
// Make newform 1216.2.a.s in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1216_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_1216_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_1216_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 := [-4, -1, 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_1216_a();"
function MakeCharacter_1216_a()
    N := 1216;
    order := 1;
    char_gens := [191, 837, 705];
    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_1216_a_Hecke(Kf)
    return MakeCharacter_1216_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, 0], [0, -1], [1, -1], [3, 0], [1, 1], [0, -1], [-3, -2], [-1, 0], [4, -1], [0, 3], [6, -2], [-4, 2], [4, 0], [-7, -1], [1, 3], [6, 1], [6, 1], [-7, 5], [2, 1], [2, 4], [-3, 4], [10, 0], [8, -6], [-6, 6], [4, -2], [-6, -4], [2, -8], [-6, 7], [2, -1], [8, 2], [12, -2], [15, 1], [5, -6], [-7, -1], [-9, 1], [-10, 8], [14, 4], [-4, 8], [0, 0], [14, -4], [4, 4], [-6, -4], [-1, 0], [6, -8], [-2, 6], [-5, 12], [-14, 1], [8, 6], [-12, 3], [11, -1], [-9, 7], [15, 0], [-8, 4], [-1, 3], [20, -4], [-3, -9], [6, -8], [-4, -9], [7, -3], [-10, -6], [-27, 1], [-20, -3], [-14, -6], [-3, 2], [10, 7], [-18, -1], [-6, -1], [-8, 8], [3, -1], [-9, 3], [-18, 7], [31, 2], [-20, 0], [4, -7], [-2, 9], [18, -4], [-19, 5], [7, 1], [14, -6], [22, 2], [20, -4], [-6, 9], [-28, 2], [24, -10], [8, 4], [3, 3], [-12, 2], [-33, 2], [23, -7], [1, -9], [-13, 3], [-8, 4], [6, 8], [20, -8], [-1, -3], [-4, -1], [4, -10], [14, -6], [6, -17], [37, 1], [-12, 8], [7, -7], [24, -4], [-8, 12], [8, 0], [-1, -4], [23, -7], [14, -12], [22, -14], [-14, -6], [-16, -2], [15, -17], [17, -3], [44, -2], [-25, 13], [-4, -18], [-7, 13], [47, 0], [-1, 5], [6, -17], [4, -11], [-18, 2], [4, 1], [24, 4], [31, -9], [2, 10], [-38, 8], [11, -8], [13, -6], [6, 6], [-3, -1], [26, 2], [-12, 0], [-7, 9], [-19, 14], [3, -6], [18, -9], [-24, -1], [-2, -3], [3, -2], [-12, -15], [-9, -3], [11, -6], [-18, 1], [8, 17], [14, -10], [-26, 4], [22, 6], [25, -7], [32, 6], [2, 3], [45, -3], [-11, -7], [-6, 20], [14, -1], [-6, 2], [-4, 13], [-38, 11], [-19, -4], [2, 13], [-12, 16], [-2, 4], [-24, 0], [36, 8], [-38, 10], [-4, -10], [-12, 4], [11, -23], [-14, 2], [19, 9], [20, -3], [-6, 5], [20, -20], [-46, 1], [-4, 8], [15, -21], [-4, 4], [-33, -9], [8, -16], [5, 11], [5, 4], [16, 9], [-6, -12], [18, 2], [3, -4], [19, -13], [14, -5], [22, -14], [32, -10], [-32, 3], [4, -6], [-7, -13], [18, -10], [20, -3], [-44, -8], [-16, 12], [13, -12], [-2, -6], [-14, -11], [-7, 24], [12, 14], [-4, 16], [6, 17], [36, -16], [-52, 6], [-29, 5], [-27, 6], [-32, -9], [-31, 4], [26, -15], [34, 7], [6, 14], [-5, 13], [52, -7], [28, 4], [28, -2], [-12, 19], [12, -6], [-10, -8], [9, -11], [46, -4], [-52, -2], [-56, -6], [33, -18], [-50, 15], [9, -7], [-10, 20], [14, 16], [22, 4], [-3, 13], [61, -7], [50, 1], [-18, 14], [-10, 16], [-21, 3], [-53, 10], [-11, -11], [5, 7], [12, -10], [34, -6], [34, -22], [-21, 23], [4, 13], [38, 9], [20, 16], [5, 11], [1, -13], [-6, -22], [-40, -4], [1, -14], [18, -28], [35, 8], [8, -25], [-10, 12], [21, 11], [-36, 6], [3, 11], [-12, 24], [8, -6], [-62, -4], [48, 8], [26, -6], [5, 11], [26, -7], [-13, -5], [-14, 21], [6, -13], [-37, 0], [26, 11], [29, -15], [10, 5], [60, -11], [33, -32], [39, -22], [-14, 8], [4, -15], [-19, -7], [66, -3], [58, 6], [9, -5], [18, -16], [20, 11], [1, -26], [2, -15], [-15, 23], [35, -23], [11, 22], [16, 14], [35, -10], [-24, -6], [-38, 0], [-31, 15], [8, 26], [-18, -2], [-70, 4], [-26, 6], [8, -34], [1, 17], [34, -5], [23, 7], [45, 13], [22, -17], [27, -24], [-6, 33], [-21, -19], [-22, 20], [-24, -1], [-28, 9], [-1, 13], [6, -22], [-77, 1], [-34, 4], [-36, 14], [-12, 8], [-40, 16], [-54, -10], [19, -9], [42, -20], [-41, 4], [5, 21], [16, 5], [-3, -30], [-50, -3], [14, 20], [14, 21], [18, 8], [68, -13], [-20, -19], [-4, 0], [-21, 5], [-15, 15], [26, 19], [-52, 3], [9, -7], [5, 5], [23, -3], [14, -41], [20, -16], [7, -8], [56, 13], [-26, 7], [65, -2], [60, -17], [30, -24], [-42, 17], [20, 0], [22, -8], [12, -7], [-24, 10], [-41, 17], [-44, 12], [12, -6], [53, 7], [38, -18], [-42, 0], [-20, -8], [-12, -13], [-32, -8], [21, 10], [-44, -6], [-17, 21], [-10, 0], [-48, 10], [-26, 17], [31, 7], [-24, 8], [-6, 40], [58, 16], [-74, 4], [44, -6], [8, -11], [-19, -18], [-8, -5], [3, -3], [-9, 21], [-48, 13], [16, -24], [-73, 3], [-29, 10], [36, 0], [14, 24], [-57, -1], [-27, 8], [15, -5], [-28, -1], [-56, -18], [48, 17], [-30, 14], [-57, 19], [-56, 8], [6, 32], [6, -17], [76, -14], [3, 19], [14, 2], [22, 12], [-66, 4], [-32, 0], [-80, 4], [-18, 6], [30, -12], [-53, 19], [8, 10], [50, -18], [-50, 12], [-19, 26], [-5, -21], [-28, -8], [24, -15], [-29, -15], [-66, 18], [-43, -15], [-62, 25], [-77, 11], [-58, 11], [-6, -26], [68, 6], [14, -42], [-18, 33], [-30, 34], [58, 12], [-56, 9], [-43, 0], [-68, 9], [-12, -14], [-32, -11], [48, 3], [-17, 12], [-31, -17], [57, -10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1216_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1216_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_1216_2_a_s(:prec:=2)
    chi := MakeCharacter_1216_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_1216_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_1216_2_a_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1216_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-4, 1, 1]>,<5,R![-4, -1, 1]>,<7,R![-3, 1]>],Snew);
    return Vf;
end function;