
// Make newform 9792.2.a.dh in Magma, downloaded from the LMFDB on 04 May 2024.
// 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_dh();".
// 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_dh();".
// 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 := [-2, -11, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-8, -1, 1]];
    Rf_basisdens := [1, 1, 2];
    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_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)
    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+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], [0, 0, 0], [0, -1, -1], [-1, 0, -1], [1, 1, 0], [-1, 1, 0], [-1, 0, 0], [3, 1, 0], [0, 1, -1], [3, 2, -1], [-3, -2, -1], [-3, 2, 1], [-3, 1, -2], [3, -3, -2], [-2, 0, 2], [2, 2, 2], [8, 0, 2], [-5, -2, -1], [2, -2, -2], [-1, 2, 1], [4, 2, 4], [-5, 0, 3], [2, -2, 2], [-6, 0, -4], [6, 2, 2], [2, -2, -2], [3, -1, -4], [9, -3, 0], [-3, -4, -1], [-11, -1, 0], [3, 5, -2], [7, 3, 0], [-2, 2, 4], [-4, -2, -6], [6, 0, 0], [-4, 0, 4], [9, -3, -2], [0, 4, 0], [2, -3, -3], [-6, -1, 5], [8, 0, -4], [-1, 6, 3], [12, 2, 6], [-14, 2, -2], [0, 7, 3], [9, -2, 3], [-6, 4, 2], [1, 1, -4], [1, -5, -6], [10, 4, 8], [-1, 1, -4], [-2, 8, 6], [0, -4, -2], [2, -2, 0], [6, -6, -4], [-8, -6, -6], [-2, 1, 3], [-1, 1, 2], [-11, -4, -1], [2, -4, 4], [-24, -2, 2], [14, 0, 0], [4, 8, 0], [-1, 2, 1], [-6, -2, 6], [15, 2, 7], [9, 3, -2], [-6, 0, -8], [8, 8, 4], [-7, -5, 4], [14, 4, 0], [2, -2, 0], [-9, 4, 3], [16, 0, 2], [-10, 2, 0], [6, 8, 2], [18, 0, 0], [5, 0, -1], [9, -7, -6], [-7, 7, -2], [-20, 4, 0], [-21, 1, -4], [33, -2, -1], [3, -3, 2], [-3, 6, -1], [20, -4, -4], [6, -4, 0], [21, 3, 6], [-28, 0, -2], [2, 0, 6], [22, -2, -4], [2, 1, 1], [-1, 8, 7], [-20, -4, -12], [18, 0, 2], [12, -9, -3], [-8, 6, 0], [-15, -3, -2], [-6, 14, 6], [-25, -2, -5], [16, 6, 6], [-16, 0, -6], [18, 2, 10], [12, -12, -8], [12, -8, -4], [-3, 5, 6], [16, 4, 8], [-4, 4, -4], [14, 6, 0], [8, -2, -4], [9, -8, 1], [1, 5, 2], [4, 4, -2], [10, -2, 0], [-19, 5, 4], [7, 7, 6], [-16, -8, -8], [0, 2, 2], [-2, -1, -7], [16, 4, 10], [9, -1, 12], [-6, 2, -6], [26, -3, -1], [21, 3, 10], [2, 8, 6], [-10, 2, -2], [-33, 0, 1], [-14, -9, -5], [8, -8, -4], [24, 0, -2], [-13, 1, 4], [-3, -6, 7], [-31, -6, -9], [-1, 7, -10], [-4, 4, 8], [-7, -11, 2], [-32, -2, -4], [-22, -8, -10], [30, -2, 2], [9, 13, -6], [4, -12, 0], [-8, -5, 7], [31, -4, -5], [-29, -5, 0], [-14, -8, -16], [16, -13, -3], [3, -14, -1], [-22, 12, 4], [0, 16, 4], [6, 14, 8], [7, -2, 15], [4, -8, 2], [15, 1, 8], [-42, 3, 3], [12, -6, -6], [-2, 