
// Make newform 1369.2.a.a in Magma, downloaded from the LMFDB on 06 October 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1369_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_1369_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_1369_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_1369_a();"
function MakeCharacter_1369_a()
    N := 1369;
    order := 1;
    char_gens := [2];
    v := [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_1369_a_Hecke( : Kf := Kf)
    return MakeCharacter_1369_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], [1], [-2], [3], [3], [-6], [2], [6], [4], [4], [0], [0], [3], [-6], [3], [9], [-4], [0], [12], [-3], [-9], [6], [9], [14], [12], [3], [6], [-12], [6], [-4], [-7], [10], [18], [0], [15], [8], [3], [6], [2], [21], [-16], [-3], [20], [-6], [3], [24], [-13], [19], [-8], [-5], [6], [16], [0], [-10], [-8], [-9], [-30], [-3], [-12], [-20], [-24], [-6], [7], [-10], [-6], [-18], [0], [-13], [-32], [30], [-14], [-15], [8], [21], [15], [-16], [16], [-33], [10], [24], [15], [-30], [-10], [9], [24], [21], [26], [-18], [10], [-36], [-28], [-4], [-18], [12], [-36], [14], [-15], [3], [-6], [0], [-12], [28], [-26], [44], [-33], [42], [-22], [-21], [45], [-18], [18], [-29], [-3], [35], [30], [-33], [-24], [-2], [-44], [45], [-30], [-21], [-3], [-46], [-12], [20], [6], [15], [18], [31], [-25], [-39], [-27], [12], [33], [24], [39], [23], [8], [44], [-33], [-3], [4], [-8], [24], [0], [-24], [22], [6], [24], [18], [18], [36], [57], [12], [50], [-6], [30], [-47], [-18], [-38], [-39], [-48], [12], [-32], [-39], [-30], [-48], [25], [-14], [14], [-30], [-20], [19], [45], [-4], [30], [18], [-29], [-10], [-12], [20], [36], [-52], [-24], [-15], [-47], [-36], [66], [63], [54], [-36], [60], [57], [-33], [9], [-30], [-54], [63], [-4], [26], [37], [58], [15], [-60], [32], [-60], [36], [-26], [-33], [18], [-10], [54], [48], [-46], [-33], [18], [50], [-68], [6], [-33], [-5], [15], [-6], [-3], [6], [6], [-45], [27], [-70], [-21], [-65], [23], [-3], [1], [33], [-50], [4], [-34], [-70], [44], [30], [-21], [-6], [-54], [56], [12], [-50], [-45], [16], [-18], [-18], [-12], [-6], [-26], [-45], [-27], [12], [-27], [-12], [24], [-62], [-50], [-34], [-52], [45], [-15], [-70], [-9], [-4], [-30], [42], [36], [-6], [-22], [-16], [12], [6], [43], [40], [39], [60], [-12], [53], [72], [3], [24], [42], [-6], [34], [20], [-22], [-51], [-3], [9], [-45], [-27], [-6], [64], [63], [66], [-63], [15], [34], [-30], [12], [-2], [54], [75], [6], [81], [-15], [57], [-69], [8], [66], [45], [40], [-19], [14], [-60], [67], [40], [24], [9], [-30], [-20], [-24], [28], [6], [-3], [22], [-54], [-26], [-30], [-12], [-15], [-21], [-30], [-48], [1], [-33], [34], [-60], [-14], [46], [57], [-27], [10], [3], [68], [43], [50], [-24], [50], [66], [64], [-10], [3], [-14], [48], [-27], [82], [26], [47], [36], [52], [36], [90], [-70], [6], [-84], [-30], [90], [-3], [-60], [87], [66], [-94], [37], [40], [-16], [78], [42], [-66], [81], [-53], [-18], [-6], [-98], [-45], [76], [16], [-48], [-63], [-6], [-66], [-75], [23], [-42], [-15], [76], [-42], [28], [-56], [47], [72], [-78], [-84], [75], [-69], [-93], [-94], [60], [-48], [-78], [60], [-87], [42], [104], [46], [-12], [-48], [45], [-34], [3], [-84], [45], [-92], [-46]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1369_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1369_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_1369_2_a_a(:prec:=1)
    chi := MakeCharacter_1369_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_1369_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_1369_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1369_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![2, 1]>,<3,R![-1, 1]>],Snew);
    return Vf;
end function;