LMFDB - Newform orbit 46.2.c.a

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

0.580699

−

1.27155i

−0.959493

0.281733i

−1.66741

1.92429i

−1.34125

−

0.393828i

1.75667

−

1.12894i

0.415415

0.909632i

0.684944

0.790468i

2.14200

1.37658i

9.1

−0.959493

0.281733i

0.712591

0.822373i

0.841254

−

0.540641i

0.174863

1.21620i

−0.915415

−

0.588302i

0.260554

−

0.570534i

−0.654861

0.755750i

0.258432

−

1.79743i

−0.510424

−

1.11767i

13.1

−0.654861

−

0.755750i

−2.71616

−

1.74557i

−0.142315

0.989821i

0.985691

−

2.15836i

0.459493

3.19584i

0.381761

0.112095i

0.841254

−

0.540641i

3.08427

6.75361i

−2.27667

0.668491i

25.1

0.841254

0.540641i

−0.0530529

−

0.368991i

0.415415

0.909632i

−3.26024

−

0.957293i

0.154861

−

0.339098i

−0.297176

0.342959i

−0.142315

0.989821i

2.74514

−

0.806046i

−2.22514

−

2.56794i

27.1

0.415415

0.909632i

−0.524075

0.153882i

−0.654861

0.755750i

0.767092

0.492980i

−0.357685

−

0.412791i

−0.601808

−

4.18567i

−0.959493

−

0.281733i

−2.27279

1.46063i

−0.129769

0.902563i

29.1

0.415415

−

0.909632i

−0.524075

−

0.153882i

−0.654861

−

0.755750i

0.767092

−

0.492980i

−0.357685

0.412791i

−0.601808

4.18567i

−0.959493

0.281733i

−2.27279

−

1.46063i

−0.129769

−

0.902563i

31.1

−0.142315

0.989821i

0.580699

1.27155i

−0.959493

−

0.281733i

−1.66741

−

1.92429i

−1.34125

0.393828i

1.75667

1.12894i

0.415415

−

0.909632i

0.684944

−

0.790468i

2.14200

−

1.37658i

35.1

0.841254

−

0.540641i

−0.0530529

0.368991i

0.415415

−

0.909632i

−3.26024

0.957293i

0.154861

0.339098i

−0.297176

−

0.342959i

−0.142315

−

0.989821i

2.74514

0.806046i

−2.22514

2.56794i

39.1

−0.654861

0.755750i

−2.71616

1.74557i

−0.142315

−

0.989821i

0.985691

2.15836i

0.459493

−

3.19584i

0.381761

−

0.112095i

0.841254

0.540641i

3.08427

−

6.75361i

−2.27667

−

0.668491i

41.1

−0.959493

−

0.281733i

0.712591

−

0.822373i

0.841254

0.540641i

0.174863

−

1.21620i

−0.915415

0.588302i

0.260554

0.570534i

−0.654861

−

0.755750i

0.258432

1.79743i

−0.510424

1.11767i

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.a	✓	10
3.b	odd	2	1		414.2.i.f		10
4.b	odd	2	1		368.2.m.b		10
23.c	even	11	1	inner	46.2.c.a	✓	10
23.c	even	11	1		1058.2.a.m		5
23.d	odd	22	1		1058.2.a.l		5
69.g	even	22	1		9522.2.a.bu		5
69.h	odd	22	1		414.2.i.f		10
69.h	odd	22	1		9522.2.a.bp		5
92.g	odd	22	1		368.2.m.b		10
92.g	odd	22	1		8464.2.a.bx		5
92.h	even	22	1		8464.2.a.bw		5

