
// Make newform 1911.2.a.s in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1911_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_1911_2_a_s();".
// 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_1911_2_a_s();".
// 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 := [6, 4, -6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [-3, 0, 1, 0], [-1, -4, 0, 1], [3, 2, -1, 0]];
    Rf_basisdens := [1, 1, 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_1911_a();"
function MakeCharacter_1911_a()
    N := 1911;
    order := 1;
    char_gens := [638, 1522, 1471];
    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_1911_a_Hecke(Kf)
    return MakeCharacter_1911_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, 0, 0], [-1, 0, 0, 0], [1, 0, 1, 0], [0, 0, 0, 0], [0, -1, 0, -1], [-1, 0, 0, 0], [0, 2, 0, 0], [-1, -1, 1, -1], [1, -1, 1, 1], [1, -1, 1, -1], [-1, 3, 1, -1], [4, -2, 2, -2], [4, -1, -2, -1], [1, -1, -1, -1], [-1, -2, -1, 0], [1, 1, 1, 1], [6, -3, 0, -1], [-2, -4, 0, 0], [-6, 2, -2, -2], [2, -3, 2, -3], [1, 5, -3, 1], [1, 5, -1, 1], [-1, 2, -1, 0], [3, -2, 3, -2], [5, -3, 1, 1], [10, -2, -2, 0], [-8, 2, 0, 2], [4, 0, -2, 2], [2, -6, 0, -2], [-1, 1, -1, 1], [8, 2, 0, 2], [0, -2, 0, 2], [-2, 5, -2, -1], [-6, 0, 2, 0], [-4, -1, -4, 1], [0, -2, 4, 2], [4, -2, 6, -2], [-4, -2, 0, -2], [-15, 0, -3, 2], [4, 0, 4, 2], [1, -7, 1, -1], [2, -4, 4, 0], [12, -2, 6, -4], [2, -4, 0, 4], [-8, -7, 4, -1], [4, -4, 0, 0], [1, -3, -1, -3], [-3, -3, -1, -3], [2, -3, -4, -1], [6, 0, -4, 0], [-3, -5, -3, -1], [6, 5, -6, 1], [-3, 5, -7, 1], [2, -4, 2, -4], [14, 2, 2, 0], [11, 1, -1, 3], [12, 0, 0, -2], [8, 4, 4, 0], [11, 3, 3, -1], [-12, 7, 0, 1], [-8, 8, 0, 4], [5, -2, 5, -2], [1, 3, 3, -1], [10, 0, -2, 0], [10, 0, 0, -4], [-2, 11, -2, 1], [-14, 2, 2, 2], [11, -5, 3, -5], [4, 6, -2, 0], [-3, -5, 5, -1], [16, 3, -2, -1], [-6, 1, -6, 1], [-2, 0, 6, 4], [-2, 0, -4, 4], [-2, -4, 2, 0], [-14, 1, 4, -1], [6, 0, 0, 0], [-3, -13, 5, -1], [14, -1, 2, 5], [1, -3, -3, 1], [2, -4, 2, 4], [12, -8, -2, -4], [-4, 13, -4, 1], [-4, -10, -2, 2], [-14, 0, -6, 4], [-7, 3, 5, -3], [10, -5, 10, -3], [0, 0, -2, -4], [0, -3, 6, -3], [6, -6, -2, 6], [-6, 8, -6, 0], [-3, 10, -3, 0], [-10, 12, 6, 0], [20, 0, 2, -2], [-24, 12, 4, 4], [-22, 0, 2, -4], [17, 0, -7, 0], [16, 4, 4, -2], [-6, -4, 2, -4], [12, -12, -2, 0], [-1, 1, -3, 5], [18, 1, 6, 3], [0, -8, 0, 4], [-3, -5, -3, -1], [-3, -3, 7, 1], [-2, -6, 0, -6], [-9, 2, 3, -4], [9, -4, 9, 0], [-15, -1, -3, 5], [10, 0, 0, -4], [-2, -4, 6, 0], [0, 14, -2, -2], [-12, 11, 0, 5], [12, 0, -4, 4], [-2, 8, 2, 0], [23, 3, -1, 3], [0, 4, -4, 4], [-18, -14, 6, -2], [22, 2, 4, -2], [7, 3, 7, 5], [-15, 3, -7, 7], [11, -5, -9, -1], [22, 4, -2, 2], [2, 5, 2, -7], [-19, 3, -5, 3], [-5, -1, -5, -5], [-2, 2, -4, 6], [2, 16, 2, -4], [6, -10, 2, -2], [17, -3, 1, 1], [-34, 2, -6, -2], [22, -3, -2, -3], [17, -9, 