5, -7], [-17, 17, 6], [-29, -1, -2], [2, -6, 0], [3, -2, -1], [10, -12, 2], [0, -14, -12], [-9, 7, -4], [2, 2, -6], [10, -2, 4], [20, -5, 9], [27, 6, 13], [-5, -6, -13], [-26, -6, -6], [39, 6, -5], [20, 8, -6], [-7, -1, -4], [24, -7, -1], [-5, 5, -6], [3, -11, 2], [-23, 3, 0], [0, 4, -12], [14, 5, 11], [27, 3, 8], [29, 9, 6], [-21, 3, -4], [18, 12, 2], [1, 8, 11], [-14, -2, 8], [-8, -6, -14], [34, -6, 2], [-5, 6, 7], [-45, -3, -2], [8, 8, 14], [23, 2, -7], [0, 14, 0], [-4, 4, 0], [15, 5, 4], [-30, -4, 4], [1, 7, 2], [8, 8, 2], [16, 2, -4], [25, 12, -9], [16, 4, 18], [-8, 4, -16], [-2, 1, -1], [11, 6, -3], [-17, 3, 6], [9, -3, 14], [8, 12, 6], [-8, -10, 12], [-2, -4, -6], [22, -10, -4], [-41, 9, 4], [40, 0, 4], [52, 4, 2], [24, 10, 0], [19, -4, -1], [16, 16, -4], [-28, -1, -3], [10, -6, -14], [-21, -5, 8], [44, 0, 0], [-4, 3, 17], [6, -2, 10], [-33, -1, -6], [-1, -8, -9], [-26, -4, 0], [7, 4, 3], [14, -14, 0], [0, -4, -14], [-7, -5, 12], [-10, 9, 5], [-7, 3, -2], [-20, -8, 8], [11, 13, 4], [-46, -2, 0], [-26, 0, -6], [-8, -2, -4], [22, 6, 14], [-32, -8, -8], [-8, -8, -18], [36, 5, 5], [-3, -3, 0], [14, -14, 0], [27, -19, -10], [4, 12, 4], [35, 5, 14], [-15, 5, 10], [13, -1, -12], [-22, -11, 5], [-55, 6, 3], [-35, -1, -2], [3, 5, -2], [-10, -6, 12], [3, -5, 14], [40, 8, 2], [-42, -4, 2], [10, 4, 0], [-18, 0, -20], [-14, 22, 8], [-7, -14, -3], [6, 14, 0], [-4, 11, 11], [14, -16, 0], [25, 4, -3], [8, 16, 18], [-13, 4, 5], [17, 4, -5], [13, -3, 6], [51, -11, -4], [-20, -4, -10], [16, 18, -12], [38, -2, 4], [-20, -10, -12], [-17, -8, -15], [31, 9, 22], [-29, 3, 2], [-30, 20, 10], [34, -6, -8], [-3, 1, -4], [12, 16, -2], [29, -9, 0], [17, -9, 6], [4, 20, 18], [6, 0, 14], [9, 16, -7], [-11, -2, -5], [-40, 4, 2], [-4, 6, 10], [-4, -4, -20], [-24, -8, -6], [19, 6, 7], [3, 9, 22], [12, -10, -4], [-16, -17, -9], [-25, 1, 6], [-10, 18, -4], [-49, -5, -12], [-3, 0, -17], [-2, 7, 1], [54, 4, 6], [-2, 2, 6], [-13, 5, 14], [-28, -16, -16], [1, 7, 10], [-16, 2, -8], [-17, 0, -9], [7, 13, -14], [8, 8, 20], [30, -14, -6], [16, -8, 2], [-29, -6, 11], [-50, -10, -4], [-2, -4, -12], [15, -6, -7], [31, -6, -5], [19, 7, -10], [19, 5, 8], [-18, -2, 4], [-21, 5, -6], [-18, -6, -8], [-38, 11, -1], [10, 8, 16], [-6, -4, 8], [38, 2, -12], [68, 4, 2], [-8, 14, 8], [5, -11, -4], [6, -16, -16], [-21, 3, -2], [-26, -4, 6], [-46, 0, 2], [27, 6, 9], [-32, -15, -3], [-35, 8, 11], [44, -11, -3], [23, -2, 1], [-24, 8, -2], [10, 6, 24], [5, -3, -4], [-19, -9, 12], [17, -5, 0], [-14, -16, -4], [-1, 3, 0], [5, -9, 0], [-20, 12, 8], [-50, 7, 5], [41, 13, 8], [-28, 