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

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{10} + 4T_{3}^{9} + 5T_{3}^{8} - 2T_{3}^{7} + 25T_{3}^{6} + T_{3}^{5} + 4T_{3}^{4} + 16T_{3}^{3} + 9T_{3}^{2} + 3T_{3} + 1 $ T3^10 + 4*T3^9 + 5*T3^8 - 2*T3^7 + 25*T3^6 + T3^5 + 4*T3^4 + 16*T3^3 + 9*T3^2 + 3*T3 + 1 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} + 4 T^{9} + \cdots + 1 $ T^10 + 4T^9 + 5T^8 - 2T^7 + 25T^6 + T^5 + 4T^4 + 16T^3 + 9T^2 + 3T + 1
$5$	$ T^{10} + 6 T^{9} + \cdots + 529 $ T^10 + 6T^9 + 14T^8 + 29T^7 + 86T^6 + 153T^5 + 126T^4 - 201T^3 + 587T^2 - 713*T + 529
$7$	$ T^{10} - 3 T^{9} + \cdots + 1 $ T^10 - 3T^9 + 20T^8 - 71T^7 + 125T^6 - 78T^5 + 36T^4 - 9T^3 + 5T^2 - 4*T + 1
$11$	$ T^{10} + 12 T^{9} + \cdots + 109561 $ T^10 + 12T^9 + 56T^8 + 188T^7 + 683T^6 + 1596T^5 + 6942T^4 + 21264T^3 + 32165T^2 + 331*T + 109561
$13$	$ T^{10} + 14 T^{9} + \cdots + 1 $ T^10 + 14T^9 + 86T^8 + 258T^7 + 543T^6 + 617T^5 + 476T^4 + 218T^3 + 82T^2 + 15*T + 1
$17$	$ T^{10} - 15 T^{9} + \cdots + 214369 $ T^10 - 15T^9 + 137T^8 - 746T^7 + 2786T^6 - 6612T^5 + 14260T^4 - 1677T^3 - 30373T^2 + 151401*T + 214369
$19$	$ T^{10} - 2 T^{9} + \cdots + 4489 $ T^10 - 2T^9 + 37T^8 + 14T^7 + 995T^6 - 3915T^5 + 7566T^4 - 29850T^3 + 46423T^2 + 25393*T + 4489
$23$	$ T^{10} + T^{9} + \cdots + 6436343 $ T^10 + T^9 + 78T^8 + 34T^7 + 2938T^6 + 837T^5 + 67574T^4 + 17986T^3 + 949026T^2 + 279841T + 6436343
$29$	$ T^{10} + 8 T^{9} + \cdots + 4489 $ T^10 + 8T^9 - 2T^8 + 259T^7 + 4657T^6 + 19095T^5 + 42012T^4 + 53451T^3 + 41959T^2 + 10050*T + 4489
$31$	$ T^{10} + 21 T^{9} + \cdots + 20529961 $ T^10 + 21T^9 + 232T^8 + 1946T^7 + 12717T^6 + 55802T^5 + 127139T^4 - 116920T^3 - 140568T^2 - 244674*T + 20529961
$37$	$ T^{10} - 28 T^{9} + \cdots + 49857721 $ T^10 - 28T^9 + 509T^8 - 6167T^7 + 61950T^6 - 449701T^5 + 2308058T^4 - 6246149T^3 + 4537724T^2 + 55393545*T + 49857721
$41$	$ T^{10} + \cdots + 172475689 $ T^10 + 31T^9 + 521T^8 + 5998T^7 + 47910T^6 + 242254T^5 + 811556T^4 + 3313523T^3 + 17754927T^2 + 62342351*T + 172475689
