LMFDB - Newform orbit 46.2.c.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [46,2,Mod(3,46)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(46, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([16]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("46.3");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 46 = 2 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	46.c (of order $11$, degree $10$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.367311849298$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\Q(\zeta_{22})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a primitive root of unity $\zeta_{22}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \zeta_{22} q^{2} + (\zeta_{22}^{9} + \zeta_{22}^{7} + \cdots - \zeta_{22}) q^{3} + \zeta_{22}^{2} q^{4} + ( - \zeta_{22}^{9} + \cdots - \zeta_{22}^{3}) q^{5} + (\zeta_{22}^{9} + \zeta_{22}^{7} + \cdots - 1) q^{6}+ \cdots + (\zeta_{22}^{8} + \zeta_{22}^{7} + \cdots + 1) q^{99}+O(q^{100}) $ q + z * q^2 + (z^9 + z^7 + z^4 - z) * q^3 + z^2 * q^4 + (-z^9 - z^7 - z^5 - z^3) * q^5 + (z^9 + z^7 - z^6 + 2z^5 - z^4 + z^3 - 2z^2 + z - 1) * q^6 + (-2z^8 + z^7 - z^6 + 2z^5 - 3z^4 + 2z^3 - z^2 + z - 2) * q^7 + z^3 * q^8 + (-2z^9 + z^8 - 3z^7 + 2z^6 - 3z^5 + 2z^4 - 3z^3 + z^2 - 2z) q^9 + (-z^9 - z^7 - z^5 - z^3 + z^2 - z + 1) * q^10 + (z^8 - z^6 - z^5 + z^4 + z^3 - z) * q^11 + (z^9 + z^6 - z^3 - 1) * q^12 + (z^9 + 3z^8 - z^7 + 3z^6 + z^5 - z^3 + z^2 - z + 1) * q^13 + (-2z^9 + z^8 - z^7 + 2z^6 - 3z^5 + 2z^4 - z^3 + z^2 - 2z) q^14 + (-z^9 + z^8 + z^6 + 2z^5 + z^4 + 2z^3 + z^2 + 2z + 1) q^15 + z^4 * q^16 + (z^9 + 2z^7 - 3z^6 - z^4 + z^3 + 3z - 2) q^17 + (-z^9 - z^8 - z^6 - z^4 - z^3 - 2z + 2) q^18 + (z^9 - 3z^8 + 3z^7 - z^6 - z^4 - 3z^2 - 1) q^19 + (-z^9 - z^7 - z^5 + 1) * q^20 + (3z^9 - 5z^8 + 3z^7 - 3z^6 + 5z^5 - 3z^4 + z^2 + 2z + 1) q^21 + (z^9 - z^7 - z^6 + z^5 + z^4 - z^2) * q^22 + (z^9 - z^8 - z^7 - 2z^6 + 3z^3 - 2z^2 + 3z + 1) * q^23 + (z^9 - z^8 + 2z^7 - z^6 + z^5 - 2z^4 + z^3 - z^2 - 1) * q^24 + (3z^9 - z^8 + 2z^7 - 2z^6 + z^5 - 3z^4 - 3z^2 + 4z - 3) * q^25 + (4z^9 - 2z^8 + 4z^7 + z^5 - 2z^4 + 2z^3 - 2z^2 + 2z - 1) q^26 + (-z^9 + 4z^8 - 4z^7 + z^6 + z^4 + z^3 + z^2 + z + 1) * q^27 + (-z^9 + z^8 - z^6 + z^4 - z^3 - 2z + 2) q^28 + (-4z^9 - 2z^7 + 3z^6 - 2z^5 + 3z^4 - 3z^3 + 2z^2 - 3z + 2) * q^29 + (z^8 + 3z^6 + 3z^4 + 3z^2 + 1) q^30 + (-3z^9 + 3z^8 + 4z^6 - 4z^5 + 4z^4 - 3z^3 + 4z^2 - 4z + 4) * q^31 + z^5 * q^32 + (-3z^9 + z^8 + z^7 + z^6 - 3z^5 + 2z^2 - 2z) * q^33 + (z^9 + z^8 - 2z^7 - z^6 + z^3 + 2z^2 - z - 1) * q^34 + (z^9 + z^8 + z^6 - 2z^5 + 2z^4 - z^3 - z - 1) * q^35 + (-2z^9 + z^8 - 2z^7 + z^6 - 2z^5 - z^3 - z^2 + z + 1) q^36 + (-z^9 + 3z^8 - 2z^7 + 4z^6 + 2z^5 + 4z^4 - 2z^3 + 3z^2 - z) q^37 + (-2z^9 + 2z^8 - z^6 - z^4 - 2z^3 - z^2 - 1) q^38 + (-4z^7 - 3z^6 - z^5 - 4z^4 - z^3 - 3z^2 - 4z) q^39 + (-z^9 - z^7 - z^5 + z^4 - z^3 + z^2 + 1) * q^40 + (z^9 - 3z^7 - 3z^5 + z^3 - z + 1) * q^41 + (-2z^9 + 2z^6 - 3z^4 + 4z^3 - z^2 + 4z - 3) q^42 + (3z^9 - 8z^8 + 3z^7 + 5z^5 - 2z^4 + 3z^3 - 3z^2 + 2z - 5) * q^43 + (z^9 - 2z^8 + 2z^5 - z^4 - z^2 + z - 1) * q^44 + (6z^9 - 3z^8 + 6z^7 - 5z^6 + 5z^5 - 6z^4 + 3z^3 - 6z^2 - 5) * q^45 + (-2z^8 - z^7 - z^6 + z^5 + 2z^4 - z^3 + 2z^2 + 2z - 1) * q^46 + (-2z^9 + 4z^7 - 2z^6 + 2z^5 - 4z^4 + 2z^2 - 3) * q^47 + (z^8 - z^5 - z^2 - 1) * q^48 + (-4z^9 + 4z^8 - 4z^7 - 4z^5 + 3z^4 + 3z^3 - 3z^2 - 3z + 4) * q^49 + (2z^9 - z^8 + z^7 - 2z^6 - 3z^4 + z^2 - 3) q^50 + (-2z^9 + z^7 + z^5 - 2z^3 + z - 1) * q^51 + (2z^9 + 4z^7 - 3z^6 + 2z^5 - 2z^4 + 2z^3 - 2z^2 + 3z - 4) * q^52 + (-3z^7 + z^4 - 3z) * q^53 + (3z^9 - 3z^8 + z^6 + 2z^4 + 2z^2 + 1) * q^54 + (z^9 + z^5 + z) * q^55 + (z^8 - 2z^7 + z^6 - z^3 - z^2 + z + 1) q^56 + (-3z^9 - 2z^8 - 3z^7 + 2z^6 - 