
// Make newform 3381.2.a.t in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3381_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_3381_2_a_t();".
// 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_3381_2_a_t();".
// 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 := [-1, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [-1, 2]];
    Rf_basisdens := [1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_3381_a();"
function MakeCharacter_3381_a()
    N := 3381;
    order := 1;
    char_gens := [2255, 346, 442];
    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_3381_a_Hecke(Kf)
    return MakeCharacter_3381_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 := [[0, -1], [1, 0], [1, -1], [0, 0], [4, 0], [0, 2], [5, -1], [-5, -1], [1, 0], [0, 2], [2, 2], [0, 2], [2, 4], [1, -3], [4, 0], [-3, -1], [-4, 4], [0, -2], [3, -1], [-8, 0], [2, -4], [3, 3], [-4, 0], [-1, 1], [-4, -2], [-2, 4], [9, 1], [8, 4], [-6, -4], [5, -5], [6, -2], [-8, -4], [1, -1], [0, 4], [-5, -3], [-16, 0], [8, -2], [-6, 6], [-4, -4], [0, -2], [12, 4], [-6, 8], [6, -2], [4, 6], [4, 6], [-3, -3], [18, -2], [8, 8], [18, -2], [6, -4], [-14, 0], [-4, 4], [-24, 2], [6, 2], [22, 0], [2, 2], [-4, 2], [22, -2], [-2, 8], [9, 7], [1, -3], [-1, -3], [2, -2], [-16, 4], [-14, 4], [16, -6], [10, 6], [8, -10], [8, -8], [28, -2], [4, 6], [2, 2], [-19, -3], [16, 2], [1, 13], [-30, 2], [5, 7], [2, 0], [-9, 13], [-4, -6], [4, -8], [-22, -4], [8, 0], [-10, -8], [-8, 8], [-8, -12], [6, 4], [4, 2], [-8, 6], [-4, -4], [2, -10], [16, 8], [24, 0], [40, 0], [10, -10], [28, -4], [12, 2], [31, 1], [9, -11], [-20, 2], [18, -2], [-31, -5], [-6, 6], [-13, 1], [23, 3], [-12, 2], [-8, 4], [-34, 0], [-12, 4], [20, 10], [-12, 12], [-36, -2], [-27, 3], [-11, 9], [-5, -5], [11, 17], [-9, -13], [-12, 0], [-14, -4], [14, 10], [16, -2], [16, -6], [5, 19], [-20, 0], [18, -10], [23, 1], [-6, -12], [-4, 0], [-17, 7], [-22, 0], [0, -12], [-2, -10], [21, 13], [22, 8], [-20, -10], [10, -12], [-43, 3], [7, 3], [3, -7], [-12, -14], [10, 6], [30, -8], [-30, 2], [22, -6], [12, -10], [26, -6], [-22, 0], [34, 4], [16, -4], [-4, 0], [42, -4], [7, -15], [6, -6], [-40, 4], [19, -9], [4, -20], [15, -17], [2, 12], [-14, 12], [9, 15], [-28, 12], [-17, 13], [-30, 10], [14, -14], [-3, -13], [4, -12], [-6, 2], [-20, -10], [28, 10], [-14, 0], [10, -18], [-10, -12], [46, 6], [-20, -6], [-2, -10], [-11, -9], [-36, 0], [-10, 8], [45, -3], [-24, -10], [-10, -10], [38, 2], [22, -16], [0, -14], [-12, 12], [15, -3], [-50, 0], [-13, -1], [14, 4], [-32, -12], [-26, -8], [-28, -8], [29, 1], [-2, -4], [-4, -8], [29, -1], [2, 8], [-22, 4], [1, -21], [4, 12], [-11, -1], [14, -18], [2, 8], [16, 2], [-8, -4], [-8, -18], [-9, -9], [4, -12], [30, 4], [10, 6], [34, 0], [-2, 12], [-43, 13], [-28, -16], [-16, -24], [34, 