [12], [18], [-20], [-8], [-16], [-6], [-6], [-4], [6], [-28], [18], [20], [0], [28], [8], [-26], [20], [18], [12], [-2], [14], [-12], [-28], [-30], [-24], [-26], [6], [24], [-26], [-18], [16], [-4], [-14], [36], [30], [-26], [0], [-10], [30], [-34], [-38], [36], [-34], [-26], [22], [-8], [12], [-38], [22], [20], [-28], [-26], [-22], [6], [-20], [-36], [20], [6], [4], [2], [8], [-46], [12], [-14], [8], [34], [22], [-38], [-6], [24], [8], [-26], [8], [-44], [24], [-28], [-10], [-26], [-28], [16], [20], [-34], [-32], [26], [28], [14], [-44], [-10], [26], [-38], [18], [-6], [-46], [-4], [-2], [6], [-40], [52], [14], [-4], [26], [-38], [-8], [14], [-20], [-32], [32], [26], [-20], [-6], [-4], [-6], [36], [30], [22], [60], [-4], [54], [48], [12], [-38], [18], [-6], [-20], [14], [-4], [12], [46], [-14], [22], [-20], [-14], [8], [-36], [-56], [2], [-24], [12], [-52], [-38], [4], [-42], [-28], [12], [46], [-26], [-14], [24], [-44], [42], [-48], [34], [2], [28], [-22], [16], [-36], [-18], [58], [-14], [-12], [-54], [32], [44], [-62], [52], [14], [18], [-54], [36], [58], [14], [16], [-46], [-70], [-42], [-22], [54], [-2], [38], [60], [-50], [-30], [54], [52], [-48], [-2], [16], [-64], [-18], [68], [-32], [-42], [44], [-76], [-24], [-32], [-68], [-12], [-54], [10], [-26], [70], [-8], [-68], [48], [-38], [50], [20], [-14], [42], [-44], [-4], [8], [28], [58], [46], [-20], [4], [32], [38], [36], [-74], [30], [-8], [34], [48], [-4], [-38], [-8], [30], [56], [12], [-10], [-10], [4], [-20], [30], [18], [-26], [-36], [40], [-66], [-76], [-52], [54], [-40], [0], [10], [-28], [36], [-56], [-72], [-74], [40], [-52], [54], [-30], [-38], [-64], [88], [-2], [28], [28], [0], [10], [-66], [40], [-18], [36], [-48], [-26], [-20], [46], [2], [6], [80], [30], [10], [56], [-2], [90], [-60], [44], [26], [28], [-8], [52], [-6], [-4], [16], [84], [86], [66], [50], [48], [30], [-78], [92], [-64], [-34], [-48], [-60], [-20], [78], [36], [-60], [-82], [50], [38], [74], [-38], [48], [0], [62], [84], [-12], [10], [4], [-64], [-12], [6], [-24], [36], [-50], [-20], [4], [78], [-18], [36], [64], [0], [66], [34], [14], [58], [40], [70], [2], [58], [80], [58], [-92], [-22], [-24], [-64], [-14], [-60], [-28], [-44], [-20], [-50], [-52], [-42], [68], [22], [92], [-6], [48], [-66], [42], [38], [26], [-54], [-60], [-24], [18], [10], [-84], [-52], [46], [-84], [54], [-26], [-74], [20], [-18], [-56], [-22], [-4], [62], [-38], [-36], [-86], [-44], [42]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1088_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1088_2_a_g();".
// 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_g(: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_g();".
// 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_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1088_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<5,R![0, 1]>,<7,R![-2, 1]>],Snew);
    return Vf;
end function;