5z^5 + 5z^4 - 2z^3 + 3z^2 + 2z + 3) q^57 + (-4z^9 + 2z^8 - z^7 + 2z^6 - z^5 + z^4 - 2z^3 + z^2 - 2z + 4) q^58 + (2z^8 + z^7 + 2z^6 - 6z^3 + 4z^2 - 4z + 6) q^59 + (z^9 + 3z^7 + 3z^5 + 3z^3 + z) q^60 + (-2z^9 + 2z^8 - z^5 - z^4 + z^3 - z^2 - z) * q^61 + (3z^8 + z^7 - z^6 + z^5 + z^3 - z^2 + z + 3) q^62 + (5z^9 + 2z^6 - 5z^5 + 7z^4 - 7z^3 + 5z^2 - 2z) q^63 + z^6 * q^64 + (-3z^9 - z^8 + z^7 + 3z^6 + 5z^4 + 2z^3 + 4z^2 + 2z + 5) * q^65 + (-2z^9 + 4z^8 - 2z^7 - 3z^5 + 3z^4 - z^3 + z^2 - 3z + 3) * q^66 + (z^9 + 5z^8 - 3z^7 + 3z^6 - 5z^5 - z^4 + z^2 + 4z + 1) q^67 + (2z^9 - 3z^8 - z^6 + z^5 + 3z^3 - 2z^2 - 1) * q^68 + (3z^9 + z^8 + 8z^7 - 4z^6 + 6z^5 - 2z^4 + 6z^3 - 5z^2) q^69 + (2z^9 - z^8 + 2z^7 - 3z^6 + 3z^5 - 2z^4 + z^3 - 2z^2 - 1) * q^70 + (z^9 - z^8 - 3z^7 + 3z^6 + z^5 - z^4 - z^2 - 1) * q^71 + (-z^9 - z^7 - 2z^5 + z^4 - 3z^3 + 3z^2 - z + 2) q^72 + (3z^9 - 4z^8 + 4z^7 - 3z^6 - 6z^4 + 5z^3 - 2z^2 + 5z - 6) * q^73 + (2z^9 - z^8 + 3z^7 + 3z^6 + 3z^5 - z^4 + 2z^3 - z + 1) q^74 + (-5z^9 - 5z^7 - 2z^6 + z^5 + z^4 - z^3 - z^2 + 2z + 5) * q^75 + (2z^8 - 3z^7 + 2z^6 - 3z^5 - 3z^3 + 2z^2 - 3z + 2) q^76 + (5z^9 - 5z^8 + 7z^5 - 4z^4 - 2z^3 - 4z^2 + 7z) q^77 + (-4z^8 - 3z^7 - z^6 - 4z^5 - z^4 - 3z^3 - 4z^2) q^78 + (-6z^8 + 5z^7 - 6z^6 - z^3 - 4z^2 + 4z + 1) q^79 + (-z^9 - z^7 + z^2 + 1) * q^80 + (-z^9 + 2z^8 - 4z^6 + 4z^3 - 2z + 1) * q^81 + (z^9 - 4z^8 + z^7 - 4z^6 + z^5 + z^3 - 2z^2 + 2z - 1) * q^82 + (-3z^9 + 4z^8 - z^7 - 5z^6 + 3z^5 - 5z^4 - z^3 + 4z^2 - 3z) q^83 + (-2z^9 + 2z^8 + 2z^6 - 5z^5 + 6z^4 - 3z^3 + 6z^2 - 5z + 2) * q^84 + (2z^8 - 2z^7 + 2z^6 - z^5 - z^3 + 2z^2 - 2z + 2) q^85 + (-5z^9 + 3z^7 + 2z^6 + z^5 - z^2 - 2z - 3) * q^86 + (3z^9 + z^8 + z^7 + 3z^6 + z^5 + z^4 + 3z^3 + 4z - 4) * q^87 + (-z^9 - z^8 + z^7 + z^6 - z^4 - 1) * q^88 + (4z^9 + 4z^7 + 7z^5 + 3z^3 - 3z^2 - 7) q^89 + (3z^9 + z^7 - z^6 - 3z^4 - 6z^2 + z - 6) q^90 + (-5z^9 - 3z^7 + 3z^6 - 3z^5 + 3z^4 + 5z^2 - 3) * q^91 + (-2z^9 - z^8 - z^7 + z^6 + 2z^5 - z^4 + 2z^3 + 2z^2 - z) * q^92 + (z^9 + 5z^6 - 5z^5 - z^2 - 8) * q^93 + (-2z^9 + 6z^8 - 4z^7 + 4z^6 - 6z^5 + 2z^4 + 2z^2 - 5z + 2) * q^94 + (3z^9 + 3z^8 + 3z^7 + 5z^5 - z^4 + 2z^3 - 2z^2 + z - 5) * q^95 + (z^9 - z^6 - z^3 - z) * q^96 + (3z^9 - 3z^8 + 5z^7 - 12z^6 + 5z^5 - 3z^4 + 3z^3 + 2z - 2) * q^97 + (-4z^7 - z^5 + 7z^4 - 7z^3 + z^2 + 4) q^98 + (z^8 + z^7 + 2z^6 - z^5 - z^4 - z^3 + 2z^2 + z + 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + q^{2} - q^{4} - 4 q^{5} - 7 q^{7} + q^{8} - 19 q^{9} + 4 q^{10} - 2 q^{11} - 11 q^{12} + 2 q^{13} - 15 q^{14} + 11 q^{15} - q^{16} - 9 q^{17} + 19 q^{18} + 2 q^{19} + 7 q^{20} + 33 q^{21} + 2 q^{22}+ \cdots + 6 q^{99}+O(q^{100}) $ 10 * q + q^2 - q^4 - 4 * q^5 - 7 * q^7 + q^8 - 19 * q^9 + 4 * q^10 - 2 * q^11 - 11 * q^12 + 2 * q^13 - 15 * q^14 + 11 * q^15 - q^16 - 9 * q^17 + 19 * q^18 + 2 * q^19 + 7 * q^20 + 33 * q^21 + 2 * q^22 + 21 * q^23 - 11 * q^25 + 9 * q^26 + 15 * q^28 - 2 * q^29 + 11 * q^31 + q^32 - 11 * q^33 - 13 * q^34 - 17 * q^35 + 3 * q^36 - 18 * q^37 - 13 * q^38 + 4 * q^40 + 5 * q^41 - 22 * q^42 - 21 * q^43 - 2 * q^44 - 10 * q^45 - 10 * q^46 - 22 * q^47 - 11 * q^48 + 24 * q^49 - 22 * q^50 - 11 * q^51 - 20 * q^52 - 7 * q^53 + 11 * q^54 + 3 * q^55 + 7 * q^56 + 11 * q^57 + 24 * q^58 + 43 * q^59 + 11 * q^60 - 3 * q^61 + 33 * q^62 - 23 * q^63 - q^64 + 41 * q^65 + 11 * q^66 - q^67 + 2 * q^68 + 33 * q^69 + 6 * q^70 - 11 * q^71 + 8 * q^72 - 28 * q^73 + 18 * q^74 + 44 * q^75 + 2 * q^76 + 30 * q^77 + 34 * q^79 + 7 * q^80 + 13 * q^81 + 6 * q^82 - 3 * q^83 - 11 * q^84 + 8 * q^85 - 34 * q^86 - 33 * q^87 - 9 * q^88 - 49 * q^89 - 45 * q^90 - 52 * q^91 - q^92 - 88 * q^93 - 11 * q^94 - 36 * q^95 + 16 * q^97 + 20 * q^98 + 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/46\mathbb{Z}\right)^\times$.