4], [28, 4], [44, 6], [48, 8], [-46, -4], [-24, -18], [19, -5], [-56, -2], [-13, -5], [28, 0], [44, 10], [3, 5], [-16, 4], [-29, 3], [24, 4], [4, -6], [-15, -11], [33, -7], [-40, -2], [21, 9], [52, -4], [-16, -14], [17, -13], [40, 4], [-24, 12], [-46, -10], [2, 14], [22, 6], [-20, 2], [30, 8], [36, -4], [30, -18], [4, 8], [7, 11], [12, 12], [42, -4], [21, 15], [10, -30], [52, -10], [-10, -20], [0, 20], [2, -28], [9, 13], [-44, 10], [-44, 10], [11, -13], [32, 20], [-22, 16], [-20, -14], [6, 0], [13, -23], [-43, -1], [-15, 19], [-41, 11], [-8, -14], [30, -20], [9, 13], [-12, 30], [21, -27], [-38, -4], [-8, 24], [0, -8], [-18, -16], [-50, -12], [0, 12], [24, -12], [-37, -13], [18, -14], [-36, -22], [6, 10], [56, 4], [24, -2], [-63, 3], [-8, -16], [48, -6], [7, 17], [44, 8], [24, 22], [-22, 14], [20, 18], [-7, -1], [-33, 23], [-34, -12], [-28, 8], [-12, 8], [30, 16], [-39, -13], [-31, -15], [24, 0], [-7, -3], [-44, -10], [32, 20], [14, -20], [-50, -2], [44, -6], [12, -8], [-35, -1], [-39, 15], [-16, 12], [12, -12], [-64, 2], [-56, -4], [12, 8], [20, -2], [-2, -12], [-21, 23], [14, -28], [-18, 12], [-48, -8], [-41, -11], [-8, -6], [77, 1], [44, 16], [-28, -20], [53, -9], [38, 8], [22, -8], [-12, 20], [-4, 40], [59, -1], [48, 0], [12, -26], [31, -23], [-4, 18], [-43, -11], [52, -14], [3, -15], [30, -4], [-59, 5], [59, -11], [48, -12], [-32, 26], [54, 2], [-20, -28], [-21, 9], [-8, -28], [-28, 6], [-20, -2], [9, 25], [-4, 30], [-16, -30], [-18, -2], [2, 6], [8, -10], [-48, -4], [-60, -2], [20, 22], [12, 20], [-6, 30], [36, -16], [-54, 16], [-62, -8], [-21, 3], [50, 0], [-16, 0], [10, 6], [-32, 8], [19, 21], [-71, -7], [30, 16], [44, 16], [24, -8], [6, 16], [15, -7], [-12, -6], [3, -7], [-19, 11], [-6, -6], [-22, -8], [23, 3], [48, 0], [46, -18], [-22, 16], [59, -9], [0, 32], [-60, -6], [64, -2], [-36, 28], [-54, -2], [-48, -8], [4, 18], [-33, -25], [-25, -7], [-55, -19], [6, 24], [-54, -8], [-4, -30], [27, -21], [-59, -9], [46, 20], [0, -16], [-62, 12], [0, 30], [-11, 9], [4, -12], [32, 6], [-58, -8], [24, -20], [11, -25], [12, 10], [-8, 22], [4, -8], [14, 14], [-9, -15], [6, -2], [85, -1], [40, 14], [4, 20], [-36, -4], [6, 32], [-34, 4], [26, 22], [58, 0], [-10, -6], [-68, 8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3381_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3381_2_a_t();".
// 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_3381_2_a_t(:prec:=2)
    chi := MakeCharacter_3381_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_3381_2_a_t();".
// 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_3381_2_a_t( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3381_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-5, 0, 1]>,<5,R![-4, -2, 1]>,<11,R![-4, 1]>,<13,R![-20, 0, 1]>],Snew);
    return Vf;
end function;