
// Make newform 2496.2.a.bg in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2496_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_2496_2_a_bg();".
// 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_2496_2_a_bg();".
// 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_2496_a();"
function MakeCharacter_2496_a()
    N := 2496;
    order := 1;
    char_gens := [703, 1093, 833, 769];
    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_2496_a_Hecke(Kf)
    return MakeCharacter_2496_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], [-1, 0], [1, 1], [-3, 1], [0, -2], [1, 0], [0, -2], [1, 1], [-2, -2], [4, -2], [-7, 1], [0, -2], [-7, 1], [2, -2], [-4, -2], [4, 2], [2, 4], [-8, -2], [-1, 3], [-10, 0], [0, 2], [0, 4], [6, 4], [-9, 3], [12, -2], [-10, 0], [-2, -6], [2, -2], [-6, 4], [2, 4], [2, 6], [-14, -2], [-7, -7], [-8, 4], [-1, -5], [3, -5], [8, 2], [-9, -1], [-8, 2], [2, -4], [-10, -2], [-12, 6], [10, 6], [-6, 0], [1, -7], [-6, -2], [0, -4], [5, 5], [12, 6], [6, 0], [-24, 2], [6, -8], [12, 6], [-16, 0], [12, 6], [-28, 0], [-4, 6], [-3, -7], [-10, -4], [-11, -3], [26, -2], [-9, 11], [-15, -3], [22, -2], [14, 8], [1, -7], [11, -1], [-14, 4], [20, 0], [12, 10], [-19, -3], [-4, 6], [-24, -4], [2, -4], [-3, 13], [8, 6], [-14, -8], [-8, -2], [15, 3], [0, 10], [10, -2], [2, -4], [8, -2], [-12, -6], [6, -6], [14, 2], [-21, 7], [32, 2], [1, 1], [25, 5], [28, -4], [-30, 0], [1, -7], [8, 8], [27, -1], [16, -4], [-27, 5], [-18, -8], [20, 8], [-2, 16], [-4, 16], [-1, -13], [2, 2], [4, 6], [-2, -6], [-2, -12], [14, -4], [11, 7], [12, 0], [-18, 8], [8, -4], [4, 2], [-11, 13], [-15, -11], [-5, 3], [0, -2], [31, -5], [-16, 0], [-16, 2], [16, 12], [24, -2], [-40, 2], [-12, -6], [-16, 6], [-13, -9], [10, -8], [22, -8], [-2, -2], [6, 10], [-10, 8], [-11, 1], [-30, -4], [30, 2], [8, 6], [13, 5], [6, -4], [-33, -5], [37, 1], [-6, 16], [-42, 0], [-27, -11], [-13, -9], [-2, 2], [-24, 6], [-12, 2], [-14, -8], [28, -6], [12, -2], [-8, 4], [-24, 14], [-16, -2], [24, 2], [10, -2], [-34, -6], [18, 14], [38, 2], [-12, -8], [9, -7], [8, 22], [43, -1], [2, 4], [0, -6], [-3, -3], [-16, -12], [-9, 19], [-34, -4], [-12, 0], [10, 8], [0, 2], [-26, 0], [4, -26], [-4, -22], [18, -6], [36, -10], [30, -6], [-10, -4], [-7, 5], [35, 7], [42, -2], [42, -4], [17, 5], [-18, 14], [8, 22], [-9, -5], [-28, 6], [10, 0], [20, -2], [-3, 21], [-28, 6], [-12, 6], [8, 10], [-34, 4], [-36, 0], [15, 3], [-2, -2], [36, -2], [2, 16], [6, 24], [-25, -5], [6, 18], [25, -7], [-4, 0], [-18, 8], [0, 18], [-42, 0], [4, -14], [-19, -11], [-6, -26], [7, -21], [-54, -2], [-4, -14], [-32, -10], [-8, -4], [-4, 2], [-24, -2], [10, -24], [4, 8], [-22, 0], [-14, -12], [-13, -25], [22, 4], [-51, -7], [15, -5], [-15, -23], [48, 