
// Make newform 1089.2.a.k in Magma, downloaded from the LMFDB on 16 April 2024.
// 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_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_1089_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_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)
    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+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 := [[2], [0], [-4], [1], [0], [-2], [-4], [-3], [-2], [-6], [-5], [3], [2], [12], [-2], [-6], [10], [3], [-1], [0], [-11], [11], [-6], [-12], [5], [10], [-7], [18], [1], [-6], [-13], [6], [-8], [16], [16], [-16], [-1], [25], [18], [-24], [-6], [-23], [-8], [-5], [8], [-21], [-21], [-17], [0], [-18], [-18], [-6], [-14], [2], [14], [10], [14], [8], [-11], [12], [-11], [-12], [-19], [-24], [-10], [20], [-11], [5], [2], [15], [12], [-4], [-8], [7], [16], [-26], [-18], [31], [28], [21], [-26], [-2], [-18], [-17], [37], [-4], [-20], [18], [6], [16], [-24], [22], [-40], [14], [23], [32], [-6], [6], [29], [-2], [20], [8], [-28], [-12], [-25], [15], [-4], [46], [8], [-1], [8], [-13], [24], [-4], [-32], [24], [-37], [4], [-10], [-46], [-5], [-13], [-12], [34], [11], [-50], [26], [6], [-12], [-30], [41], [20], [19], [5], [24], [11], [-36], [4], [10], [48], [-17], [38], [-27], [10], [-11], [-16], [11], [4], [-45], [-30], [-45], [-2], [49], [22], [33], [-36], [-5], [18], [23], [-42], [-54], [-34], [13], [-2], [42], [-6], [4], [-49], [26], [0], [-20], [19], [-50], [38], [-16], [58], [23], [-26], [44], [43], [53], [-32], [54], [-42], [-30], [60], [57], [15], [2], [-32], [5], [14], [-21], [-30], [48], [42], [-58], [61], [12], [22], [42], [21], [33], [-11], [-20], [-16], [-63], [-24], [-36], [55], [-21], [48], [21], [38], [-24], [-34], [11], [6], [12], [-18], [-11], [-12], [24], [23], [-18], [26], [36], [12], [-8], [-52], [-26], [-47], [-59], [-50], [37], [-22], [9], [-24], [44], [14], [8], [68], [-3], [15], [26], [-6], [32], [-56], [-60], [-46], [-47], [28], [-48], [41], [-30], [44], [22], [12], [-72], [-53], [16], [-42], [29], [-81], [-60], [-35], [-66], [-6], [-13], [-72], [50], [-45], [-11], [-4], [53], [-67], [48], [-62], [-65], [72], [-46], [47], [52], [-58], [13], [-60], [-39], [42], [1], [26], [-44], [-56], [18], [-12], [66], [58], [-11], [44], [-66], [35], [75], [-18], [-41], [56], [-40], [-59], [-66], [9], [-78], [15], [42], [-16], [-24], [-76], [-48], [41], [46], [8], [37], [-2], [-53], [67], [-50], [89], [56], [-31], [-12], [-43], [-60], [-28], [-35], [0], [-13], [26], [-12], [84], [-29], [8], [49], [29], [2], [48], [42], [23], [10], [-16], [-95], [-1], [84], [-6], [56], [9], [-18], [24], [-65], [-24], [12], [64], [52], [40], [-37], [60], [60], [-40], [-33], [-26], [18], [92], [15], [72], [-21], [14], [-4], [-80], [25], [-68], [-80], [47], [-90], [45], [68], [-6], [35], [-50], [13], [-24], [0], [74], [-76], [16], [-79], [-94], [-80], [-35], [90], [-26], [-79], [-84], [73], [-62], [14], [2], [60], [48], [-4], [-48], [81], [62], [-76], [58], [-18], [32], [-76], [89], [-22], [-48], [76], [-63], [-62], [-12], [-74], [43], [78], [-34], [55], [6], [-42], [26], [25], [-70]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1089_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1089_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_1089_2_a_k(: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_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_1089_2_a_k( : 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![4, 1]>,<7,R![-1, 1]>],Snew);
    return Vf;
end function;