
// Make newform 1089.2.a.b in Magma, downloaded from the LMFDB on 15 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1089_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_1089_2_a_b();".
// 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_1089_2_a_b();".
// 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_1089_a();"
function MakeCharacter_1089_a()
    N := 1089;
    order := 1;
    char_gens := [848, 244];
    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_1089_a_Hecke( : Kf := Kf)
    return MakeCharacter_1089_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 := [[-2], [0], [-1], [2], [0], [-4], [-2], [0], [1], [0], [7], [3], [-8], [6], [-8], [6], [-5], [-12], [-7], [3], [-4], [10], [-6], [-15], [-7], [2], [-16], [18], [-10], [-9], [-8], [-18], [7], [-10], [-10], [-2], [-7], [4], [-12], [-6], [15], [7], [-17], [-4], [-2], [0], [-12], [19], [18], [15], [24], [-30], [8], [23], [2], [14], [-10], [28], [2], [-18], [-4], [24], [-8], [-12], [-1], [-13], [7], [22], [28], [-30], [21], [-20], [-17], [26], [-5], [1], [15], [-2], [-2], [30], [-20], [22], [-18], [-11], [-40], [11], [-35], [12], [12], [-11], [27], [20], [23], [-8], [20], [-26], [-15], [3], [16], [8], [-8], [-2], [4], [0], [28], [33], [-28], [44], [-40], [-2], [22], [16], [-18], [-25], [7], [33], [29], [7], [41], [10], [37], [-14], [-42], [16], [17], [2], [-25], [-15], [3], [36], [-50], [4], [-23], [-22], [12], [-20], [6], [32], [-53], [0], [38], [22], [39], [-52], [25], [5], [-14], [8], [-15], [-24], [12], [43], [4], [-22], [-12], [-12], [-10], [30], [-8], [42], [27], [34], [32], [-47], [27], [-39], [-8], [-38], [10], [-39], [-10], [22], [32], [16], [5], [55], [-2], [13], [-44], [20], [8], [-58], [-51], [-42], [51], [30], [-48], [24], [-50], [2], [-31], [34], [-3], [18], [-12], [21], [-2], [-41], [-42], [14], [60], [18], [18], [-40], [25], [47], [-15], [-36], [0], [-8], [-48], [-27], [39], [-28], [-30], [47], [-68], [12], [72], [-39], [68], [-60], [15], [29], [-12], [70], [-54], [0], [-28], [52], [-71], [20], [-22], [32], [49], [58], [-15], [-36], [-55], [-37], [41], [-32], [-36], [-15], [-56], [-60], [-52], [28], [30], [34], [32], [2], [-33], [-10], [-6], [-20], [22], [-78], [-33], [2], [-4], [48], [-50], [6], [42], [40], [45], [3], [46], [-6], [17], [-57], [34], [40], [-8], [59], [57], [-10], [-52], [12], [-56], [-43], [-52], [-62], [-28], [3], [-6], [18], [-35], [70], [-77], [52], [-36], [-18], [-54], [-40], [-23], [-79], [30], [22], [66], [-72], [20], [-4], [-13], [-17], [-63], [45], [-60], [-84], [24], [-70], [-18], [89], [48], [10], [-35], [-38], [-86], [20], [68], [73], [-58], [-91], [-26], [-13], [-45], [-1], [48], [4], [-22], [-78], [70], [-56], [48], [-93], [25], [4], [7], [-38], [29], [57], [60], [-13], [-59], [10], [67], [-37], [-48], [57], [28], [3], [18], [36], [50], [54], [75], [62], [-22], [31], [82], [42], [-3], [-50], [3], [-11], [48], [-14], [-72], [-57], [0], [34], [-20], [98], [13], [-20], [-58], [-14], [-30], [-18], [-22], [-39], [-38], [38], [-40], [-39], [72], [-7], [16], [-23], [5], [41], [55], [-17], [-87], [56], [70], [-30], [-68], [-58], [-50], [-49], [-48], [42], [-20], [-42], [-42], [52], [44], [25], [6], [-62], [-4], [-2], [82], [63], [-40], [-57], [-38], [54], [25], [-88], [72], [-50], [-86], [-3], [81], [70], [-53], [-80]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1089_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1089_2_a_b();".
// 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_1089_2_a_b(:prec:=1)
    chi := MakeCharacter_1089_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_1089_2_a_b();".
// 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_1089_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1089_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![2, 1]>,<5,R![1, 1]>,<7,R![-2, 1]>],Snew);
    return Vf;
end function;