LMFDB - Newform orbit 138.2.e.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [138,2,Mod(13,138)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(138, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 14]))

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

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

chi := DirichletCharacter("138.13");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 138 = 2 \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	138.e (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:	$1.10193554789$
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:	$\mathrm{SU}(2)[C_{11}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/138\mathbb{Z}\right)^\times$.

$n$	$47$	$97$
$\chi(n)$	$1$	$-\zeta_{22}$

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} $

13.1

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

0.654861

0.755750i

0.841254

0.540641i

−0.142315

0.989821i

0.810827

−

1.77546i

0.142315

0.989821i

0.439490

0.129046i

−0.841254

0.540641i

0.415415

0.909632i

1.87279

−

0.549899i

25.1

−0.841254

−

0.540641i

−0.142315

−

0.989821i

0.415415

0.909632i

−4.24593

−

1.24672i

−0.415415

0.909632i

−2.17208

2.50672i

0.142315

−

0.989821i

−0.959493

0.281733i

2.89788

3.34433i

31.1

0.142315

−

0.989821i

0.415415

0.909632i

−0.959493

−

0.281733i

1.59283

1.83823i

0.959493

−

0.281733i

1.56130

1.00339i

−0.415415

0.909632i

−0.654861

0.755750i

2.04620

−

1.31501i

49.1

0.142315

0.989821i

0.415415

−

0.909632i

−0.959493

0.281733i

1.59283

−

1.83823i

0.959493

0.281733i

1.56130

−

1.00339i

−0.415415

−

0.909632i

−0.654861

−

0.755750i

2.04620

1.31501i

55.1

0.959493

−

0.281733i

−0.654861

−

0.755750i

0.841254

−

0.540641i

−0.592229

−

4.11904i

−0.841254

−

0.540641i

−1.22301

2.67803i

0.654861

−

0.755750i

−0.142315

0.989821i

−1.72871

−

3.78534i

73.1

−0.415415

−

0.909632i

−0.959493

0.281733i

−0.654861

0.755750i

2.43450

1.56456i

0.654861

0.755750i

0.394306

2.74246i

0.959493

0.281733i

0.841254

−

0.540641i

0.411844

−

2.86444i

85.1

0.654861

−

0.755750i

0.841254

−

0.540641i

−0.142315

−

0.989821i

0.810827

1.77546i

0.142315

−

0.989821i

0.439490

−

0.129046i

−0.841254

−

0.540641i

0.415415

−

0.909632i

1.87279

0.549899i

121.1

−0.415415

0.909632i

−0.959493

−

0.281733i

−0.654861

−

0.755750i

2.43450

−

1.56456i

0.654861

−

0.755750i

0.394306

−

2.74246i

0.959493

−

0.281733i

0.841254

0.540641i

0.411844

2.86444i

127.1

−0.841254

0.540641i

−0.142315

0.989821i

0.415415

−

0.909632i

−4.24593

1.24672i

−0.415415

−

0.909632i

−2.17208

−

2.50672i

0.142315

0.989821i

−0.959493

−

0.281733i

2.89788

−

3.34433i

133.1

0.959493

0.281733i

−0.654861

0.755750i

0.841254

0.540641i

−0.592229

4.11904i

−0.841254

0.540641i

−1.22301

−

2.67803i

0.654861

0.755750i

−0.142315

−

0.989821i

−1.72871

3.78534i

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	138.2.e.c	✓	10
3.b	odd	2	1		414.2.i.b		10
23.c	even	11	1	inner	138.2.e.c	✓	10
23.c	even	11	1		3174.2.a.z		5
23.d	odd	22	1		3174.2.a.y		5
69.g	even	22	1		9522.2.a.ca		5
69.h	odd	22	1		414.2.i.b		10
69.h	odd	22	1		9522.2.a.bv		5

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
138.2.e.c	✓	10	1.a	even	1	1	trivial
138.2.e.c	✓	10	23.c	even	11	1	inner
414.2.i.b		10	3.b	odd	2	1
414.2.i.b		10	69.h	odd	22	1
3174.2.a.y		5	23.d	odd	22	1
3174.2.a.z		5	23.c	even	11	1
9522.2.a.bv		5	69.h	odd	22	1
9522.2.a.ca		5	69.g	even	22	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{10} + 33T_{5}^{7} - 165T_{5}^{6} + 506T_{5}^{5} + 2178T_{5}^{4} - 14399T_{5}^{3} + 45012T_{5}^{2} - 66792T_{5} + 64009 $ T5^10 + 33*T5^7 - 165*T5^6 + 506*T5^5 + 2178*T5^4 - 14399*T5^3 + 45012*T5^2 - 66792*T5 + 64009 acting on $S_{2}^{\mathrm{new}}(138, [\chi])$.

