
// Make newform 9072.2.a.cg in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_9072_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_9072_2_a_cg();".
// 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_9072_2_a_cg();".
// 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 := [-3, 12, -11, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [3, -11, 0, 1], [-12, 22, 1, -2]];
    Rf_basisdens := [1, 1, 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_9072_a();"
function MakeCharacter_9072_a()
    N := 9072;
    order := 1;
    char_gens := [1135, 6805, 3809, 2593];
    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_9072_a_Hecke(Kf)
    return MakeCharacter_9072_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, 0], [-1, 1, 0, 0], [1, 0, 0, 0], [-2, 1, 0, 0], [-1, 0, -1, 0], [0, -1, 1, -1], [1, -1, 0, 1], [0, 0, 1, -1], [-3, 0, -2, 1], [1, -1, -2, -2], [1, -1, 0, 0], [2, 1, 2, 2], [1, 1, -1, 2], [-1, 1, -1, -1], [0, -4, -1, -3], [-3, 0, 1, 3], [-6, 0, -3, 1], [4, -1, 1, -3], [-1, 0, 0, 1], [-5, -1, 0, -3], [5, 1, 3, 3], [-2, -1, 0, 1], [0, -3, 1, -2], [-3, -2, 1, -1], [-1, -1, 5, 3], [4, -4, 1, -3], [-9, -2, 0, -2], [-7, -1, 1, -1], [-4, 4, 0, 2], [2, -3, -3, -2], [7, 0, 2, 3], [3, -1, -1, 4], [11, 3, 0, 1], [-6, 1, -5, 0], [9, 2, 0, -1], [-9, 7, 3, 3], [0, -5, -2, -7], [-1, -2, 1, -2], [-9, 3, 0, 2], [-5, 3, -1, -3], [-11, 1, 2, 2], [-5, 3, -2, 6], [-5, -2, -1, -7], [-3, 2, 2, 1], [0, 8, 1, 5], [-3, 6, 1, 4], [9, -4, -5, 0], [-1, -6, 1, -3], [1, 0, 3, -2], [-7, 7, 3, 4], [7, -2, -5, -2], [-8, 2, -4, 5], [0, -5, -5, -5], [14, 0, 1, -4], [-3, 3, 5, 7], [-5, 2, 0, 3], [6, 1, 1, 4], [-8, 7, 0, 1], [0, -1, 2, -9], [-1, 7, 0, 4], [-15, -2, 2, 3], [8, 1, 3, -9], [-2, -5, 7, 4], [1, 4, 3, 3], [-12, -7, -1, -8], [-8, -5, -2, 5], [11, -6, 3, -1], [-3, 5, 6, 3], [-12, -4, -4, -2], [-2, 4, -4, 1], [0, -9, 2, -1], [-1, -1, 4, -6], [-5, -5, 0, -10], [-5, 6, 1, -5], [-3, 5, 0, -4], [-5, 5, 5, 9], [13, -3, 4, -2], [-6, -9, 0, 4], [-2, 8, -3, 4], [-2, 0, 3, 7], [7, 3, -1, -3], [-19, -2, 0, 1], [7, 0, 8, 8], [-3, 2, 4, -7], [-12, 4, -1, 8], [-8, 5, -3, 3], [19, -3, 2, -7], [6, -5, 6, -7], [4, 0, 7, 1], [-1, 5, 7, 2], [-3, -3, -5, -11], [-5, 10, -2, 11], [-9, 3, 5, 8], [-8, 4, 7, 2], [-10, 1, -5, 2], [-8, -2, 1, -5], [-9, 4, -6, -4], [5, -6, -2, 3], [5, 8, 1, 6], [-18, 0, 8, 3], [-31, 0, 2, 1], [-2, -3, -12, -2], [5, 2, 4, -8], [11, -6, -11, -5], [10, -2, 