LMFDB - Newform orbit 690.2.m.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [690,2,Mod(31,690)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(690, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 0, 6]))

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

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

chi := DirichletCharacter("690.31");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 690 = 2 \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	690.m (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:	$5.50967773947$
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 $\left(\mathbb{Z}/690\mathbb{Z}\right)^\times$.

$n$	$277$	$461$	$511$
$\chi(n)$	$1$	$1$	$\zeta_{22}^{4}$

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} $

31.1

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

−0.142315

0.989821i

0.415415

0.909632i

−0.959493

−

0.281733i

0.654861

0.755750i

−0.959493

0.281733i

2.31329

1.48666i

0.415415

−

0.909632i

−0.654861

0.755750i

−0.841254

0.540641i

121.1

0.415415

−

0.909632i

−0.959493

−

0.281733i

−0.654861

−

0.755750i

−0.841254

0.540641i

−0.654861

0.755750i

0.0867074

−

0.603063i

−0.959493

0.281733i

0.841254

0.540641i

0.142315

0.989821i

151.1

−0.654861

−

0.755750i

0.841254

0.540641i

−0.142315

0.989821i

−0.415415

0.909632i

−0.142315

−

0.989821i

−1.88745

−

0.554206i

0.841254

−

0.540641i

0.415415

0.909632i

0.959493

−

0.281733i

211.1

0.415415

0.909632i

−0.959493

0.281733i

−0.654861

0.755750i

−0.841254

−

0.540641i

−0.654861

−

0.755750i

0.0867074

0.603063i

−0.959493

−

0.281733i

0.841254

−

0.540641i

0.142315

−

0.989821i

271.1

−0.959493

−

0.281733i

−0.654861

0.755750i

0.841254

0.540641i

0.142315

−

0.989821i

0.841254

−

0.540641i

−1.24302

−

2.72183i

−0.654861

−

0.755750i

−0.142315

−

0.989821i

−0.415415

0.909632i

301.1

0.841254

0.540641i

−0.142315

−

0.989821i

0.415415

0.909632i

0.959493

0.281733i

0.415415

−

0.909632i

0.730471

−

0.843008i

−0.142315

0.989821i

−0.959493

0.281733i

0.654861

0.755750i

331.1

−0.959493

0.281733i

−0.654861

−

0.755750i

0.841254

−

0.540641i

0.142315

0.989821i

0.841254

0.540641i

−1.24302

2.72183i

−0.654861

0.755750i

−0.142315

0.989821i

−0.415415

−

0.909632i

361.1

−0.654861

0.755750i

0.841254

−

0.540641i

−0.142315

−

0.989821i

−0.415415

−

0.909632i

−0.142315

0.989821i

−1.88745

0.554206i

0.841254

0.540641i

0.415415

−

0.909632i

0.959493

0.281733i

541.1

0.841254

−

0.540641i

−0.142315

0.989821i

0.415415

−

0.909632i

0.959493

−

0.281733i

0.415415

0.909632i

0.730471

0.843008i

−0.142315

−

0.989821i

−0.959493

−

0.281733i

0.654861

−

0.755750i

601.1

−0.142315

−

0.989821i

0.415415

−

0.909632i

−0.959493

0.281733i

0.654861

−

0.755750i

−0.959493

−

0.281733i

2.31329

−

1.48666i

0.415415

0.909632i

−0.654861

−

0.755750i

−0.841254

−

0.540641i

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	690.2.m.a	✓	10
23.c	even	11	1	inner	690.2.m.a	✓	10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
690.2.m.a	✓	10	1.a	even	1	1	trivial
690.2.m.a	✓	10	23.c	even	11	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{10} - 11T_{7}^{7} + 22T_{7}^{6} + 143T_{7}^{5} - 121T_{7}^{3} + 363T_{7}^{2} - 121T_{7} + 121 $ T7^10 - 11*T7^7 + 22*T7^6 + 143*T7^5 - 121*T7^3 + 363*T7^2 - 121*T7 + 121 acting on $S_{2}^{\mathrm{new}}(690, [\chi])$.