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} + T^{9} + \cdots + 1 $ T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$5$	$ T^{10} + 33 T^{7} + \cdots + 64009 $ T^10 + 33T^7 - 165T^6 + 506T^5 + 2178T^4 - 14399T^3 + 45012T^2 - 66792*T + 64009
$7$	$ T^{10} + 2 T^{9} + \cdots + 529 $ T^10 + 2T^9 + 15T^8 - 3T^7 + 93T^6 - 210T^5 + 504T^4 - 1192T^3 + 3336T^2 - 2392*T + 529
$11$	$ T^{10} - 11 T^{9} + \cdots + 64009 $ T^10 - 11T^9 + 99T^8 - 605T^7 + 2882T^6 - 10384T^5 + 21780T^4 - 17908T^3 + 4961T^2 - 30613*T + 64009
$13$	$ T^{10} + 13 T^{9} + \cdots + 1 $ T^10 + 13T^9 + 59T^8 + 96T^7 - 6T^6 - 254T^5 + 317T^4 + 40T^3 + 58T^2 + 6*T + 1
$17$	$ T^{10} - 11 T^{8} + \cdots + 33860761 $ T^10 - 11T^8 + 253T^7 - 110T^6 + 7117T^5 + 69575T^4 - 457743T^3 + 2630056T^2 + 3328468T + 33860761
$19$	$ T^{10} + 2 T^{9} + \cdots + 978121 $ T^10 + 2T^9 + 26T^8 - 355T^7 + 313T^6 - 3532T^5 + 50158T^4 - 155533T^3 + 333732T^2 - 658674*T + 978121
$23$	$ T^{10} + 10 T^{9} + \cdots + 6436343 $ T^10 + 10T^9 + 100T^8 + 593T^7 + 3532T^6 + 16301T^5 + 81236T^4 + 313697T^3 + 1216700T^2 + 2798410*T + 6436343
$29$	$ T^{10} + 27 T^{9} + \cdots + 591361 $ T^10 + 27T^9 + 344T^8 + 2589T^7 + 14122T^6 + 54231T^5 + 141938T^4 + 241123T^3 + 288149T^2 + 424488*T + 591361
$31$	$ T^{10} + 18 T^{9} + \cdots + 529 $ T^10 + 18T^9 + 148T^8 + 959T^7 + 5327T^6 + 17885T^5 + 42420T^4 + 59637T^3 + 54569T^2 + 8786*T + 529
$37$	$ T^{10} + T^{9} + \cdots + 157609 $ T^10 + T^9 + 34T^8 - 285T^7 + 1530T^6 - 10185T^5 + 75439T^4 - 161391T^3 + 88947T^2 - 43273T + 157609
$41$	$ T^{10} + 16 T^{9} + \cdots + 38809 $ T^10 + 16T^9 + 223T^8 + 1511T^7 + 8380T^6 + 25686T^5 + 26757T^4 - 89889T^3 - 9368T^2 + 24231*T + 38809
$43$	$ T^{10} - 20 T^{9} + \cdots + 978121 $ T^10 - 20T^9 + 191T^8 - 1675T^7 + 15900T^6 - 106294T^5 + 407097T^4 - 892863T^3 + 1577436T^2 - 1561631*T + 978121
$47$	$ (T^{5} - 55 T^{3} + \cdots + 253)^{2} $ (T^5 - 55T^3 - 22T^2 + 352*T + 253)^2
$53$	$ T^{10} + T^{9} + \cdots + 31956409 $ T^10 + T^9 + 67T^8 - 1143T^7 + 13003T^6 - 117281T^5 + 886183T^4 - 4558619T^3 + 16282839T^2 - 34703767T + 31956409
$59$	$ T^{10} + T^{9} + \cdots + 1 $ T^10 + T^9 - 21T^8 - 65T^7 + 199T^6 + 672T^5 + 1046T^4 + 188T^3 + 177T^2 - 10T + 1
$61$	$ T^{10} + 34 T^{9} + \cdots + 39601 $ T^10 + 34T^9 + 507T^8 + 4291T^7 + 23189T^6 + 88298T^5 + 245136T^4 + 406440T^3 + 409388T^2 + 133728*T + 39601
$67$	$ T^{10} - 8 T^{9} + \cdots + 10042561 $ T^10 - 8T^9 + 64T^8 + 1644T^7 + 10311T^6 + 50942T^5 + 273716T^4 + 1122064T^3 + 3290875T^2 + 7228489*T + 10042561
$71$	$ T^{10} + 22 T^{9} + \cdots + 64009 $ T^10 + 22T^9 + 374T^8 + 6028T^7 + 56584T^6 + 257224T^5 + 589633T^4 + 573661T^3 + 431607T^2 - 200376*T + 64009
$73$	$ T^{10} - 31 T^{9} + \cdots + 20151121 $ T^10 - 31T^9 + 389T^8 - 2852T^7 + 17924T^6 - 103896T^5 + 500817T^4 - 1763172T^3 + 5198262T^2 - 11428994*T + 20151121
$79$	$ T^{10} + \cdots + 650199001 $ T^10 - 32T^9 + 496T^8 - 4751T^7 + 34783T^6 - 259159T^5 + 2343870T^4 - 17829723T^3 + 84028209T^2 - 253256068*T + 650199001
$83$	$ T^{10} + \cdots + 6146089609 $ T^10 - 33T^9 + 627T^8 - 7876T^7 + 92510T^6 - 836693T^5 + 10096119T^4 - 87644172T^3 + 728881131T^2 - 3440844330*T + 6146089609
$89$	$ T^{10} + \cdots + 4197614521 $ T^10 + 23T^9 + 221T^8 + 1772T^7 + 25884T^6 + 244289T^5 - 49675T^4 - 8214260T^3 + 149279439T^2 - 717732542*T + 4197614521
$97$	$ T^{10} + \cdots + 16124682289 $ T^10 + T^9 + 221T^8 + 6172T^7 + 27380T^6 + 621204T^5 + 9899638T^4 + 4919091T^3 + 618026861T^2 + 5369222189T + 16124682289
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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:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	138.e (of order \(11\), degree \(10\), minimal)