$n$	$5$
$\chi(n)$	$-\zeta_{22}$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

3.1

0.142315	+	0.989821i
0.959493	−	0.281733i
0.654861	+	0.755750i
−0.841254	−	0.540641i
−0.415415	−	0.909632i
−0.415415	+	0.909632i
0.142315	−	0.989821i
−0.841254	+	0.540641i
0.654861	−	0.755750i
0.959493	+	0.281733i

0.142315

0.989821i

0.817178

−

1.78937i

−0.959493

0.281733i

−0.357685

0.412791i

1.88745

0.554206i

−3.96028

2.54512i

−0.415415

−

0.909632i

−0.569485

−

0.657220i

−0.459493

−

0.295298i

9.1

0.959493

−

0.281733i

−1.80075

−

2.07817i

0.841254

−

0.540641i

0.459493

3.19584i

−2.31329

−

1.48666i

−0.497033

1.08835i

0.654861

−

0.755750i

−0.649167

4.51506i

1.34125

2.93694i

13.1

0.654861

0.755750i

−0.512546

−

0.329393i

−0.142315

0.989821i

0.154861

−

0.339098i

−0.0867074

−

0.603063i

−1.97611

−

0.580239i

−0.841254

0.540641i

−1.09204

−

2.39124i

0.357685

−

0.105026i

25.1

−0.841254

−

0.540641i

0.425839

2.96177i

0.415415

0.909632i

−1.34125

−

0.393828i

1.24302

−

2.72183i

2.81051

−

3.24350i

0.142315

−

0.989821i

−5.71228

1.67728i

0.915415

1.05645i

27.1

−0.415415

−

0.909632i

1.07028

−

0.314261i

−0.654861

0.755750i

−0.915415

−

0.588302i

−0.730471

−

0.843008i

0.122916

0.854902i

0.959493

0.281733i

−1.47703

0.949230i

−0.154861

1.07708i

29.1

−0.415415

0.909632i

1.07028

0.314261i

−0.654861

−

0.755750i

−0.915415

0.588302i

−0.730471

0.843008i

0.122916

−

0.854902i

0.959493

−

0.281733i

−1.47703

−

0.949230i

−0.154861

−

1.07708i

31.1

0.142315

−

0.989821i

0.817178

1.78937i

−0.959493

−

0.281733i

−0.357685

−

0.412791i

1.88745

−

0.554206i

−3.96028

−

2.54512i

−0.415415

0.909632i

−0.569485

0.657220i

−0.459493

0.295298i

35.1

−0.841254

0.540641i

0.425839

−

2.96177i

0.415415

−

0.909632i

−1.34125

0.393828i

1.24302

2.72183i

2.81051

3.24350i

0.142315

0.989821i

−5.71228

−

1.67728i

0.915415

−

1.05645i

39.1

0.654861

−

0.755750i

−0.512546

0.329393i

−0.142315

−

0.989821i

0.154861

0.339098i

−0.0867074

0.603063i

−1.97611

0.580239i

−0.841254

−

0.540641i

−1.09204

2.39124i

0.357685

0.105026i

41.1

0.959493

0.281733i

−1.80075

2.07817i

0.841254

0.540641i

0.459493

−

3.19584i

−2.31329

1.48666i

−0.497033

−

1.08835i

0.654861

0.755750i

−0.649167

−

4.51506i

1.34125

−

2.93694i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
23.c	even	11	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	46.2.c.b	✓	10
3.b	odd	2	1		414.2.i.c		10
4.b	odd	2	1		368.2.m.a		10
23.c	even	11	1	inner	46.2.c.b	✓	10
23.c	even	11	1		1058.2.a.j		5
23.d	odd	22	1		1058.2.a.k		5
69.g	even	22	1		9522.2.a.bw		5
69.h	odd	22	1		414.2.i.c		10
69.h	odd	22	1		9522.2.a.bz		5
92.g	odd	22	1		368.2.m.a		10
92.g	odd	22	1		8464.2.a.bu		5
92.h	even	22	1		8464.2.a.bv		5

