
// Make newform 1224.2.a.i in Magma, downloaded from the LMFDB on 03 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1224_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_1224_2_a_i();".
// 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_1224_2_a_i();".
// 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_1224_a();"
function MakeCharacter_1224_a()
    N := 1224;
    order := 1;
    char_gens := [919, 613, 137, 649];
    v := [1, 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_1224_a_Hecke( : Kf := Kf)
    return MakeCharacter_1224_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]];
    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, 0], [-2, 0], [1, -1], [-1, 1], [0, 2], [-1, 0], [-2, -2], [-1, 1], [-2, 0], [-1, 1], [-2, -4], [-2, 0], [-6, 2], [-4, -4], [2, 0], [-10, -2], [-2, 4], [-12, 0], [-7, -1], [6, 4], [5, 3], [-6, 2], [12, -2], [2, 0], [0, 6], [-8, 0], [-3, 3], [10, 0], [-6, -4], [-2, 2], [1, -9], [12, -2], [-9, 1], [-6, 0], [10, -2], [6, -8], [-11, -5], [13, 3], [-2, 0], [-10, -2], [10, 0], [4, 4], [6, 4], [2, -4], [11, 5], [1, -1], [10, -2], [-1, 1], [-20, -2], [6, 8], [0, -8], [-10, 4], [0, 4], [0, 10], [2, 6], [10, -4], [0, -8], [6, -12], [-2, -8], [1, -1], [10, 0], [-12, 8], [17, -1], [10, -8], [-2, 8], [-6, 2], [-6, 8], [7, 1], [6, 0], [6, -8], [-18, 2], [17, -1], [8, 2], [-17, -7], [10, 6], [12, -6], [22, -4], [-22, -4], [-6, 0], [23, 1], [-16, 10], [-13, -11], [-24, -2], [15, -7], [4, 0], [-10, 8], [24, 6], [-14, -8], [4, 12], [-10, 14], [-27, 3], [41, -1], [-6, 2], [-27, -5], [-21, -3], [2, 0], [6, 8], [-16, -4], [-30, 0], [-5, -3], [-8, 6], [2, 10], [-18, 8], [1, -1], [-4, 2], [-10, -10], [6, 8], [-20, 12], [30, -4], [-13, -3], [6, 0], [-34, 0], [17, 7], [2, 14], [-10, -8], [15, -15], [-32, 0], [-6, -12], [8, 12], [22, -8], [2, -16], [-26, 8], [21, -13], [13, 11], [8, -10], [6, 4], [-7, 7], [12, 4], [-18, -8], [-30, -6], [33, 7], [-29, -3], [12, -2], [-12, -10], [12, 2], [4, -14], [-9, -7], [2, -8], [34, 4], [7, 9], [-50, 0], [-27, 11], [23, 1], [-26, -8], [3, 13], [2, 16], [-10, 8], [10, 18], [4, 12], [-14, 8], [-10, 8], [-20, 8], [9, -9], [-39, -9], [-21, -3], [-24, -8], [18, 4], [26, 0], [-10, 16], [1, -17], [28, -2], [2, -2], [-14, 2], [-34, 0], [9, -1], [15, 1], [38, -4], [-14, 0], [30, -8], [-44, 8], [-40, -6], [-5, -3], [36, 2], [-46, -2], [-14, -12], [19, 13], [38, 0], [6, -22], [28, 6], [-24, 8], [-3, 19], [-18, 12], [14, 8], [38, 10], [32, -10], [26, -8], [12, -16], [22, -12], [5, 27], [2, -16], [15, 1], [-10, 22], [-20, 2], [-7, -1], [34, -12], [6, -12], [26, 0], [38, -8], [16, -8], [-26, -8], [1, -9], [38, 0], [0, 22], [-4, 0], [36, 10], [-36, 4], [-18, 6], [-38, -4], [4, -16], [-22, 8], [-28, 2], [21, 3], [26, 2], [-25, 1], [10, 8], [-16, 16], [-10, -24], [-17, -7], [-40, 6], [46, -8], [15, 17], [-18, -16], [19, 5], [-12, -8], [16, -22], [-30, 4], [-5, 5], [-38, -2], [-37, 5], [-18, 24], [15, 1], [-34, -6], [8, -22], [40, 4], [-14, 6], [38, 4], [10, 12], [33, 15], [-18, 2], [11, -3], [28, 8], [44, -12], [-20, -10], [-6, -4], [-15, -17], [-41, 9], [3, -11], [14, -2], [-50, -6], [-10, -8], [-30, -12], [10, -18], [-10, 4], [36, 10], [-16, -4], [6, -4], [-1, -7], [-10, -24], [-8, -10], [-17, -7], [60, -8], [-6, -8], [6, 12], [22, 24], [-20, 8], [-54, 0], [-2, 24], [21, 3], [18, -8], [22, -4], [-32, -4], [10, 16], [22, 2], [-38, 8], [-22, 22], [-38, -6], [16, 18], [-4, -6], [-18, 14], [-12, -28], [-1, 1], [-29, 21], [-42, -8], [23, -7], [-40, 8], [10, -8], [38, 16], [-18, 10], [8, 18], [-42, 8], [9, 7], [-40, 10], [-45, 13], [-54, -8], [-26, 0], [20, -4], [24, -2], [7, -15], [-2, 6], [44, -6], [-52, 2], [-67, 3], [65, -9], [7, -15], [10, -8], [32, -4], [-10, -4], [0, -24], [-18, -8], [-21, -19], [14, 16], [-34, -16], [18, -6], [20, -12], [-22, -16], [-18, -10], [-39, 15], [-42, 4], [-10, -24], [25, 23], [-30, -16], [-30, 0], [0, -32], [-10, 24], [18, -16], [27, 5], [-19, 19], [-27, 27], [-10, 24], [-66, -12], [-6, -28], [15, -15], [-28, 16], [25, -9], [-73, 1], [-26, 0], [66, 4], [-30, 0], [-2, -6], [-2, 0], [-26, 16], [10, -20], [24, 16], [32, 14], [51, -11], [-22, 8], [36, 24], [53, 11], [14, -8], [42, -6], [38, 4], [-38, 8], [-1, 9], [12, 14], [-36, -18], [62, -6], [45, 3], [10, 12], [-14, -10], [-30, 8], [-10, 8], [-8, 16], [27, -11], [22, -10], [-14, -24], [-18, -16], [12, 4], [22, 4], [18, -22], [1, -17], [-17, -7], [-16, 22], [16, 16], [46, 4], [-11, 3], [11, 13], [-56, -2], [-26, -24], [-6, 0], [10, -4], [14, 8], [11, -27], [2, -12], [-27, -13], [59, -3], [18, -18], [28, -2], [-29, -3], [-24, -8], [-42, 20], [-30, -4], [24, -12], [-16, -4], [2, -26], [34, 0], [8, -16], [30, -8], [9, -1], [-24, -18], [26, 24], [-4, 38], [-28, 28], [22, -8], [42, 16], [-45, 5], [-12, 30], [4, 30], [-58, -18], [33, -25], [2, 16], [12, -6], [-32, 28], [-41, 17], [2, 24], [6, 8], [11, 37], [-21, 13], [-58, -8], [68, 12], [26, -16], [10, -16], [-51, -13], [-46, 10], [-38, -8], [42, -8], [41, -25], [-14, -28], [24, 4], [-41, -23]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1224_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1224_2_a_i();".
// 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_1224_2_a_i(:prec:=2)
    chi := MakeCharacter_1224_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_1224_2_a_i();".
// 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_1224_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1224_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<7,R![-4, -2, 1]>],Snew);
    return Vf;
end function;