8, -14], [-25, -11, -6], [-17, 0, 5], [-17, -7, 20], [50, 5, 13], [15, -26, -13], [35, 1, 10], [-34, -6, 2], [-28, 18, 8], [1, 18, -9], [5, 13, -6], [14, 10, -8], [-54, -2, 4], [-20, 8, 8], [-13, -13, 14], [10, -8, -18], [19, 14, 7], [-1, 9, 16], [-32, -2, 14], [20, 4, 0], [-7, 11, -12], [-23, -9, -18], [-22, 1, -9], [-12, 0, -20], [10, -18, 2], [12, -12, -14], [-2, -12, 2], [34, -5, 15], [48, -20, -10], [-28, -12, -20], [-51, -4, -9], [-23, 3, 6], [31, -6, 1], [-37, -13, -14], [4, 18, -12], [-3, 21, 6], [-32, 7, -9], [16, 22, 20], [27, -2, 13], [-39, -1, -4], [12, -12, -8], [-30, 13, 13], [5, -5, -14], [22, 16, 20], [-10, -10, 20], [34, 14, 0], [-28, 10, 8], [2, 13, -9], [-8, -16, 2], [-19, 3, 10], [54, -8, -10], [-26, 6, -6], [12, -20, 0], [-8, 2, 4], [-18, 12, 10], [36, 0, -6], [-19, -10, 5], [-2, 10, 20], [31, -9, 4], [6, 4, 0], [-2, 8, -12], [-35, -4, 1], [50, 4, -8], [16, -24, 0], [-41, -11, 10], [-21, -15, -2], [2, 0, -4], [34, 8, 20], [54, 10, 18], [-14, 12, -14], [-27, -1, 14], [-1, -2, 27], [-32, 9, 7], [-1, 10, -19], [7, -29, -18], [-16, 2, 14], [50, 2, 10], [3, -4, 1], [26, -29, -9], [-16, 0, -4], [36, 4, 18], [-44, -10, 12], [22, -8, -10], [11, -19, 0], [-23, -1, -20], [-6, 23, 3], [1, 23, 6], [-30, 10, 4], [38, -2, 4], [9, 6, 15], [-3, -8, -21], [34, 0, -8], [14, -8, 8], [20, 12, 18], [4, -6, 2], [35, 5, -10], [1, 3, -14], [-4, 16, 10], [-3, 17, 2], [-18, -18, -8], [4, -18, 8], [-12, -28, -4], [18, 2, -2], [-12, -17, 13], [-42, 2, -2], [-29, -21, 2], [36, -4, 4], [20, 13, 7], [61, 1, 8], [-20, -6, 20], [10, -6, -8], [84, 0, 6], [-24, 20, -6], [40, -4, -2], [-15, -4, 19], [-36, -16, -10], [28, -14, -14], [-17, -18, 9], [-46, -2, 0], [41, -12, -17], [-13, 21, -8], [10, 2, -16], [-53, -9, -8], [18, 18, 28], [-25, 3, -22], [27, 1, 18], [-27, 2, 11], [16, -24, -22], [-11, 2, 7], [8, -6, 4], [-19, -1, -30], [-23, 22, 9], [-32, -11, -7], [32, -16, -12], [-71, -10, -9], [46, -4, 12], [-18, -8, 12], [-4, -4, 4], [-24, -20, -18]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9792_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9792_2_a_dh();".
// 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_dh(:prec:=3)
    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_dh();".
// 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_dh( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9792_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![6, -14, -1, 1]>,<7,R![-12, -10, 2, 1]>,<11,R![8, -8, -3, 1]>,<13,R![-12, -8, 3, 1]>,<19,R![4, 16, -9, 1]>,<23,R![74, -30, -1, 1]>],Snew);
    return Vf;
end function;