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
46.2.c.b	✓	10	1.a	even	1	1	trivial
46.2.c.b	✓	10	23.c	even	11	1	inner
368.2.m.a		10	4.b	odd	2	1
368.2.m.a		10	92.g	odd	22	1
414.2.i.c		10	3.b	odd	2	1
414.2.i.c		10	69.h	odd	22	1
1058.2.a.j		5	23.c	even	11	1
1058.2.a.k		5	23.d	odd	22	1
8464.2.a.bu		5	92.g	odd	22	1
8464.2.a.bv		5	92.h	even	22	1
9522.2.a.bw		5	69.g	even	22	1
9522.2.a.bz		5	69.h	odd	22	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{10} + 11T_{3}^{8} + 55T_{3}^{6} - 99T_{3}^{5} + 242T_{3}^{4} - 242T_{3}^{3} - 121T_{3}^{2} + 121T_{3} + 121 $ T3^10 + 11*T3^8 + 55*T3^6 - 99*T3^5 + 242*T3^4 - 242*T3^3 - 121*T3^2 + 121*T3 + 121 acting on $S_{2}^{\mathrm{new}}(46, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{10} - T^{9} + \cdots + 1 $ T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$3$	$ T^{10} + 11 T^{8} + \cdots + 121 $ T^10 + 11T^8 + 55T^6 - 99T^5 + 242T^4 - 242T^3 - 121T^2 + 121*T + 121
$5$	$ T^{10} + 4 T^{9} + \cdots + 1 $ T^10 + 4T^9 + 16T^8 + 53T^7 + 102T^6 + 111T^5 + 70T^4 + 27T^3 + 9T^2 + 3*T + 1
$7$	$ T^{10} + 7 T^{9} + \cdots + 1849 $ T^10 + 7T^9 + 16T^8 + 35T^7 + 509T^6 + 2100T^5 + 3942T^4 + 4791T^3 + 4519T^2 + 2494*T + 1849
$11$	$ T^{10} + 2 T^{9} + \cdots + 1 $ T^10 + 2T^9 + 26T^8 + 74T^7 + 159T^6 + 32T^5 + 20T^4 + 40T^3 + 3T^2 - 5*T + 1
$13$	$ T^{10} - 2 T^{9} + \cdots + 436921 $ T^10 - 2T^9 + 48T^8 - 316T^7 + 1039T^6 - 6973T^5 + 28488T^4 - 29322T^3 + 6042T^2 - 156657*T + 436921
$17$	$ T^{10} + 9 T^{9} + \cdots + 1 $ T^10 + 9T^9 + 81T^8 + 454T^7 + 2282T^6 + 5116T^5 + 4332T^4 + 631T^3 + 47T^2 + 5*T + 1
$19$	$ T^{10} - 2 T^{9} + \cdots + 139129 $ T^10 - 2T^9 + 15T^8 + 146T^7 + 709T^6 + 1321T^5 + 16630T^4 + 16240T^3 + 81227T^2 + 203285*T + 139129
$23$	$ T^{10} - 21 T^{9} + \cdots + 6436343 $ T^10 - 21T^9 + 210T^8 - 1528T^7 + 9758T^6 - 52491T^5 + 224434T^4 - 808312T^3 + 2555070T^2 - 5876661*T + 6436343
$29$	$ T^{10} + 2 T^{9} + \cdots + 529 $ T^10 + 2T^9 + 48T^8 - 25T^7 - 61T^6 + 109T^5 + 8688T^4 + 41653T^3 + 56477T^2 + 7222*T + 529
$31$	$ T^{10} - 11 T^{9} + \cdots + 2076481 $ T^10 - 11T^9 + 110T^8 - 946T^7 + 6831T^6 - 37048T^5 + 191301T^4 - 775368T^3 + 2273832T^2 - 3487220*T + 2076481
$37$	$ T^{10} + 18 T^{9} + \cdots + 14645929 $ T^10 + 18T^9 + 225T^8 + 2279T^7 + 17328T^6 + 82697T^5 + 255864T^4 + 506611T^3 + 1024714T^2 + 5315703*T + 14645929
$41$	$ T^{10} - 5 T^{9} + \cdots + 1985281 $ T^10 - 5T^9 + 25T^8 - 70T^7 + 482T^6 + 1726T^5 + 5736T^4 + 57351T^3 + 9299T^2 - 74677*T + 1985281
$43$	$ T^{10} + 21 T^{9} + \cdots + 53333809 $ T^10 + 21T^9 + 232T^8 + 780T^7 - 4564T^6 - 75659T^5 + 91741T^4 + 226258T^3 + 4711312T^2 + 8442268*T + 53333809
$47$	$ (T^{5} + 11 T^{4} + \cdots + 3883)^{2} $ (T^5 + 11T^4 - 66T^3 - 726T^2 + 385T + 3883)^2