$43$	$ T^{10} - 11 T^{9} + \cdots + 7027801 $ T^10 - 11T^9 + 66T^8 - 418T^7 + 2948T^6 - 6391T^5 + 34243T^4 - 302500T^3 + 472384T^2 - 174966*T + 7027801
$47$	$ (T^{5} - 9 T^{4} + \cdots - 529)^{2} $ (T^5 - 9T^4 - 82T^3 + 922T^2 - 1679T - 529)^2
$53$	$ T^{10} + 21 T^{9} + \cdots + 31236921 $ T^10 + 21T^9 + 144T^8 + 1242T^7 + 18063T^6 + 130734T^5 + 1326051T^4 + 6267942T^3 + 19040022T^2 + 33500466*T + 31236921
$59$	$ T^{10} + 5 T^{9} + \cdots + 4489 $ T^10 + 5T^9 - 41T^8 + 213T^7 + 2935T^6 - 8425T^5 + 21499T^4 - 26023T^3 + 26547T^2 - 15611*T + 4489
$61$	$ T^{10} + \cdots + 349727401 $ T^10 - 37T^9 + 489T^8 - 2792T^7 + 21244T^6 - 363034T^5 + 3206746T^4 - 12725509T^3 + 57402101T^2 - 89334677*T + 349727401
$67$	$ T^{10} + 13 T^{9} + \cdots + 94109401 $ T^10 + 13T^9 + 202T^8 + 844T^7 - 4087T^6 - 109660T^5 - 185165T^4 + 2503046T^3 + 16810214T^2 - 28229910*T + 94109401
$71$	$ T^{10} - 49 T^{9} + \cdots + 8300161 $ T^10 - 49T^9 + 1180T^8 - 17890T^7 + 188065T^6 - 1399520T^5 + 7374669T^4 - 26803890T^3 + 52079724T^2 + 26614678*T + 8300161
$73$	$ T^{10} + 8 T^{9} + \cdots + 7921 $ T^10 + 8T^9 + 42T^8 + 83T^7 - 150T^6 - 3543T^5 + 2478T^4 + 8241T^3 + 25701T^2 + 18245*T + 7921
$79$	$ T^{10} - 8 T^{9} + \cdots + 17161 $ T^10 - 8T^9 + 9T^8 - 226T^7 + 1720T^6 + 2498T^5 + 4029T^4 + 5289T^3 + 12347T^2 - 13493*T + 17161
$83$	$ T^{10} + \cdots + 667137241 $ T^10 + 7T^9 + 49T^8 + 750T^7 + 18571T^6 - 135323T^5 - 1721815T^4 - 6308725T^3 + 171600108T^2 - 123385133*T + 667137241
$89$	$ T^{10} + 13 T^{9} + \cdots + 26739241 $ T^10 + 13T^9 + 147T^8 + 1944T^7 + 10620T^6 + 21262T^5 + 236839T^4 + 504236T^3 + 1524064T^2 - 12824080*T + 26739241
$97$	$ T^{10} + 32 T^{9} + \cdots + 5031049 $ T^10 + 32T^9 + 628T^8 + 7754T^7 + 71897T^6 + 453584T^5 + 1913770T^4 + 5305112T^3 + 9849335T^2 + 10710325*T + 5031049
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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)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

