
// Make newform 1386.2.a.f in Magma, downloaded from the LMFDB on 20 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_f();".
// 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_f();".
// 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_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], [-2], [-1], [-1], [-4], [0], [4], [-4], [-2], [-10], [-6], [0], [-4], [-10], [14], [-10], [-8], [8], [4], [4], [16], [-4], [-10], [6], [-12], [2], [12], [-14], [14], [-16], [-8], [-6], [20], [-22], [16], [10], [24], [8], [-4], [-12], [14], [-8], [-6], [18], [-14], [-4], [-14], [-8], [-10], [-6], [8], [8], [26], [-2], [24], [-14], [-28], [6], [-30], [0], [0], [16], [6], [-6], [18], [-20], [-34], [12], [32], [-2], [0], [18], [-34], [8], [-14], [18], [18], [-10], [-4], [30], [10], [16], [-10], [-28], [-4], [-6], [-38], [-32], [12], [-14], [-12], [12], [28], [-16], [-12], [-38], [42], [16], [-2], [4], [-30], [-12], [-14], [-28], [-42], [42], [-12], [12], [-24], [-24], [2], [-38], [-2], [8], [18], [22], [-6], [-46], [-20], [-38], [-10], [-12], [-12], [42], [6], [-10], [-6], [46], [-8], [-52], [-16], [20], [30], [12], [4], [34], [20], [-26], [-26], [-40], [50], [8], [20], [-14], [-30], [4], [-12], [14], [-40], [-22], [42], [-4], [28], [48], [4], [-56], [10], [-52], [0], [-12], [-34], [-24], [-30], [22], [-26], [-4], [36], [26], [18], [-52], [2], [-24], [6], [-14], [-38], [12], [-10], [4], [20], [40], [4], [-2], [48], [-24], [-38], [22], [-34], [-2], [-16], [38], [60], [40], [54], [-12], [6], [22], [18], [4], [4], [-14], [24], [-34], [-28], [10], [22], [50], [4], [30], [18], [-18], [-6], [8], [54], [-12], [-14], [-56], [36], [52], [66], [58], [28], [18], [-16], [0], [34], [18], [48], [44], [-36], [-14], [-8], [8], [62], [22], [60], [-26], [-14], [8], [26], [-16], [36], [-2], [2], [-40], [-72], [-10], [46], [60], [-56], [22], [-28], [-28], [-62], [56], [-4], [10], [-52], [-74], [-32], [0], [-60], [-8], [-60], [14], [30], [22], [66], [4], [-74], [42], [32], [46], [-40], [76], [-6], [-60], [-34], [-70], [-20], [-48], [4], [-36], [-20], [36], [36], [-2], [34], [-38], [0], [-50], [10], [-6], [60], [-74], [-32], [-74], [18], [4], [68], [76], [-42], [-88], [32], [64], [-50], [12], [-22], [32], [62], [24], [18], [18], [76], [0], [60], [-70], [-24], [6], [-54], [-16], [50], [-16], [-8], [-82], [-22], [-16], [-66], [-24], [-18], [-50], [30], [-40], [4], [-20], [-30], [-30], [-80], [58], [52], [22], [-54], [36], [56], [-14], [-60], [-34], [-8], [-72], [-54], [-28], [-34], [-46], [36], [-10], [72], [22], [-64], [54], [0], [-50], [88], [72], [-60], [-66], [-10], [60], [-32], [10], [-32], [-6], [88], [90], [24], [50], [-6], [48], [24], [-64], [-8], [-2], [-46], [-48], [-66], [28], [14], [-16], [98], [84], [30], [-50], [-62], [16], [46], [-16], [46], [-20], [54], [-40], [-2], [0], [78], [32], [-42], [-4], [40], [46], [-90], [-94], [6], [12], [-2], [-68], [-84], [38], [-50], [-64], [-74], [102], [16], [-62], [-96], [52], [40], [-14], [-38], [56], [-6], [98], [36]];
    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_f();".
// 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_f(:prec:=1)
    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_f();".
// 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_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1386_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<13,R![4, 1]>,<17,R![0, 1]>],Snew);
    return Vf;
end function;