$53$	$ T^{10} + 7 T^{9} + \cdots + 1849 $ T^10 + 7T^9 + 16T^8 + 112T^7 + 289T^6 - 1244T^5 + 5251T^4 + 6166T^3 + 87712T^2 + 16684*T + 1849
$59$	$ T^{10} - 43 T^{9} + \cdots + 8300161 $ T^10 - 43T^9 + 859T^8 - 9921T^7 + 71941T^6 - 383173T^5 + 1810601T^4 - 4817141T^3 + 5467155T^2 + 947849*T + 8300161
$61$	$ T^{10} + 3 T^{9} + \cdots + 529 $ T^10 + 3T^9 + 53T^8 + 104T^7 + 884T^6 - 32T^5 - 426T^4 - 2719T^3 + 9377T^2 + 4393*T + 529
$67$	$ T^{10} + T^{9} + \cdots + 157609 $ T^10 + T^9 - 54T^8 + 1244T^7 + 35531T^6 - 176054T^5 + 529585T^4 - 861882T^3 + 715276T^2 - 353330T + 157609
$71$	$ T^{10} + 11 T^{9} + \cdots + 2076481 $ T^10 + 11T^9 + 110T^8 + 946T^7 + 6831T^6 + 37048T^5 + 191301T^4 + 775368T^3 + 2273832T^2 + 3487220*T + 2076481
$73$	$ T^{10} + 28 T^{9} + \cdots + 34774609 $ T^10 + 28T^9 + 366T^8 + 3351T^7 + 22174T^6 + 84589T^5 + 317586T^4 + 1349477T^3 + 6301101T^2 + 19017825*T + 34774609
$79$	$ T^{10} + \cdots + 118613881 $ T^10 - 34T^9 + 903T^8 - 15478T^7 + 192248T^6 - 1679030T^5 + 10509027T^4 - 44558515T^3 + 119824563T^2 - 181868809*T + 118613881
$83$	$ T^{10} + \cdots + 923126689 $ T^10 + 3T^9 - 277T^8 - 3042T^7 + 46501T^6 + 174879T^5 + 3259721T^4 + 8208253T^3 + 68309180T^2 + 92698533*T + 923126689
$89$	$ T^{10} + 49 T^{9} + \cdots + 380689 $ T^10 + 49T^9 + 1191T^8 + 16020T^7 + 124936T^6 + 545634T^5 + 1243335T^4 + 1438440T^3 + 1566448T^2 + 772484*T + 380689
$97$	$ T^{10} + \cdots + 788093329 $ T^10 - 16T^9 - 96T^8 - 686T^7 + 38113T^6 + 295184T^5 + 1368790T^4 - 10824124T^3 + 112376147T^2 - 186152063*T + 788093329
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 46 = 2 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	46.c (of order \(11\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{22} q^{2} + (\zeta_{22}^{9} + \zeta_{22}^{7} + \cdots - \zeta_{22}) q^{3} + \zeta_{22}^{2} q^{4} + ( - \zeta_{22}^{9} + \cdots - \zeta_{22}^{3}) q^{5} + (\zeta_{22}^{9} + \zeta_{22}^{7} + \cdots - 1) q^{6}+ \cdots + (\zeta_{22}^{8} + \zeta_{22}^{7} + \cdots + 1) q^{99}+O(q^{100}) \) q + z * q^2 + (z^9 + z^7 + z^4 - z) * q^3 + z^2 * q^4 + (-z^9 - z^7 - z^5 - z^3) * q^5 + (z^9 + z^7 - z^6 + 2z^5 - z^4 + z^3 - 2z^2 + z - 1) * q^6 + (-2z^8 + z^7 - z^6 + 2z^5 - 3z^4 + 2z^3 - z^2 + z - 2) * q^7 + z^3 * q^8 + (-2z^9 + z^8 - 3z^7 + 2z^6 - 3z^5 + 2z^4 - 3z^3 + z^2 - 2z) q^9 + (-z^9 - z^7 - z^5 - z^3 + z^2 - z + 1) * q^10 + (z^8 - z^6 - z^5 + z^4 + z^3 - z) * q^11 + (z^9 + z^6 - z^3 - 1) * q^12 + (z^9 + 3z^8 - z^7 + 3z^6 + z^5 - z^3 + z^2 - z + 1) * q^13 + (-2z^9 + z^8 - z^7 + 2z^6 - 3z^5 + 2z^4 - z^3 + z^2 - 2z) q^14 + (-z^9 + z^8 + z^6 + 2z^5 + z^4 + 2z^3 + z^2 + 2z + 1) q^15 + z^4 * q^16 + (z^9 + 2z^7 - 3z^6 - z^4 + z^3 + 3z - 2) q^17 + (-z^9 - z^8 - z^6 - z^4 - z^3 - 2z + 2) q^18 + (z^9 - 3z^8 + 3z^7 - z^6 - z^4 - 3z^2 - 1) q^19 + (-z^9 - z^7 - z^5 + 1) * q^20 + (3z^9 - 5z^8 + 3z^7 - 3z^6 + 5z^5 - 3z^4 + z^2 + 2z + 1) q^21 + (z^9 - z^7 - z^6 + z^5 + z^4 - z^2) * q^22 + (z^9 - z^8 - z^7 - 2z^6 + 3z^3 - 2z^2 + 3z + 1) * q^23 + (z^9 - z^8 + 2z^7 - z^6 + z^5 - 2z^4 + z^3 - z^2 - 1) * q^24 + (3z^9 - z^8 + 2z^7 - 2z^6 + z^5 - 3z^4 - 3z^2 + 4z - 3) * q^25 + (4z^9 - 2z^8 + 4z^7 + z^5 - 2z^4 + 2z^3 - 2z^2 + 2z - 1) q^26 + (-z^9 + 4z^8 - 4z^7 + z^6 + z^4 + z^3 + z^2 + z + 1) * q^27 + (-z^9 + z^8 - z^6 + z^4 - z^3 - 2z + 2) q^28 + (-4z^9 - 2z^7 + 3z^6 - 2z^5 + 3z^4 - 3z^3 + 2z^2 - 3z + 2) * q^29 + (z^8 + 3z^6 + 3z^4 + 3z^2 + 1) q^30 + (-3z^9 + 3z^8 + 4z^6 - 4z^5 + 4z^4 - 3z^3 + 4z^2 - 4z + 4) * q^31 + z^5 * q^32 + (-3z^9 + z^8 + z^7 + z^6 - 3z^5 + 2z^2 - 2z) * q^33 + (z^9 + z^8 - 2z^7 - z^6 + z^3 + 2z^2 - z - 1) * q^34 + (z^9 + z^8 + z^6 - 2z^5 + 2z^4 - z^3 - z - 1) * q^35 + (-2z^9 + z^8 - 2z^7 + z^6 - 2z^5 - z^3 - z^2 + z + 1) q^36 + (-z^9 + 3z^8 - 2z^7 + 4z^6 + 2z^5 + 4z^4 - 2z^3 + 3z^2 - z) q^37 + (-2z^9 + 2z^8 - z^6 - z^4 - 2z^3 - z^2 - 1) q^38 + (-4z^7 - 3z^6 - z^5 - 4z^4 - z^3 - 3z^2 - 4z) q^39 + (-z^9 - z^7 - z^5 + z^4 - z^3 + z^2 + 1) * q^40 + (z^9 - 3z^7 - 3z^5 + z^3 - z + 1) * q^41 + (-2z^9 + 2z^6 - 3z^4 + 4z^3 - z^2 + 4z - 3) q^42 + (3z^9 - 8z^8 + 3z^7 + 5z^5 - 2z^4 + 3z^3 - 3z^2 + 2z - 5) * q^43 + (z^9 - 2z^8 + 2z^5 - z^4 - z^2 + z - 1) * q^44 + (6z^9 - 3z^8 + 6z^7 - 5z^6 + 5z^5 - 6z^4 + 3z^3 - 6z^2 - 5) * q^45 + (-2z^8 - z^7 - z^6 + z^5 + 2z^4 - z^3 + 2z^2 + 2z - 1) * q^46 + (-2z^9 + 4z^7 - 2z^6 + 2z^5 - 4z^4 + 2z^2 - 3) * q^47 + (z^8 - z^5 - z^2 - 1) * q^48 + (-4z^9 + 4z^8 - 4z^7 - 4z^5 + 3z^4 + 3z^3 - 3z^2 - 3z + 4) * q^49 + (2z^9 - z^8 + z^7 - 2z^6 - 3z^4 + z^2 - 3) q^50 + (-2z^9 + z^7 + z^5 - 2z^3 + z - 1) * q^51 + (2z^9 + 4z^7 - 3z^6 + 2z^5 - 2z^4 + 2z^3 - 2z^2 + 3z - 4) * q^52 + (-3z^7 + z^4 - 3z) * q^53 + (3z^9 - 3z^8 + z^6 + 2z^4 + 2z^2 + 1) * q^54 + (z^9 + z^5 + z) * q^55 + (z^8 - 2z^7 + z^6 - z^3 - z^2 + z + 1) q^56 + (-3z^9 - 2z^8 - 3z^7 + 2z^6 - 5z^5 + 5z^4 - 2z^3 + 3z^2 + 2z + 3) q^57 + (-4z^9 + 2z^8 - z^7 + 2z^6 - z^5 + z^4 - 2z^3 + z^2 - 2z + 4) q^58 + (2z^8 + z^7 + 2z^6 - 6z^3 + 4z^2 - 4z + 6) q^59 + (z^9 + 3z^7 + 3z^5 + 3z^3 + z) q^60 + (-2z^9 + 2z^8 - z^5 - z^4 + z^3 - z^2 - z) * q^61 + (3z^8 + z^7 - z^6 + z^5 + z^3 - z^2 + z + 3) q^62 + (5z^9 + 2z^6 - 5z^5 + 7z^4 - 7z^3 + 5z^2 - 2z) q^63 + z^6 * q^64 + (-3z^9 - z^8 + z^7 + 3z^6 + 5z^4 + 2z^3 + 4z^2 + 2z + 5) * q^65 + (-2z^9 + 4z^8 - 2z^7 - 3z^5 + 3z^4 - z^3 + z^2 - 3z + 3) * q^66 + (z^9 + 5z^8 - 3z^7 + 3z^6 - 5z^5 - z^4 + z^2 + 4z + 1) q^67 + (2z^9 - 3z^8 - z^6 + z^5 + 3z^3 - 2z^2 - 1) * q^68 + (3z^9 + z^8 + 8z^7 - 4z^6 + 6z^5 - 2z^4 + 6z^3 - 5z^2) q^69 + (2z^9 - z^8 + 2z^7 - 3z^6 + 3z^5 - 2z^4 + z^3 - 2z^2 - 1) * q^70 + (z^9 - z^8 - 3z^7 + 3z^6 + z^5 - z^4 - z^2 - 1) * q^71 + (-z^9 - z^7 - 2z^5 + z^4 - 3z^3 + 3z^2 - z + 2) q^72 + (3z^9 - 4z^8 + 4z^7 - 3z^6 - 6z^4 + 