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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.a

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
$3$	\( T^{10} + 4 T^{9} + \cdots + 1 \) T^10 + 4T^9 + 5T^8 - 2T^7 + 25T^6 + T^5 + 4T^4 + 16T^3 + 9T^2 + 3T + 1
$5$	\( T^{10} + 6 T^{9} + \cdots + 529 \) T^10 + 6T^9 + 14T^8 + 29T^7 + 86T^6 + 153T^5 + 126T^4 - 201T^3 + 587T^2 - 713*T + 529
$7$	\( T^{10} - 3 T^{9} + \cdots + 1 \) T^10 - 3T^9 + 20T^8 - 71T^7 + 125T^6 - 78T^5 + 36T^4 - 9T^3 + 5T^2 - 4*T + 1
$11$	\( T^{10} + 12 T^{9} + \cdots + 109561 \) T^10 + 12T^9 + 56T^8 + 188T^7 + 683T^6 + 1596T^5 + 6942T^4 + 21264T^3 + 32165T^2 + 331*T + 109561
$13$	\( T^{10} + 14 T^{9} + \cdots + 1 \) T^10 + 14T^9 + 86T^8 + 258T^7 + 543T^6 + 617T^5 + 476T^4 + 218T^3 + 82T^2 + 15*T + 1
$17$	\( T^{10} - 15 T^{9} + \cdots + 214369 \) T^10 - 15T^9 + 137T^8 - 746T^7 + 2786T^6 - 6612T^5 + 14260T^4 - 1677T^3 - 30373T^2 + 151401*T + 214369
$19$	\( T^{10} - 2 T^{9} + \cdots + 4489 \) T^10 - 2T^9 + 37T^8 + 14T^7 + 995T^6 - 3915T^5 + 7566T^4 - 29850T^3 + 46423T^2 + 25393*T + 4489
$23$	\( T^{10} + T^{9} + \cdots + 6436343 \) T^10 + T^9 + 78T^8 + 34T^7 + 2938T^6 + 837T^5 + 67574T^4 + 17986T^3 + 949026T^2 + 279841T + 6436343
$29$	\( T^{10} + 8 T^{9} + \cdots + 4489 \) T^10 + 8T^9 - 2T^8 + 259T^7 + 4657T^6 + 19095T^5 + 42012T^4 + 53451T^3 + 41959T^2 + 10050*T + 4489
$31$	\( T^{10} + 21 T^{9} + \cdots + 20529961 \) T^10 + 21T^9 + 232T^8 + 1946T^7 + 12717T^6 + 55802T^5 + 127139T^4 - 116920T^3 - 140568T^2 - 244674*T + 20529961
$37$	\( T^{10} - 28 T^{9} + \cdots + 49857721 \) T^10 - 28T^9 + 509T^8 - 6167T^7 + 61950T^6 - 449701T^5 + 2308058T^4 - 6246149T^3 + 4537724T^2 + 55393545*T + 49857721
$41$	\( T^{10} + \cdots + 172475689 \) T^10 + 31T^9 + 521T^8 + 5998T^7 + 47910T^6 + 242254T^5 + 811556T^4 + 3313523T^3 + 17754927T^2 + 62342351*T + 172475689
$43$	\( T^{10} - 11 T^{9} + \cdots + 7027801 \) T^10 - 11T^9 + 66T^8 - 418T^7 + 2948T^6 - 6391T^5 + 34243T^4 - 302500T^3 + 472384T^2 - 174966*T + 7027801
$47$	\( (T^{5} - 9 T^{4} + \cdots - 529)^{2} \) (T^5 - 9T^4 - 82T^3 + 922T^2 - 1679T - 529)^2
$53$	\( T^{10} + 21 T^{9} + \cdots + 31236921 \) T^10 + 21T^9 + 144T^8 + 1242T^7 + 18063T^6 + 130734T^5 + 1326051T^4 + 6267942T^3 + 19040022T^2 + 33500466*T + 31236921
$59$	\( T^{10} + 5 T^{9} + \cdots + 4489 \) T^10 + 5T^9 - 41T^8 + 213T^7 + 2935T^6 - 8425T^5 + 21499T^4 - 26023T^3 + 26547T^2 - 15611*T + 4489
$61$	\( T^{10} + \cdots + 349727401 \) T^10 - 37T^9 + 489T^8 - 2792T^7 + 21244T^6 - 363034T^5 + 3206746T^4 - 12725509T^3 + 57402101T^2 - 89334677*T + 349727401
$67$	\( T^{10} + 13 T^{9} + \cdots + 94109401 \) T^10 + 13T^9 + 202T^8 + 844T^7 - 4087T^6 - 109660T^5 - 185165T^4 + 2503046T^3 + 16810214T^2 - 28229910*T + 94109401
$71$	\( T^{10} - 49 T^{9} + \cdots + 8300161 \) T^10 - 49T^9 + 1180T^8 - 17890T^7 + 188065T^6 - 1399520T^5 + 7374669T^4 - 26803890T^3 + 52079724T^2 + 26614678*T + 8300161
$73$	\( T^{10} + 8 T^{9} + \cdots + 7921 \) T^10 + 8T^9 + 42T^8 + 83T^7 - 150T^6 - 3543T^5 + 2478T^4 + 8241T^3 + 25701T^2 + 18245*T + 7921
$79$	\( T^{10} - 8 T^{9} + \cdots + 17161 \) T^10 - 8T^9 + 9T^8 - 226T^7 + 1720T^6 + 2498T^5 + 4029T^4 + 5289T^3 + 12347T^2 - 13493*T + 17161
$83$	\( T^{10} + \cdots + 667137241 \) T^10 + 7T^9 + 49T^8 + 750T^7 + 18571T^6 - 135323T^5 - 1721815T^4 - 6308725T^3 + 171600108T^2 - 123385133*T + 667137241
$89$	\( T^{10} + 13 T^{9} + \cdots + 26739241 \) T^10 + 13T^9 + 147T^8 + 1944T^7 + 10620T^6 + 21262T^5 + 236839T^4 + 504236T^3 + 1524064T^2 - 12824080*T + 26739241
$97$	\( T^{10} + 32 T^{9} + \cdots + 5031049 \) T^10 + 32T^9 + 628T^8 + 7754T^7 + 71897T^6 + 453584T^5 + 1913770T^4 + 5305112T^3 + 9849335T^2 + 10710325*T + 5031049
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\(n\)	\(5\)
\(\chi(n)\)	\(-\zeta_{22}\)