2, -1], [6, -3, -2, -2], [13, -11, -3, -3], [-6, -8, 4, -6], [-16, -11, -2, -12], [-11, -14, 3, -2], [-15, 6, -5, 8], [-2, -5, -4, 2], [2, 2, -10, -7], [-12, 9, 7, 2], [-18, 6, -2, 9], [5, 6, -3, 3], [-25, 7, -9, 9], [-14, 9, 6, 7], [15, -3, 3, -9], [-7, 2, -8, 5], [10, 3, -3, -10], [15, 4, -2, 1], [-6, -8, -3, 0], [2, -3, -3, 2], [6, 6, 6, 2], [6, 0, 0, 6], [13, 6, -2, 9], [5, -3, 0, -2], [23, -8, -5, 0], [5, 4, -3, 11], [-20, -3, -2, -11], [-13, -1, -3, -3], [28, -1, -4, -1], [6, 4, -6, 2], [-28, 8, -1, 7], [-31, 0, 1, -2], [-5, 5, 12, 10], [0, 1, -6, 11], [20, -11, 3, -3], [20, -6, 4, -1], [-31, 5, -9, 3], [-12, 4, 2, -12], [7, 4, 4, -3], [5, 5, 3, -11], [-21, 0, 3, -6], [-1, 1, 7, 3], [0, -6, 5, -5], [-5, -12, 2, -2], [-16, -7, 4, 3], [12, 3, 0, -1], [-9, 15, 0, 6], [-25, -3, -5, -2], [22, -6, -5, -1], [1, 0, 0, 6], [9, 0, -2, 1], [-2, 1, 1, 6], [7, 8, 4, 15], [-14, 1, 0, -7], [-21, -7, 2, 8], [-22, -4, 2, -1], [24, 10, -1, 2], [-44, 5, -4, 7], [6, -5, -14, -1], [11, 8, -4, -8], [-18, 8, -7, 1], [7, -7, 3, -1], [10, 0, -4, -4], [8, -1, 12, -3], [15, 3, -1, -7], [15, -6, -7, 3], [-8, -5, 7, 10], [-9, 4, 7, -4], [-7, 12, -10, 4], [-7, -4, -5, -4], [18, -11, 3, -2], [-1, -17, 7, -2], [-23, 1, -8, -2], [18, -4, -3, -11], [23, -7, -4, -4], [-22, -4, 3, -1], [39, -7, -2, -7], [20, -12, -4, -6], [15, 5, -4, -7], [10, 0, -5, -9], [15, -5, -1, -1], [-9, 1, -10, 0], [-10, 9, -6, 5], [10, 9, -4, 4], [25, -1, 11, -9], [-7, -6, -1, 11], [-19, 11, -3, 6], [5, -12, 3, -3], [12, -5, 5, -4], [-3, -17, -1, -12], [18, -6, -4, 11], [-7, -3, -2, -19], [9, 11, 1, 11], [14, 7, -3, -9], [15, -14, 3, -18], [-17, 4, -5, 0], [25, -10, 1, -2], [9, -12, -8, -5], [-18, 13, 6, 10], [22, 13, 2, 4], [4, 7, -1, 14], [-6, -4, -4, 6], [10, 0, 10, 5], [18, 6, -11, 0], [30, 5, 12, 12], [20, 1, -8, -2], [21, 1, 5, 3], [-8, -2, 0, 8], [36, 5, 1, 4], [-6, 13, 12, 13], [11, 8, 5, 11], [24, -2, 7, 1], [3, -9, 3, 2], [18, -1, -6, -11], [-17, 0, 0, 11], [11, 4, -12, -13], [2, 8, 8, 4], [10, 2, 5, -10], [-21, 11, 6, 8], [44, 6, -1, 4], [-6, 13, -2, 11], [48, -5, -1, -6], [-20, 0, -7, -1], [-16, -1, -3, 6], [5, -3, 4, 0], [-19, -15, 0, -10], [6, 4, -4, -10], [-8, 14, 4, 12], [1, 9, 8, 9], [14, 10, 2, -2], [-23, 5, -10, 4], [2, 6, -2, -20], [-4, -22, 5, -13], [-15, 1, 7, 8], [-17, 9, 3, -1], [10, 5, 4, 18], [-11, 4, 14, -3], [34, 5, -9, -8], [-7, -8, -6, -19], [12, 8, -6, -11], [18, -9, -8, 