0], [46, -4], [38, 12], [-16, -12], [14, -14], [-14, -12], [4, -2], [16, -4], [-25, -13], [20, 10], [-28, -8], [-22, -4], [-32, -14], [-9, 11], [14, -18], [2, -10], [-30, -16], [38, -6], [-36, 16], [-64, -2], [11, -9], [6, 6], [-15, 5], [-50, 0], [-15, 17], [6, 30], [22, 24], [-35, -3], [-46, 0], [26, 4], [0, 2], [40, -10], [-34, -8], [33, -7], [6, 28], [-6, -24], [-34, 2], [56, 4], [-54, -4], [2, -8], [-13, 7], [4, -32], [13, -19], [45, -3], [13, -15], [-14, 12], [-32, -10], [3, -17], [-36, -10], [-26, 2], [62, 0], [-5, -25], [52, 6], [-24, 14], [70, -4], [48, -16], [-18, 26], [49, 5], [-2, 34], [-20, -6], [-15, 29], [-12, 4], [-28, -6], [39, 3], [-37, 11], [20, -10], [-4, 26], [52, 0], [11, -17], [-26, 24], [-16, -6], [-18, -12], [-28, 0], [46, 4], [-36, 16], [-39, 5], [2, -24], [9, 17], [30, 2], [-4, 0], [8, -12], [56, 2], [-34, 14], [-10, -4], [10, -24], [22, 8], [-48, 12], [1, 17], [-22, 12], [-44, -8], [56, -14], [30, 8], [-40, -6], [0, -18], [4, -2], [48, -18], [38, 18], [22, -12], [64, 6], [3, -21], [-21, 7], [-20, -14], [71, -1], [-1, -5], [16, -24], [-4, -10], [-14, -12], [-28, -10], [-28, -8], [20, -22], [-67, 5], [-42, 20], [-8, -2], [-59, 13], [24, 18], [42, -10], [-12, 10], [-26, -24], [7, 27], [22, 2], [29, -11], [22, -18], [-66, -8], [27, -13], [-24, 18], [38, -12], [7, -37], [-12, 6], [39, 19], [2, -18], [0, 30], [30, -24], [16, 26], [4, -18], [-20, 14], [-8, 14], [8, -18], [72, -10], [60, 8], [40, -14], [6, -6], [44, 6], [-47, -15], [15, -9], [-24, -22], [-26, 10], [22, -22], [52, -18], [32, 30], [4, -8], [-18, -28], [-26, 4], [-16, 0], [-26, -12], [10, 16], [64, 10], [23, 11], [-3, 13], [9, 5], [-9, -37], [-33, -17], [14, 4], [55, 3], [52, -6], [21, -3], [-44, 8], [8, -22], [-1, -5], [-52, -14], [-4, -8], [4, -2], [12, 10], [-33, 3], [-20, -6], [-8, 4], [-43, 21], [44, 2], [-20, 26], [-11, -15], [-33, 11], [17, -23], [56, -4], [30, -8], [7, 19], [-7, -35], [-12, 26], [-46, 0], [-12, -22], [44, -8], [50, -2], [70, 4], [-16, -18], [46, 8], [8, -20], [-73, 3], [-54, 2], [-10, -8], [-8, -10], [-2, -16], [26, 14], [-14, -32], [-57, -5], [40, 24], [51, 15], [-5, 27], [26, -10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2496_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2496_2_a_bg();".
// 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_2496_2_a_bg(:prec:=2)
    chi := MakeCharacter_2496_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_2496_2_a_bg();".
// 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_2496_2_a_bg( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2496_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-4, -2, 1]>,<7,R![4, 6, 1]>,<11,R![-20, 0, 1]>,<17,R![-20, 0, 1]>,<19,R![-4, -2, 1]>],Snew);
    return Vf;
end function;