
// Make newform 1386.2.a.n in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1386_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_1386_2_a_n();".
// 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_1386_2_a_n();".
// 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
        poly := [-10, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_1386_a();"
function MakeCharacter_1386_a()
    N := 1386;
    order := 1;
    char_gens := [155, 199, 1135];
    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_1386_a_Hecke(Kf)
    return MakeCharacter_1386_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 := [[-1, 0], [0, 0], [0, 1], [-1, 0], [1, 0], [2, 0], [-4, -1], [2, 1], [0, 2], [-4, 0], [6, 1], [2, 2], [4, 1], [0, -2], [-2, -3], [2, -2], [8, -2], [10, 0], [-2, 2], [8, 0], [4, -3], [2, -2], [2, 1], [-10, -2], [2, -4], [6, 0], [-2, 5], [8, 2], [-8, 0], [-2, 0], [2, 6], [-2, -1], [18, 0], [2, -3], [-2, 0], [8, 4], [4, -3], [-4, -4], [0, 2], [6, 2], [10, -2], [8, 3], [12, -2], [-2, 4], [-24, 0], [10, 3], [-12, -2], [-2, -3], [2, -3], [-16, -3], [-6, 0], [0, -4], [4, -1], [20, -2], [-14, 2], [-6, -6], [-8, -1], [0, -2], [-24, 0], [-10, -4], [-6, -3], [-6, -6], [-18, -1], [14, 5], [14, 2], [6, -6], [10, -6], [-26, 2], [8, 2], [-10, 6], [-14, 2], [-10, 2], [2, -1], [-10, 0], [12, -8], [6, 1], [-6, 0], [12, 3], [2, -8], [24, 1], [4, 4], [-2, -12], [-10, 6], [-34, 0], [0, -2], [2, -2], [6, 0], [30, -4], [-2, 8], [-16, -4], [-8, 8], [-4, 8], [-8, -2], [4, -4], [2, -6], [24, -6], [-24, -5], [-14, -2], [18, -7], [6, 8], [-16, -8], [8, 8], [2, -7], [-14, -4], [0, 4], [22, -2], [12, -6], [-12, 9], [-12, 0], [16, 1], [-8, 4], [-6, -8], [-8, -4], [36, -4], [4, -10], [0, -12], [16, 6], [6, -3], [18, 8], [-4, -6], [0, -7], [-18, 10], [2, 2], [6, 6], [12, -6], [18, 0], [34, 2], [-26, 1], [-18, -7], [18, -4], [16, -4], [-16, 4], [12, -2], [-2, 10], [0, 3], [16, 3], [28, -7], [-22, 9], [-12, 1], [-50, 2], [2, -3], [2, 16], [-28, 6], [4, -12], [16, -9], [6, 9], [6, -4], [-40, 5], [-24, -2], [-24, 2], [-36, 4], [6, 0], [12, -8], [-16, 0], [22, 10], [16, 2], [8, -8], [-2, -10], [-20, -7], [34, -2], [-28, 8], [-14, 6], [-32, -8], [20, -4], [12, 4], [-26, -7], [-36, -8], [30, 6], [30, 4], [56, -1], [-4, -14], [20, 3], [-8, 4], [14, -10], [6, -7], [14, 12], [-8, 16], [10, 0], [44, -2], [-30, -2], [-28, -8], [6, 7], [-2, -18], [40, -5], [-8, 4], [-8, 5], [-38, 0], [-8, 8], [6, -10], [12, -10], [-6, -2], [52, -2], [-6, -10], [-20, 7], [16, 4], [30, -6], [42, 2], [-38, -2], [4, 5], [28, 2], [-34, 0], [56, 2], [-12, 7], [-4, 15], [40, -4], [-8, -19], [14, 9], [16, 8], [6, 18], [-32, 4], [-38, -2], [28, 7], [-16, -12], [-16, 14], [-24, 6], [-26, -6], [-6, -10], [24, -3], [-8, 8], [14, -4], [-34, 4], [40, -6], [-6, 12], [36, 4], [42, -3], [4, -12], [-26, 4], [36, 0], [-36, 10], [8, -10], [6, 2], [-18, -1], [48, -8], [38, -2], [-4, -8], [-24, 6], [14, 16], [-14, -16], [-4, 4], [-26, -3], [26, 10], [-10, -1], [14, -11], [26, 0], [56, 1], [12, 6], [-66, 1], [-16, -6], [24, 12], [34, -10], [-22, -8], [4, 15], [-12, -4], [22, -12], [-2, 6], [44, -2], [-30, 14], [-6, -11], [48, 7], [12, -1], [-16, 8], [28, 4], [14, 4], [46, -2], [62, 4], [36, 6], [-26, -10], [-2, -20], [40, 0], [24, -8], [-28, 17], [-12, 4], [-10, -2], [26, -6], [52, 3], [46, 1], [-12, 0], [62, 0], [30, -8], [-74, -1], [-60, 0], [36, -4], [12, 0], [34, -6], [2, 17], [-48, 6], [-8, 12], [-34, 12], [-10, -11], [-48, 9], [-38, 2], [12, 16], [54, 0], [14, -1], [-8, 0], [18, 4], [-10, -11], [-32, 7], [18, 13], [34, -7], [48, -1], [-26, -8], [50, -6], [26, -2], [-36, 8], [24, 8], [52, 4], [0, 13], [8, -8], [16, 8], [-12, 24], [-34, -4], [26, -4], [14, -2], [24, 12], [-12, 9], [-36, -10], [-54, 6], [-18, -6], [-6, -24], [-58, 3], [-34, 0], [-18, 2], [40, 6], [-46, -2], [-26, 2], [-4, -12], [0, -12], [-6, 6], [78, 0], [-10, 16], [38, -16], [-40, -6], [6, -17], [28, 0], [48, 12], [-10, -10], [-8, -25], [2, -20], [-28, -8], [-42, -4], [-48, 12], [-34, -10], [84, -2], [-14, 20], [12, 12], [-40, -15], [52, 12], [-24, -12], [-12, 3], [10, -7], [-52, 12], [-6, -10], [-56, -8], [28, 8], [-48, -9], [74, 3], [-18, -13], [-14, -2], [-48, -12], [12, -16], [-16, -3], [-32, -20], [48, 12], [4, 16], [-2, -4], [-2, -14], [-26, 6], [-18, -4], [28, 8], [-20, 16], [-10, 18], [-40, -16], [16, 4], [10, -4], [-24, 0], [-70, 2], [-32, 13], [76, 3], [20, 13], [12, -23], [-12, -4], [-50, -2], [74, 2], [46, 3], [-66, 5], [-16, 6], [-32, 5], [32, 14], [26, -13], [-26, 12], [-52, -9], [20, 0], [-60, 2], [-4, -8], [26, -2], [24, 20], [2, 20], [26, 18], [-30, -8], [-66, 12], [48, 0], [16, 20], [-30, 12], [2, 10], [40, -6], [-2, 0], [-18, 6], [-16, 12], [-32, 16], [36, 21], [28, 8], [4, -16], [-64, 8], [-42, -2], [76, -7], [-74, 6], [30, 5], [-38, 16], [-24, 4], [-58, 12], [-42, 12], [44, 0], [-58, 3], [-30, -10], [4, -19], [86, -2], [30, -12], [8, -12], [-20, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1386_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1386_2_a_n();".
// 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_1386_2_a_n(:prec:=2)
    chi := MakeCharacter_1386_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_1386_2_a_n();".
// 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_1386_2_a_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1386_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-10, 0, 1]>,<13,R![-2, 1]>,<17,R![6, 8, 1]>],Snew);
    return Vf;
end function;