
// Make newform 9792.2.a.cz in Magma, downloaded from the LMFDB on 14 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_9792_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_9792_2_a_cz();".
// 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_9792_2_a_cz();".
// 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_9792_a();"
function MakeCharacter_9792_a()
    N := 9792;
    order := 1;
    char_gens := [7039, 5509, 8705, 9217];
    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_9792_a_Hecke( : Kf := Kf)
    return MakeCharacter_9792_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], [1, 1], [0, 0], [1, -1], [-3, 1], [-1, 0], [3, -3], [-5, 1], [2, -4], [2, -2], [0, 2], [1, 1], [-3, 3], [-6, -2], [2, 4], [-2, -2], [-4, -2], [4, 0], [4, -4], [-2, -4], [-6, 6], [6, -2], [-4, 2], [-8, 2], [16, -2], [-9, -3], [-3, 3], [12, -2], [3, -1], [-7, -5], [-17, 1], [10, -4], [-6, 2], [2, 4], [-8, 0], [1, -3], [-10, -2], [7, 5], [-11, 5], [6, -2], [-6, 0], [-10, 2], [-18, 4], [9, -7], [-16, 0], [-2, 6], [-3, 7], [-9, 9], [-6, 0], [1, 1], [4, -4], [-12, 6], [12, -8], [0, 6], [0, -8], [19, -1], [-7, -5], [-6, 0], [16, -2], [6, 6], [-4, 2], [12, 8], [0, 0], [8, 10], [-18, 0], [-17, -7], [14, -12], [-12, 8], [-9, -1], [-10, 8], [4, 4], [12, -4], [-10, 4], [-12, 0], [-4, 4], [0, 14], [4, 6], [-11, -11], [-7, -3], [-20, 8], [-27, 1], [-24, 0], [19, 3], [12, 4], [26, 2], [-6, -12], [1, 5], [2, -4], [0, 16], [-6, -6], [-9, 13], [-2, -6], [6, 6], [-2, 6], [-27, -1], [20, 2], [33, 1], [-20, 0], [12, -18], [28, 0], [6, 8], [-26, -2], [-16, -2], [0, -12], [-27, 9], [-12, 16], [-38, 4], [-26, 10], [10, -16], [-6, 6], [-7, -3], [34, 4], [-10, 6], [-7, 3], [11, 7], [-24, 4], [6, -6], [-15, -7], [-26, 14], [-21, 3], [30, -12], [-9, 3], [7, 1], [12, -16], [0, 6], [14, 12], [1, -5], [8, 0], [22, -4], [-13, -3], [8, 8], [10, 6], [7, -9], [26, -4], [-3, 17], [-24, 6], [0, -4], [16, -10], [-39, 9], [-30, -6], [-15, 1], [-24, 8], [-9, -15], [-38, 8], [-3, -9], [-2, 12], [6, 0], [12, 0], [-20, 4], [34, 0], [-30, -4], [15, -15], [21, -1], [26, -14], [-9, 13], [9, 3], [-21, 7], [-22, 0], [30, 0], [-12, 0], [20, -22], [-45, 1], [18, 10], [2, -4], [13, 7], [-12, -12], [10, 0], [-12, 6], [-14, 4], [26, 2], [-25, 15], [19, -7], [23, 7], [25, -21], [-9, -13], [-4, 24], [3, 7], [-25, -3], [13, -15], [-33, -3], [28, 8], [2, -24], [16, 14], [-6, -18], [38, -16], [2, 16], [1, -9], [6, -12], [12, -20], [2, 0], [-4, -16], [-21, -11], [-20, 10], [43, -11], [6, -16], [-14, 0], [12, -18], [-46, -4], [-30, 14], [-21, 7], [28, 12], [3, -21], [-3, -15], [40, -4], [30, 0], [32, 0], [-28, 8], [17, 17], [-4, 16], [-42, 12], [6, -16], [6, -30], [12, -24], [-3, -9], [-12, -2], [7, 5], [28, 10], [-31, 19], [24, -18], [-5, 3], [64, 0], [20, 2], [2, -18], [-2, -10], [-46, -8], [39, 3], [11, -7], [-29, 17], [0, -4], [21, 9], [18, -14], [12, 20], [-30, 4], [24, 4], [30, -6], [-44, 6], [-69, -1], [45, 3], [-8, 32], [9, 15], [36, -8], [-3, 15], [49, -3], [15, 3], [3, 9], [-18, 2], [-17, 17], [9, 7], [-24, 24], [-21, 27], [42, 4], [8, 