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

Introduction

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Modular forms

Varieties

Fields

Representations

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Database

Properties

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Newspace parameters

Newform invariants

$q$-expansion

Character values

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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	138.2.e.c

Level	$138$
Weight	$2$
Character orbit	138.e
Analytic conductor	$1.102$
Analytic rank	$0$
Dimension	$10$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(1.10193554789\)
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 13.1
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
$3$	\( T^{10} + T^{9} + \cdots + 1 \) T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$5$	\( T^{10} + 33 T^{7} + \cdots + 64009 \) T^10 + 33T^7 - 165T^6 + 506T^5 + 2178T^4 - 14399T^3 + 45012T^2 - 66792*T + 64009
$7$	\( T^{10} + 2 T^{9} + \cdots + 529 \) T^10 + 2T^9 + 15T^8 - 3T^7 + 93T^6 - 210T^5 + 504T^4 - 1192T^3 + 3336T^2 - 2392*T + 529
$11$	\( T^{10} - 11 T^{9} + \cdots + 64009 \) T^10 - 11T^9 + 99T^8 - 605T^7 + 2882T^6 - 10384T^5 + 21780T^4 - 17908T^3 + 4961T^2 - 30613*T + 64009
$13$	\( T^{10} + 13 T^{9} + \cdots + 1 \) T^10 + 13T^9 + 59T^8 + 96T^7 - 6T^6 - 254T^5 + 317T^4 + 40T^3 + 58T^2 + 6*T + 1
$17$	\( T^{10} - 11 T^{8} + \cdots + 33860761 \) T^10 - 11T^8 + 253T^7 - 110T^6 + 7117T^5 + 69575T^4 - 457743T^3 + 2630056T^2 + 3328468T + 33860761
$19$	\( T^{10} + 2 T^{9} + \cdots + 978121 \) T^10 + 2T^9 + 26T^8 - 355T^7 + 313T^6 - 3532T^5 + 50158T^4 - 155533T^3 + 333732T^2 - 658674*T + 978121
$23$	\( T^{10} + 10 T^{9} + \cdots + 6436343 \) T^10 + 10T^9 + 100T^8 + 593T^7 + 3532T^6 + 16301T^5 + 81236T^4 + 313697T^3 + 1216700T^2 + 2798410*T + 6436343
$29$	\( T^{10} + 27 T^{9} + \cdots + 591361 \) T^10 + 27T^9 + 344T^8 + 2589T^7 + 14122T^6 + 54231T^5 + 141938T^4 + 241123T^3 + 288149T^2 + 424488*T + 591361
$31$	\( T^{10} + 18 T^{9} + \cdots + 529 \) T^10 + 18T^9 + 148T^8 + 959T^7 + 5327T^6 + 17885T^5 + 42420T^4 + 59637T^3 + 54569T^2 + 8786*T + 529