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} - T^{9} + \cdots + 1 $ T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$7$	$ T^{10} - 11 T^{7} + \cdots + 121 $ T^10 - 11T^7 + 22T^6 + 143T^5 - 121T^3 + 363T^2 - 121T + 121
$11$	$ T^{10} + 2 T^{9} + \cdots + 1 $ T^10 + 2T^9 + 15T^8 + 151T^7 + 412T^6 + 307T^5 + 64T^4 - 147T^3 + 58T^2 - 5*T + 1
$13$	$ T^{10} + 33 T^{8} + \cdots + 958441 $ T^10 + 33T^8 + 33T^7 + 1320T^6 + 3410T^5 + 19481T^4 + 15367T^3 + 213444T^2 - 1001517T + 958441
$17$	$ T^{10} - 16 T^{9} + \cdots + 39601 $ T^10 - 16T^9 + 157T^8 - 1049T^7 + 5234T^6 - 19581T^5 + 54939T^4 - 110949T^3 + 150484T^2 - 117808*T + 39601
$19$	$ T^{10} - 18 T^{9} + \cdots + 17161 $ T^10 - 18T^9 + 137T^8 - 563T^7 + 1466T^6 - 3277T^5 + 7242T^4 - 11105T^3 + 15926T^2 - 14803*T + 17161
$23$	$ T^{10} + T^{9} + \cdots + 6436343 $ T^10 + T^9 - 21T^8 - 131T^7 + 529T^6 + 3279T^5 + 12167T^4 - 69299T^3 - 255507T^2 + 279841T + 6436343
$29$	$ T^{10} - 22 T^{9} + \cdots + 64009 $ T^10 - 22T^9 + 220T^8 - 1540T^7 + 7513T^6 - 15103T^5 + 30250T^4 + 149072T^3 + 274912T^2 + 203159*T + 64009
$31$	$ T^{10} - 8 T^{9} + \cdots + 978121 $ T^10 - 8T^9 - 46T^8 + 390T^7 + 4338T^6 - 51666T^5 + 155741T^4 + 15563T^3 + 4281293T^2 - 569664*T + 978121
$37$	$ T^{10} + 16 T^{9} + \cdots + 351649 $ T^10 + 16T^9 + 102T^8 + 290T^7 + 361T^6 + 1772T^5 + 5593T^4 - 53226T^3 + 228958T^2 - 305395*T + 351649
$41$	$ T^{10} + \cdots + 248787529 $ T^10 + 9T^9 + 4T^8 - 2417T^7 - 13294T^6 + 23761T^5 + 2140301T^4 + 15152779T^3 + 58586465T^2 + 133518445*T + 248787529
$43$	$ T^{10} - 2 T^{9} + \cdots + 77070841 $ T^10 - 2T^9 + 37T^8 - 844T^7 + 2711T^6 - 35507T^5 + 386868T^4 - 426114T^3 + 5314169T^2 + 35967563*T + 77070841
$47$	$ (T^{5} + 24 T^{4} + \cdots - 4643)^{2} $ (T^5 + 24T^4 + 182T^3 + 317T^2 - 1514T - 4643)^2
$53$	$ T^{10} - 2 T^{9} + \cdots + 978121 $ T^10 - 2T^9 + 125T^8 - 1097T^7 - 1535T^6 - 17588T^5 + 206974T^4 + 1014886T^3 + 2591856T^2 + 2486346*T + 978121
$59$	$ T^{10} + 22 T^{9} + \cdots + 4695889 $ T^10 + 22T^9 + 462T^8 + 5863T^7 + 63767T^6 + 506847T^5 + 2492842T^4 + 7230839T^3 + 12025101T^2 + 10535954*T + 4695889
$61$	$ T^{10} + \cdots + 148035889 $ T^10 - 13T^9 + 235T^8 - 2065T^7 + 38417T^6 - 200705T^5 + 1439634T^4 - 6838245T^3 + 15323014T^2 + 130685747*T + 148035889
$67$	$ T^{10} - 2 T^{9} + \cdots + 212521 $ T^10 - 2T^9 + 136T^8 + 751T^7 + 6517T^6 + 56200T^5 + 243824T^4 + 418719T^3 + 112720T^2 - 179790*T + 212521
$71$	$ T^{10} + \cdots + 1113757129 $ T^10 + 45T^9 + 1090T^8 + 17513T^7 + 203501T^6 + 1799216T^5 + 12627902T^4 + 70242899T^3 + 291205025T^2 + 798415652*T + 1113757129
$73$	$ T^{10} - 21 T^{9} + \cdots + 2042041 $ T^10 - 21T^9 + 265T^8 - 2463T^7 + 10924T^6 - 4762T^5 + 34266T^4 - 192334T^3 + 296825T^2 + 312951*T + 2042041
$79$	$ T^{10} + \cdots + 11335647961 $ T^10 - 66T^9 + 2079T^8 - 41118T^7 + 574640T^6 - 6110478T^5 + 51721571T^4 - 345692523T^3 + 1709437059T^2 - 5629761313*T + 11335647961
$83$	$ T^{10} + \cdots + 2327773009 $ T^10 - 15T^9 + 159T^8 + 2290T^7 + 8495T^6 - 608213T^5 + 15637923T^4 - 111427211T^3 + 317932954T^2 - 751640013*T + 2327773009
$89$	$ T^{10} - 29 T^{9} + \cdots + 44742721 $ T^10 - 29T^9 + 511T^8 - 6899T^7 + 74099T^6 - 577158T^5 + 3260712T^4 - 12982118T^3 + 34530021T^2 - 56267868*T + 44742721
$97$	$ T^{10} + \cdots + 392951329 $ T^10 + 27T^9 + 179T^8 + 1412T^7 + 41083T^6 + 235709T^5 + 2760785T^4 + 17651923T^3 + 88450300T^2 + 256172629*T + 392951329
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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:	\( N \)	\(=\)	\( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	690.m (of order \(11\), degree \(10\), minimal)