7], [31, -8, -6, -11], [2, 3, 2, -2], [17, -11, -5, 1], [33, -6, -9, -8], [16, 17, -7, -4], [24, 8, 3, -8], [-22, 6, -13, 1], [-58, 9, 0, 5], [-9, -15, -1, -15], [-12, 15, 5, -1], [13, -1, 19, 9], [-23, -1, -7, -3], [-1, -9, 14, -4], [3, 10, 6, 0], [2, 16, -3, 11], [18, -1, -9, -5], [-20, -5, 5, 8], [-19, -16, 1, -4], [4, 8, -10, 0], [20, 0, 9, 4], [-13, -1, -17, -1], [17, -9, -5, -18], [-24, 7, 2, -12], [-4, -4, -3, 1], [-20, 9, 6, 13], [-33, -3, -11, -3], [11, 12, -4, -6], [-10, 1, -16, -9], [-3, 2, 3, -1], [37, -12, 11, -8], [-50, 0, -3, 5], [-41, -5, 1, -1], [-20, 8, 11, -8], [-9, 7, -3, 10], [-21, 14, 3, -4], [-28, 23, 9, 4], [26, -24, -1, -11], [-25, 4, 0, 7], [0, 1, 9, -5], [37, -5, -1, 7], [-31, 3, -15, 7], [31, 4, 12, 3], [-17, 12, 10, 7], [46, -13, 1, -9], [-18, 4, -3, -9], [36, -6, -3, 10], [-11, -17, -6, -15], [15, -11, -6, -3], [16, -20, -7, -11], [1, -5, -14, -8], [-43, 4, -8, 7], [-18, 16, 2, 14], [-5, 5, 0, 2], [7, 18, 3, 8], [3, -6, 6, -12], [-8, 4, 3, -19], [43, -2, 8, -15], [3, -7, -1, -21], [7, -26, 5, -6], [-29, 10, -2, 3], [14, -2, 21, 7], [-11, 20, 0, 15], [-9, -2, -11, 18], [-28, -21, 1, -10], [39, 13, -4, 4], [-14, -13, 2, 7], [29, 11, 0, -4], [26, -9, 2, -5], [18, 9, 12, 17], [7, -14, -1, -15], [20, 8, 1, -14], [3, -5, 4, 0], [-41, 10, -4, 8], [26, -4, 4, -4], [46, 4, 0, 1], [13, 13, 8, 25], [7, -2, -2, -15], [5, -24, 1, -2], [-14, -10, 19, 3], [10, 11, 7, -7], [-26, 24, -4, 19], [4, -1, 3, -9], [24, -5, 1, -16], [8, -6, -5, -15], [-4, -10, 10, 10], [-35, 0, 3, 4], [53, -2, -11, -4], [-5, 13, -10, 15], [17, -9, -17, -16], [14, 0, -3, 13], [3, -15, 8, -20], [45, -11, 4, -5], [-26, 15, 1, 17], [-29, 9, -5, -9], [-7, -4, -13, -18], [9, 7, -22, -14], [33, 5, 2, -6], [17, -6, 12, -13], [7, -5, 8, 18], [8, 3, -9, 11], [-26, 19, -7, 18], [0, 10, -15, -12], [-11, 4, 11, -4], [1, 13, -1, -1], [-9, 21, -13, 3], [42, -10, -11, -4], [-2, -4, 1, 3], [18, 0, 5, 17], [-16, 20, 11, 1], [-29, 12, -8, 23], [68, 6, -1, 1], [-24, 19, 1, 2], [-18, 0, -14, -2], [-25, 27, 5, 11], [-6, -2, -4, 16], [-7, -24, -11, -27], [-7, 21, -8, 0], [-23, -7, 2, -21], [26, -2, 21, 8], [-30, 0, 5, -8], [-30, 4, -15, 9], [-22, 8, -5, -1], [18, -6, -14, -5], [24, 5, -8, -7], [19, 4, -1, 1], [-31, 10, -4, 11], [8, 4, 10, 10], [1, 7, 11, 25], [5, -7, -12, -20], [-9, -3, -13, -24], [18, 17, -1, -20], [-39, 13, -1, 7], [63, -19, -7, -10], [22, -24, -2, -14], [9, 4, 0, -25], [33, 