$37$	\( T^{10} + T^{9} + \cdots + 157609 \) T^10 + T^9 + 34T^8 - 285T^7 + 1530T^6 - 10185T^5 + 75439T^4 - 161391T^3 + 88947T^2 - 43273T + 157609
$41$	\( T^{10} + 16 T^{9} + \cdots + 38809 \) T^10 + 16T^9 + 223T^8 + 1511T^7 + 8380T^6 + 25686T^5 + 26757T^4 - 89889T^3 - 9368T^2 + 24231*T + 38809
$43$	\( T^{10} - 20 T^{9} + \cdots + 978121 \) T^10 - 20T^9 + 191T^8 - 1675T^7 + 15900T^6 - 106294T^5 + 407097T^4 - 892863T^3 + 1577436T^2 - 1561631*T + 978121
$47$	\( (T^{5} - 55 T^{3} + \cdots + 253)^{2} \) (T^5 - 55T^3 - 22T^2 + 352*T + 253)^2
$53$	\( T^{10} + T^{9} + \cdots + 31956409 \) T^10 + T^9 + 67T^8 - 1143T^7 + 13003T^6 - 117281T^5 + 886183T^4 - 4558619T^3 + 16282839T^2 - 34703767T + 31956409
$59$	\( T^{10} + T^{9} + \cdots + 1 \) T^10 + T^9 - 21T^8 - 65T^7 + 199T^6 + 672T^5 + 1046T^4 + 188T^3 + 177T^2 - 10T + 1
$61$	\( T^{10} + 34 T^{9} + \cdots + 39601 \) T^10 + 34T^9 + 507T^8 + 4291T^7 + 23189T^6 + 88298T^5 + 245136T^4 + 406440T^3 + 409388T^2 + 133728*T + 39601
$67$	\( T^{10} - 8 T^{9} + \cdots + 10042561 \) T^10 - 8T^9 + 64T^8 + 1644T^7 + 10311T^6 + 50942T^5 + 273716T^4 + 1122064T^3 + 3290875T^2 + 7228489*T + 10042561
$71$	\( T^{10} + 22 T^{9} + \cdots + 64009 \) T^10 + 22T^9 + 374T^8 + 6028T^7 + 56584T^6 + 257224T^5 + 589633T^4 + 573661T^3 + 431607T^2 - 200376*T + 64009
$73$	\( T^{10} - 31 T^{9} + \cdots + 20151121 \) T^10 - 31T^9 + 389T^8 - 2852T^7 + 17924T^6 - 103896T^5 + 500817T^4 - 1763172T^3 + 5198262T^2 - 11428994*T + 20151121
$79$	\( T^{10} + \cdots + 650199001 \) T^10 - 32T^9 + 496T^8 - 4751T^7 + 34783T^6 - 259159T^5 + 2343870T^4 - 17829723T^3 + 84028209T^2 - 253256068*T + 650199001
$83$	\( T^{10} + \cdots + 6146089609 \) T^10 - 33T^9 + 627T^8 - 7876T^7 + 92510T^6 - 836693T^5 + 10096119T^4 - 87644172T^3 + 728881131T^2 - 3440844330*T + 6146089609
$89$	\( T^{10} + \cdots + 4197614521 \) T^10 + 23T^9 + 221T^8 + 1772T^7 + 25884T^6 + 244289T^5 - 49675T^4 - 8214260T^3 + 149279439T^2 - 717732542*T + 4197614521
$97$	\( T^{10} + \cdots + 16124682289 \) T^10 + T^9 + 221T^8 + 6172T^7 + 27380T^6 + 621204T^5 + 9899638T^4 + 4919091T^3 + 618026861T^2 + 5369222189T + 16124682289
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\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(1\)	\(-\zeta_{22}\)