5z^3 - 2z^2 + 5z - 6) * q^73 + (2z^9 - z^8 + 3z^7 + 3z^6 + 3z^5 - z^4 + 2z^3 - z + 1) q^74 + (-5z^9 - 5z^7 - 2z^6 + z^5 + z^4 - z^3 - z^2 + 2z + 5) * q^75 + (2z^8 - 3z^7 + 2z^6 - 3z^5 - 3z^3 + 2z^2 - 3z + 2) q^76 + (5z^9 - 5z^8 + 7z^5 - 4z^4 - 2z^3 - 4z^2 + 7z) q^77 + (-4z^8 - 3z^7 - z^6 - 4z^5 - z^4 - 3z^3 - 4z^2) q^78 + (-6z^8 + 5z^7 - 6z^6 - z^3 - 4z^2 + 4z + 1) q^79 + (-z^9 - z^7 + z^2 + 1) * q^80 + (-z^9 + 2z^8 - 4z^6 + 4z^3 - 2z + 1) * q^81 + (z^9 - 4z^8 + z^7 - 4z^6 + z^5 + z^3 - 2z^2 + 2z - 1) * q^82 + (-3z^9 + 4z^8 - z^7 - 5z^6 + 3z^5 - 5z^4 - z^3 + 4z^2 - 3z) q^83 + (-2z^9 + 2z^8 + 2z^6 - 5z^5 + 6z^4 - 3z^3 + 6z^2 - 5z + 2) * q^84 + (2z^8 - 2z^7 + 2z^6 - z^5 - z^3 + 2z^2 - 2z + 2) q^85 + (-5z^9 + 3z^7 + 2z^6 + z^5 - z^2 - 2z - 3) * q^86 + (3z^9 + z^8 + z^7 + 3z^6 + z^5 + z^4 + 3z^3 + 4z - 4) * q^87 + (-z^9 - z^8 + z^7 + z^6 - z^4 - 1) * q^88 + (4z^9 + 4z^7 + 7z^5 + 3z^3 - 3z^2 - 7) q^89 + (3z^9 + z^7 - z^6 - 3z^4 - 6z^2 + z - 6) q^90 + (-5z^9 - 3z^7 + 3z^6 - 3z^5 + 3z^4 + 5z^2 - 3) * q^91 + (-2z^9 - z^8 - z^7 + z^6 + 2z^5 - z^4 + 2z^3 + 2z^2 - z) * q^92 + (z^9 + 5z^6 - 5z^5 - z^2 - 8) * q^93 + (-2z^9 + 6z^8 - 4z^7 + 4z^6 - 6z^5 + 2z^4 + 2z^2 - 5z + 2) * q^94 + (3z^9 + 3z^8 + 3z^7 + 5z^5 - z^4 + 2z^3 - 2z^2 + z - 5) * q^95 + (z^9 - z^6 - z^3 - z) * q^96 + (3z^9 - 3z^8 + 5z^7 - 12z^6 + 5z^5 - 3z^4 + 3z^3 + 2z - 2) * q^97 + (-4z^7 - z^5 + 7z^4 - 7z^3 + z^2 + 4) q^98 + (z^8 + z^7 + 2z^6 - z^5 - z^4 - z^3 + 2z^2 + z + 1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + q^{2} - q^{4} - 4 q^{5} - 7 q^{7} + q^{8} - 19 q^{9} + 4 q^{10} - 2 q^{11} - 11 q^{12} + 2 q^{13} - 15 q^{14} + 11 q^{15} - q^{16} - 9 q^{17} + 19 q^{18} + 2 q^{19} + 7 q^{20} + 33 q^{21} + 2 q^{22}+ \cdots + 6 q^{99}+O(q^{100}) \) 10 * q + q^2 - q^4 - 4 * q^5 - 7 * q^7 + q^8 - 19 * q^9 + 4 * q^10 - 2 * q^11 - 11 * q^12 + 2 * q^13 - 15 * q^14 + 11 * q^15 - q^16 - 9 * q^17 + 19 * q^18 + 2 * q^19 + 7 * q^20 + 33 * q^21 + 2 * q^22 + 21 * q^23 - 11 * q^25 + 9 * q^26 + 15 * q^28 - 2 * q^29 + 11 * q^31 + q^32 - 11 * q^33 - 13 * q^34 - 17 * q^35 + 3 * q^36 - 18 * q^37 - 13 * q^38 + 4 * q^40 + 5 * q^41 - 22 * q^42 - 21 * q^43 - 2 * q^44 - 10 * q^45 - 10 * q^46 - 22 * q^47 - 11 * q^48 + 24 * q^49 - 22 * q^50 - 11 * q^51 - 20 * q^52 - 7 * q^53 + 11 * q^54 + 3 * q^55 + 7 * q^56 + 11 * q^57 + 24 * q^58 + 43 * q^59 + 11 * q^60 - 3 * q^61 + 33 * q^62 - 23 * q^63 - q^64 + 41 * q^65 + 11 * q^66 - q^67 + 2 * q^68 + 33 * q^69 + 6 * q^70 - 11 * q^71 + 8 * q^72 - 28 * q^73 + 18 * q^74 + 44 * q^75 + 2 * q^76 + 30 * q^77 + 34 * q^79 + 7 * q^80 + 13 * q^81 + 6 * q^82 - 3 * q^83 - 11 * q^84 + 8 * q^85 - 34 * q^86 - 33 * q^87 - 9 * q^88 - 49 * q^89 - 45 * q^90 - 52 * q^91 - q^92 - 88 * q^93 - 11 * q^94 - 36 * q^95 + 16 * q^97 + 20 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