3, 7], [-29, 3, -1, 3], [-23, 2, 1, -2], [11, 9, -5, -3], [8, -13, 2, -1], [3, -7, 5, 1], [-2, 8, -2, 6], [-13, 1, -13, 1], [16, -2, -12, -2], [0, 7, 0, 1], [-8, 12, 8, 4], [14, -7, 2, 1], [-2, -4, 12, 4], [-2, 17, 4, -5], [-5, 1, -9, -3], [14, 10, 2, 0], [-2, 6, -6, -6], [18, -9, -6, -1], [12, 10, 10, -6], [-6, 2, -6, -4], [0, -10, 16, 2], [-8, 4, 4, 0], [-3, 9, -5, 9], [15, -9, 3, 1], [-4, 10, 0, -2], [5, 2, 5, 6], [16, -10, -6, -6], [-13, 8, 11, 4], [-16, 13, -4, 1], [19, -3, -5, 5], [18, 4, -2, -4], [2, -2, 2, 6], [-20, 1, 4, -1], [17, -2, 5, 0], [-4, -20, 0, 0], [-32, 6, -6, -2], [6, -6, -8, -2], [20, 16, -4, 2], [-36, 1, 0, -3], [10, 2, 0, 10], [-21, 17, 3, -1], [-8, 8, -10, -4], [8, 14, 4, 2], [0, -6, 0, 0], [-16, 16, 0, 4], [-4, 11, 8, 1], [-26, 0, -6, 0], [-6, 12, -4, 8], [0, -8, 4, -4], [26, -2, 2, 6], [-10, -4, -12, 0], [-11, 4, -11, -8], [-12, -1, 12, 3], [30, -4, 6, -6], [-3, 9, 1, 5], [-9, -1, 5, -5], [-10, 10, 0, -2], [13, -8, 13, -10], [-36, 2, -10, 6], [4, 3, 4, -5], [-15, 11, -17, 3], [5, -6, 5, 2], [-21, -1, 3, 1], [-16, -16, -4, 2], [-18, 4, -20, 4], [15, 3, -5, 3], [-13, 2, -13, -6], [-14, 10, -14, 6], [-2, 7, -2, 13], [-12, -6, 8, -2], [21, 5, 5, -7], [-12, 18, -10, 14], [-13, 8, -1, 2], [16, 6, 4, 4], [19, 7, 9, -1], [-11, 3, -11, 9], [-18, 3, -18, 5], [-10, 6, 10, 2], [-15, -3, -11, -3], [0, -10, 0, 0], [12, -6, -8, 6], [-18, 19, 0, 5], [3, 8, -9, -2], [-1, 5, 7, -7], [17, -3, 3, -11], [-28, 2, -4, 0], [-14, -15, -2, 5], [-26, -11, -2, -5], [-9, -17, 7, -5], [16, 0, 12, 4], [30, 7, 6, -7], [2, -4, -6, -8], [-6, -12, 6, 0], [-11, 21, -7, 9], [-16, 2, -4, -4], [-9, -3, 3, -9], [-18, -6, 2, -6], [16, -7, 16, 1], [-6, -12, 4, -12], [12, 10, -4, -2], [8, 4, 0, -4], [6, -12, -12, 4], [-14, -4, 6, 0], [-57, -6, 3, 0], [22, 0, 0, 0], [6, -3, -6, -1], [-1, 11, -13, 1], [-10, 14, 2, 6], [22, 9, -2, 5], [-10, -18, -2, 6], [0, 16, -4, -4], [14, 16, 0, 8], [-17, 6, 7, -2], [20, 6, -4, 6], [1, 15, 3, 7], [9, 6, -3, -4], [-8, 10, -4, 10], [31, -3, -5, 7], [-8, 8, 2, 12], [0, -13, 6, 7], [-40, -3, 8, 1], [38, 8, -8, 0], [-2, 18, 10, -4], [-6, -19, 6, -3], [13, -15, 5, 9], [-32, -14, 0, -2], [-40, 10, -4, 0], [5, 7, 1, -5], [9, -7, -17, -3], [11, -25, -1, -3], [-15, 9, -7, 5], [2, 4, 16, 4], [-12, 1, 6, 9], [-30, 2, 2, 2], [4, -3, 16, 3], [-5, -12, -5, 12], [2, -6, 6, -2], [-2, -10, 4, -14], [14, 8, -8, 4], [-16, 8, 12, 0], [-14, -12, 12, 4], [-5, 1, 17, 1], [4, 2, -6, -6], [-24, -8, -4, -12], [22, -13, -2, -1], [28, -2, 2, -2], [12, -18, -2, -2], [12, 2, 0, 10], [-34, 18, 2, 10], [22, 2, 14, -10], [4, -24, 4, -8], [-19, 23, -11, 11], [-44, 0, -12, 4], [-23, 13, 1, 7], [-7, 13, -3, 13], [-10, 13, 2, -9], [28, -14, -12, -6], [-20, 16, 4, -10], [29, 17, 5, -7], [-8, 12, 4, 12], [16, -7, 16, -1], [29, -26, -7, -12], [15, -1, 7, -5], [28, -22, -8, -8], [-34, 12, -2, -8], [4, -22, 4, -4], [-12, -16, 12, 4], [-17, 