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

Introduction

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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	690.2.m.a

Level	$690$
Weight	$2$
Character orbit	690.m
Analytic conductor	$5.510$
Analytic rank	$0$
Dimension	$10$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(5.50967773947\)
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:	$\mathrm{SU}(2)[C_{11}]$

\(n\)	\(277\)	\(461\)	\(511\)
\(\chi(n)\)	\(1\)	\(1\)	\(\zeta_{22}^{4}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 31.1
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
$5$	\( T^{10} - T^{9} + \cdots + 1 \) T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$7$	\( T^{10} - 11 T^{7} + \cdots + 121 \) T^10 - 11T^7 + 22T^6 + 143T^5 - 121T^3 + 363T^2 - 121T + 121
$11$	\( T^{10} + 2 T^{9} + \cdots + 1 \) T^10 + 2T^9 + 15T^8 + 151T^7 + 412T^6 + 307T^5 + 64T^4 - 147T^3 + 58T^2 - 5*T + 1
$13$	\( T^{10} + 33 T^{8} + \cdots + 958441 \) T^10 + 33T^8 + 33T^7 + 1320T^6 + 3410T^5 + 19481T^4 + 15367T^3 + 213444T^2 - 1001517T + 958441
$17$	\( T^{10} - 16 T^{9} + \cdots + 39601 \) T^10 - 16T^9 + 157T^8 - 1049T^7 + 5234T^6 - 19581T^5 + 54939T^4 - 110949T^3 + 150484T^2 - 117808*T + 39601
$19$	\( T^{10} - 18 T^{9} + \cdots + 17161 \) T^10 - 18T^9 + 137T^8 - 563T^7 + 1466T^6 - 3277T^5 + 7242T^4 - 11105T^3 + 15926T^2 - 14803*T + 17161
$23$	\( T^{10} + T^{9} + \cdots + 6436343 \) T^10 + T^9 - 21T^8 - 131T^7 + 529T^6 + 3279T^5 + 12167T^4 - 69299T^3 - 255507T^2 + 279841T + 6436343
$29$	\( T^{10} - 22 T^{9} + \cdots + 64009 \) T^10 - 22T^9 + 220T^8 - 1540T^7 + 7513T^6 - 15103T^5 + 30250T^4 + 149072T^3 + 274912T^2 + 203159*T + 64009
$31$	\( T^{10} - 8 T^{9} + \cdots + 978121 \) T^10 - 8T^9 - 46T^8 + 390T^7 + 4338T^6 - 51666T^5 + 155741T^4 + 15563T^3 + 4281293T^2 - 569664*T + 978121
$37$	\( T^{10} + 16 T^{9} + \cdots + 351649 \) T^10 + 16T^9 + 102T^8 + 290T^7 + 361T^6 + 1772T^5 + 5593T^4 - 53226T^3 + 228958T^2 - 305395*T + 351649