16], [44, 6], [22, -8], [0, 28], [-60, -2], [24, 12], [9, -7], [30, -20], [-6, 6], [-20, 16], [-12, -2], [32, 10], [37, -11], [31, 17], [26, -28], [0, -18], [28, -24], [22, -16], [0, -6], [5, 19], [59, 3], [-48, 0], [-10, 12], [53, -9], [32, 4], [31, -9], [-39, 13], [60, 0], [20, -4], [-10, -14], [32, -16], [2, -8], [34, -6], [24, 24], [10, -24], [-42, 16], [33, -5], [-16, -10], [1, -23], [25, -17], [6, 0], [-17, -23], [24, -30], [5, 5], [-16, 16], [48, -18], [13, 19], [12, -24], [-27, -15], [16, -26], [-40, 0], [5, 19], [-4, -8], [54, -4], [26, -14], [-48, 10], [-48, 8], [22, 20], [-4, 28], [18, -28], [23, 11], [-9, 9], [52, 4], [11, 3], [18, -30], [-15, 27], [8, 18], [-24, -18], [-38, -6], [26, -32], [50, -4], [25, 11], [-34, -8], [15, -9], [62, 6], [-14, -22], [24, 16], [-49, 11], [-20, 30], [-5, 7], [30, 18], [38, -2], [-12, 32], [1, -17], [27, -21], [35, -17], [30, -4], [55, 5], [15, 15], [-36, 2], [-55, 25], [5, 7], [-36, -6], [57, -17], [-86, 0], [-29, 5], [75, -7], [26, -4], [-63, 7], [20, -2], [52, -14], [38, -6], [-59, 17], [4, 26], [-24, 8], [-12, -24], [-33, 19], [-54, 6], [0, 2], [-43, 29], [-28, -4], [-20, 16], [-39, 15], [-3, 25], [-49, 19], [-28, 36], [-24, 10], [56, 4], [-12, 32], [13, -17], [-52, 2], [16, 24], [-32, -22], [23, -23], [28, -20], [3, 3], [-6, 4], [13, -39], [13, -35], [-32, -2], [36, 12], [3, 15], [30, -10], [-1, 29], [3, 9], [-30, -24], [36, -16], [-28, -20], [-2, -12], [11, -1], [-88, -4], [59, -11], [-46, -2], [52, -18], [24, -24], [-30, -28], [-20, 8], [18, 20], [52, 6], [20, 2], [-21, -15], [6, 0], [22, 16], [-22, -18], [-78, 0], [44, -38], [9, -17], [13, -5], [24, 18], [30, -8], [0, -12], [-28, -8], [5, -15], [46, 16], [59, -23], [-42, 18], [-5, 7], [-18, -30], [-26, -8], [66, -24], [41, -13], [-58, -10], [36, -18], [32, -10], [32, -28], [-7, -7], [-11, 11], [-59, 31], [-15, 5], [-64, -4], [84, 0], [0, -8], [50, 0], [-12, 2], [0, 34], [-54, 0], [-58, 6], [-23, 3], [-41, 1], [62, -16], [49, 5], [-6, -18], [-14, 0], [-84, 10], [-36, 0], [33, -21], [6, -20], [-17, 23], [80, 4], [-73, 5], [-55, -1], [94, -8], [-42, 12], [-94, 4], [30, -12], [-6, -22], [6, 36], [-14, -28], [76, 8], [-64, 0], [70, -2], [38, -12], [3, 21], [-18, -12], [3, -15], [-58, 22], [-33, 3], [-27, 3], [-22, 22], [8, 4], [96, -8], [34, 0], [-49, -7], [46, -28], [3, -33], [24, -8], [-64, 8], [-18, 8], [-82, 12], [-24, 38], [30, -44]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9792_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9792_2_a_cz();".
// 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_9792_2_a_cz(:prec:=2)
    chi := MakeCharacter_9792_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(3457) - 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_9792_2_a_cz();".
// 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_9792_2_a_cz( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9792_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, -3, 1]>,<7,R![0, 1]>,<11,R![-4, -1, 1]>,<13,R![2, 5, 1]>,<19,R![-36, -3, 1]>,<23,R![16, 9, 1]>],Snew);
    return Vf;
end function;