9, 9, 1], [22, -12, 0, 4], [2, 3, -10, -3], [-23, 11, -1, 7], [-15, -3, -3, -9], [-17, 1, 5, 5], [-30, 14, 4, 2], [32, -18, 2, -8], [-16, 14, -2, -6], [-28, -7, -4, -3], [-11, 13, 5, -3], [8, 6, -4, -6], [-40, 5, -16, 15], [2, -2, -16, -10], [3, -1, -11, -1], [14, 1, 2, 5], [4, -14, -6, 2], [-7, -6, -7, 4], [-12, 5, 0, 13], [43, -17, 3, -13], [3, -1, 3, -1], [-52, 10, 4, 6], [-48, -2, -14, 6], [2, -16, 2, 6], [8, -14, 12, -2], [-14, 7, 10, 9], [8, -18, -2, -2], [20, -6, 0, 2], [-19, 7, -17, 3], [32, -6, 2, -8], [-6, -10, 0, 10], [38, -6, 12, -2], [13, -25, 13, -17], [-6, 2, -10, -6], [31, -10, -5, 8], [-10, -12, 6, -12], [13, -8, -11, 2], [-20, 2, 2, -6], [-28, 7, -10, 7], [42, -12, -4, 4], [-2, -24, 18, 0], [-20, 2, -22, 10], [-22, -2, 8, -2], [20, 3, 2, 3], [-32, -8, -4, -12], [10, -13, -2, -3], [-20, 6, -14, -4], [14, -12, 4, -4], [12, 2, 4, 14], [13, -8, -11, -2], [18, 8, -6, -2], [15, 17, -19, -3], [-42, 2, -8, -2], [2, 1, -10, 3], [18, -18, 2, 2], [-42, 0, 4, 4], [6, -10, -18, -8], [23, -14, 11, 0], [11, 4, -1, 6], [54, 14, 0, 2], [16, -15, -8, -3], [-6, 20, 4, -4], [2, 12, 2, 6], [69, -1, 9, -3], [-22, 15, 2, 3], [-34, -8, -2, 0], [37, 9, 5, 5], [-27, 8, -27, 8], [42, -20, -2, -4], [-3, -7, 13, -11], [20, -8, 2, 14], [-38, 0, -18, 4], [-26, -19, -2, 5], [-19, 1, 5, 1], [4, -8, -12, 8], [-31, -7, -3, 5], [-2, -5, -8, -3], [4, 2, 10, 4], [-38, -2, -12, -2], [16, 10, -8, 0], [2, -4, -8, 8], [-47, 18, 1, 2], [0, 3, 0, 1], [-16, -12, 0, -4], [14, -25, 14, 1], [5, 9, 7, -3], [27, -4, 3, 14], [4, 4, -4, -12], [-28, -2, -2, -2], [16, -14, -4, -6], [16, 10, -8, 10], [46, -10, 8, 2], [15, 8, 15, -4], [30, 17, -6, 1], [-20, 24, -12, 20], [-38, 5, -14, -3], [-69, 3, -5, 3], [-16, -20, 16, -4], [-22, 16, -10, 14], [31, -1, 5, -5], [-6, -1, -18, 1], [-37, 9, -17, 5], [5, -27, 5, -7], [38, 6, 6, 10], [-29, 22, -5, 14], [-11, -20, -11, 4], [-16, 16, -20, -4], [-4, -24, 6, -4], [10, -9, 10, -7], [-7, -13, 7, -5], [3, -18, -9, -16], [-4, 6, 10, 6], [-1, -13, -1, -1], [23, 15, -13, -3], [12, -28, 4, -8], [13, 13, -19, 9], [-40, -22, 8, -4], [-6, 13, -18, 1], [0, 28, 8, -4], [-24, 5, -6, 5], [0, -22, 0, -6], [5, 11, -19, -5], [26, 14, -8, -2], [42, -11, -6, -3], [4, 10, 4, 14], [22, 2, 0, 10], [-39, -6, 9, -6], [-55, 23, 5, 9], [2, -9, 14, -11], [37, -11, -9, 1], [42, -26, -6, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1911_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1911_2_a_s();".
// 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_1911_2_a_s(:prec:=4)
    chi := MakeCharacter_1911_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_1911_2_a_s();".
// 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_1911_2_a_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1911_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![6, 5, -7, -1, 1]>,<5,R![24, 20, -10, -3, 1]>],Snew);
    return Vf;
end function;