$41$	\( T^{10} + \cdots + 248787529 \) T^10 + 9T^9 + 4T^8 - 2417T^7 - 13294T^6 + 23761T^5 + 2140301T^4 + 15152779T^3 + 58586465T^2 + 133518445*T + 248787529
$43$	\( T^{10} - 2 T^{9} + \cdots + 77070841 \) T^10 - 2T^9 + 37T^8 - 844T^7 + 2711T^6 - 35507T^5 + 386868T^4 - 426114T^3 + 5314169T^2 + 35967563*T + 77070841
$47$	\( (T^{5} + 24 T^{4} + \cdots - 4643)^{2} \) (T^5 + 24T^4 + 182T^3 + 317T^2 - 1514T - 4643)^2
$53$	\( T^{10} - 2 T^{9} + \cdots + 978121 \) T^10 - 2T^9 + 125T^8 - 1097T^7 - 1535T^6 - 17588T^5 + 206974T^4 + 1014886T^3 + 2591856T^2 + 2486346*T + 978121
$59$	\( T^{10} + 22 T^{9} + \cdots + 4695889 \) T^10 + 22T^9 + 462T^8 + 5863T^7 + 63767T^6 + 506847T^5 + 2492842T^4 + 7230839T^3 + 12025101T^2 + 10535954*T + 4695889
$61$	\( T^{10} + \cdots + 148035889 \) T^10 - 13T^9 + 235T^8 - 2065T^7 + 38417T^6 - 200705T^5 + 1439634T^4 - 6838245T^3 + 15323014T^2 + 130685747*T + 148035889
$67$	\( T^{10} - 2 T^{9} + \cdots + 212521 \) T^10 - 2T^9 + 136T^8 + 751T^7 + 6517T^6 + 56200T^5 + 243824T^4 + 418719T^3 + 112720T^2 - 179790*T + 212521
$71$	\( T^{10} + \cdots + 1113757129 \) T^10 + 45T^9 + 1090T^8 + 17513T^7 + 203501T^6 + 1799216T^5 + 12627902T^4 + 70242899T^3 + 291205025T^2 + 798415652*T + 1113757129
$73$	\( T^{10} - 21 T^{9} + \cdots + 2042041 \) T^10 - 21T^9 + 265T^8 - 2463T^7 + 10924T^6 - 4762T^5 + 34266T^4 - 192334T^3 + 296825T^2 + 312951*T + 2042041
$79$	\( T^{10} + \cdots + 11335647961 \) T^10 - 66T^9 + 2079T^8 - 41118T^7 + 574640T^6 - 6110478T^5 + 51721571T^4 - 345692523T^3 + 1709437059T^2 - 5629761313*T + 11335647961
$83$	\( T^{10} + \cdots + 2327773009 \) T^10 - 15T^9 + 159T^8 + 2290T^7 + 8495T^6 - 608213T^5 + 15637923T^4 - 111427211T^3 + 317932954T^2 - 751640013*T + 2327773009
$89$	\( T^{10} - 29 T^{9} + \cdots + 44742721 \) T^10 - 29T^9 + 511T^8 - 6899T^7 + 74099T^6 - 577158T^5 + 3260712T^4 - 12982118T^3 + 34530021T^2 - 56267868*T + 44742721
$97$	\( T^{10} + \cdots + 392951329 \) T^10 + 27T^9 + 179T^8 + 1412T^7 + 41083T^6 + 235709T^5 + 2760785T^4 + 17651923T^3 + 88450300T^2 + 256172629*T + 392951329
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