9, -13, -4], [12, 8, -19, -5], [9, -18, -17, -17], [27, -10, -6, -2], [0, -20, 6, 4], [-7, -10, -4, -3], [6, 0, -4, -2], [26, 15, 8, 9], [-16, -17, 11, 6], [-1, -28, 2, -2], [47, -8, -16, -9], [-24, 7, -3, 11], [-35, 6, -11, 3], [-31, -15, -12, -30], [15, -12, 7, 12], [7, -7, 0, -8], [43, 11, 4, 9], [5, -4, -9, 11], [-10, -32, 6, -2], [-23, 11, 0, 22], [21, 9, 15, 24], [-39, 10, 4, 5], [1, 0, 12, 13], [-1, -9, 4, 14], [-19, 6, -4, -19], [36, -19, 9, -8], [-18, 20, 12, 5], [2, -3, -7, -7], [51, 10, -4, -5], [8, 18, -11, 14], [-6, -7, -5, 14], [-21, -12, 11, 1], [27, -11, 8, -7], [-10, -5, -14, 1], [2, 26, 8, 20], [65, 5, 4, 4], [2, -7, -20, -9], [-6, 5, -24, 4], [-33, 28, -4, 13], [-6, -25, 1, -20], [-11, 38, 0, 23], [36, -20, -15, -11], [-1, 5, 3, -4], [-33, 7, 0, -2], [-4, -22, 6, 0], [-8, 8, 7, 6], [18, 13, 1, 24], [-24, 9, -14, -5], [-18, -2, 4, -4], [16, -18, 2, 7], [-27, -7, 1, -4], [4, -6, 5, 14], [-19, -18, -10, -11], [-2, 14, -13, -9], [-36, -2, -5, -3], [-4, 2, -6, 10], [11, -9, -4, -2], [-1, 3, 26, 15], [27, 7, 5, -1], [-3, -15, 4, -12], [21, -8, -16, -8], [10, 7, 6, -5], [7, 14, 15, 26], [3, 7, 0, -26], [-42, -10, -10, -21], [3, 0, -11, -7], [28, 7, -8, -5], [-9, -5, 11, 12], [7, -6, -12, -1], [-16, 4, -6, -10], [-26, -1, 2, 7], [5, -4, -22, -10], [-18, 12, 16, 7], [44, -2, 3, 9], [12, 8, 21, 15], [-5, 21, 8, 24], [-16, -3, 8, -4], [-35, -4, -4, -25], [-18, 23, 3, 27], [18, -8, 16, 13], [4, 14, 14, 16], [21, 5, 18, 9], [8, -4, 4, -10], [-28, 20, 17, -1], [-12, -5, -2, 0], [43, -6, -2, 3], [-18, -8, -5, -2], [-16, 6, -15, -4], [-14, 13, 17, -1], [-16, 23, -17, 12], [-30, -1, -17, 1], [-12, 10, -16, 16], [-8, -2, 13, -4], [-45, 17, -13, 10], [-6, 22, -4, 16], [4, 4, -11, -23], [-1, -9, -19, 1], [24, -8, -14, -20], [-21, 2, 0, 1], [1, -3, 6, -20], [15, 4, -12, -16], [22, 10, -6, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9072_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9072_2_a_cg();".
// 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_9072_2_a_cg(:prec:=4)
    chi := MakeCharacter_9072_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_9072_2_a_cg();".
// 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_9072_2_a_cg( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9072_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, -9, -8, 3, 1]>,<11,R![-15, -12, 7, 7, 1]>,<13,R![10, -27, -11, 3, 1]>],Snew);
    return Vf;
end function;