$q$-expansion

Character values

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Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	46.2.c.b

Level	$46$
Weight	$2$
Character orbit	46.c
Analytic conductor	$0.367$
Analytic rank	$0$
Dimension	$10$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(0.367311849298\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Coefficient field:	\(\Q(\zeta_{22})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 3.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{10} - T^{9} + \cdots + 1 \) T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$3$	\( T^{10} + 11 T^{8} + \cdots + 121 \) T^10 + 11T^8 + 55T^6 - 99T^5 + 242T^4 - 242T^3 - 121T^2 + 121*T + 121
$5$	\( T^{10} + 4 T^{9} + \cdots + 1 \) T^10 + 4T^9 + 16T^8 + 53T^7 + 102T^6 + 111T^5 + 70T^4 + 27T^3 + 9T^2 + 3*T + 1
$7$	\( T^{10} + 7 T^{9} + \cdots + 1849 \) T^10 + 7T^9 + 16T^8 + 35T^7 + 509T^6 + 2100T^5 + 3942T^4 + 4791T^3 + 4519T^2 + 2494*T + 1849
$11$	\( T^{10} + 2 T^{9} + \cdots + 1 \) T^10 + 2T^9 + 26T^8 + 74T^7 + 159T^6 + 32T^5 + 20T^4 + 40T^3 + 3T^2 - 5*T + 1
$13$	\( T^{10} - 2 T^{9} + \cdots + 436921 \) T^10 - 2T^9 + 48T^8 - 316T^7 + 1039T^6 - 6973T^5 + 28488T^4 - 29322T^3 + 6042T^2 - 156657*T + 436921
$17$	\( T^{10} + 9 T^{9} + \cdots + 1 \) T^10 + 9T^9 + 81T^8 + 454T^7 + 2282T^6 + 5116T^5 + 4332T^4 + 631T^3 + 47T^2 + 5*T + 1
$19$	\( T^{10} - 2 T^{9} + \cdots + 139129 \) T^10 - 2T^9 + 15T^8 + 146T^7 + 709T^6 + 1321T^5 + 16630T^4 + 16240T^3 + 81227T^2 + 203285*T + 139129
$23$	\( T^{10} - 21 T^{9} + \cdots + 6436343 \) T^10 - 21T^9 + 210T^8 - 1528T^7 + 9758T^6 - 52491T^5 + 224434T^4 - 808312T^3 + 2555070T^2 - 5876661*T + 6436343
$29$	\( T^{10} + 2 T^{9} + \cdots + 529 \) T^10 + 2T^9 + 48T^8 - 25T^7 - 61T^6 + 109T^5 + 8688T^4 + 41653T^3 + 56477T^2 + 7222*T + 529
$31$	\( T^{10} - 11 T^{9} + \cdots + 2076481 \) T^10 - 11T^9 + 110T^8 - 946T^7 + 6831T^6 - 37048T^5 + 191301T^4 - 775368T^3 + 2273832T^2 - 3487220*T + 2076481
$37$	\( T^{10} + 18 T^{9} + \cdots + 14645929 \) T^10 + 18T^9 + 225T^8 + 2279T^7 + 17328T^6 + 82697T^5 + 255864T^4 + 506611T^3 + 1024714T^2 + 5315703*T + 14645929
$41$	\( T^{10} - 5 T^{9} + \cdots + 1985281 \) T^10 - 5T^9 + 25T^8 - 70T^7 + 482T^6 + 1726T^5 + 5736T^4 + 57351T^3 + 9299T^2 - 74677*T + 1985281
$43$	\( T^{10} + 21 T^{9} + \cdots + 53333809 \) T^10 + 21T^9 + 232T^8 + 780T^7 - 4564T^6 - 75659T^5 + 91741T^4 + 226258T^3 + 4711312T^2 + 8442268*T + 53333809
$47$	\( (T^{5} + 11 T^{4} + \cdots + 3883)^{2} \) (T^5 + 11T^4 - 66T^3 - 726T^2 + 385T + 3883)^2
$53$	\( T^{10} + 7 T^{9} + \cdots + 1849 \) T^10 + 7T^9 + 16T^8 + 112T^7 + 289T^6 - 1244T^5 + 5251T^4 + 6166T^3 + 87712T^2 + 16684*T + 1849
$59$	\( T^{10} - 43 T^{9} + \cdots + 8300161 \) T^10 - 43T^9 + 859T^8 - 9921T^7 + 71941T^6 - 383173T^5 + 1810601T^4 - 4817141T^3 + 5467155T^2 + 947849*T + 8300161
$61$	\( T^{10} + 3 T^{9} + \cdots + 529 \) T^10 + 3T^9 + 53T^8 + 104T^7 + 884T^6 - 32T^5 - 426T^4 - 2719T^3 + 9377T^2 + 4393*T + 529
$67$	\( T^{10} + T^{9} + \cdots + 157609 \) T^10 + T^9 - 54T^8 + 1244T^7 + 35531T^6 - 176054T^5 + 529585T^4 - 861882T^3 + 715276T^2 - 353330T + 157609
$71$	\( T^{10} + 11 T^{9} + \cdots + 2076481 \) T^10 + 11T^9 + 110T^8 + 946T^7 + 6831T^6 + 37048T^5 + 191301T^4 + 775368T^3 + 2273832T^2 + 3487220*T + 2076481
$73$	\( T^{10} + 28 T^{9} + \cdots + 34774609 \) T^10 + 28T^9 + 366T^8 + 3351T^7 + 22174T^6 + 84589T^5 + 317586T^4 + 1349477T^3 + 6301101T^2 + 19017825*T + 34774609
$79$	\( T^{10} + \cdots + 118613881 \) T^10 - 34T^9 + 903T^8 - 15478T^7 + 192248T^6 - 1679030T^5 + 10509027T^4 - 44558515T^3 + 119824563T^2 - 181868809*T + 118613881
$83$	\( T^{10} + \cdots + 923126689 \) T^10 + 3T^9 - 277T^8 - 3042T^7 + 46501T^6 + 174879T^5 + 3259721T^4 + 8208253T^3 + 68309180T^2 + 92698533*T + 923126689
$89$	\( T^{10} + 49 T^{9} + \cdots + 380689 \) T^10 + 49T^9 + 1191T^8 + 16020T^7 + 124936T^6 + 545634T^5 + 1243335T^4 + 1438440T^3 + 1566448T^2 + 772484*T + 380689
$97$	\( T^{10} + \cdots + 788093329 \) T^10 - 16T^9 - 96T^8 - 686T^7 + 38113T^6 + 295184T^5 + 1368790T^4 - 10824124T^3 + 112376147T^2 - 186152063*T + 788093329
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\(n\)	\(5\)
\(\chi(n)\)	\(-\zeta_{22}\)