LMFDB - Newform orbit 414.2.i.c

Newspace parameters

Level:	$ N $	$=$	$ 414 = 2 \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	414.i (of order $11$, degree $10$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$3.30580664368$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\Q(\zeta_{22})$
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:	no (minimal twist has level 46)
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/414\mathbb{Z}\right)^\times$.

$n$	$47$	$235$
$\chi(n)$	$1$	$\zeta_{22}^{4}$

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.

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

55.1

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

−0.959493

0.281733i

0.841254

−

0.540641i

−0.459493

−

3.19584i

−0.497033

1.08835i

−0.654861

0.755750i

1.34125

2.93694i

73.1

0.415415

0.909632i

−0.654861

0.755750i

0.915415

0.588302i

0.122916

0.854902i

−0.959493

−

0.281733i

−0.154861

1.07708i

127.1

0.841254

−

0.540641i

0.415415

−

0.909632i

1.34125

−

0.393828i

2.81051

3.24350i

−0.142315

−

0.989821i

0.915415

−

1.05645i

163.1

0.841254

0.540641i

0.415415

0.909632i

1.34125

0.393828i

2.81051

−

3.24350i

−0.142315

0.989821i

0.915415

1.05645i

271.1

−0.959493

−

0.281733i

0.841254

0.540641i

−0.459493

3.19584i

−0.497033

−

1.08835i

−0.654861

−

0.755750i

1.34125

−

2.93694i

289.1

−0.654861

−

0.755750i

−0.142315

0.989821i

−0.154861

0.339098i

−1.97611

−

0.580239i

0.841254

−

0.540641i

0.357685

−

0.105026i

307.1

−0.142315

0.989821i

−0.959493

−

0.281733i

0.357685

0.412791i

−3.96028

−

2.54512i

0.415415

−

0.909632i

−0.459493

0.295298i

325.1

−0.142315

−

0.989821i

−0.959493

0.281733i

0.357685

−

0.412791i

−3.96028

2.54512i

0.415415

0.909632i

−0.459493

−

0.295298i

361.1

−0.654861

0.755750i

−0.142315

−

0.989821i

−0.154861

−

0.339098i

−1.97611

0.580239i

0.841254

0.540641i

0.357685

0.105026i

397.1

0.415415

−

0.909632i

−0.654861

−

0.755750i

0.915415

−

0.588302i

0.122916

−

0.854902i

−0.959493

0.281733i

−0.154861

−

1.07708i

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	414.2.i.c		10
3.b	odd	2	1		46.2.c.b	✓	10
12.b	even	2	1		368.2.m.a		10
23.c	even	11	1	inner	414.2.i.c		10
23.c	even	11	1		9522.2.a.bz		5
23.d	odd	22	1		9522.2.a.bw		5
69.g	even	22	1		1058.2.a.k		5
69.h	odd	22	1		46.2.c.b	✓	10
69.h	odd	22	1		1058.2.a.j		5
276.j	odd	22	1		8464.2.a.bv		5
276.o	even	22	1		368.2.m.a		10
276.o	even	22	1		8464.2.a.bu		5

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
46.2.c.b	✓	10	3.b	odd	2	1
46.2.c.b	✓	10	69.h	odd	22	1
368.2.m.a		10	12.b	even	2	1
368.2.m.a		10	276.o	even	22	1
414.2.i.c		10	1.a	even	1	1	trivial
414.2.i.c		10	23.c	even	11	1	inner
1058.2.a.j		5	69.h	odd	22	1
1058.2.a.k		5	69.g	even	22	1
8464.2.a.bu		5	276.o	even	22	1
8464.2.a.bv		5	276.j	odd	22	1
9522.2.a.bw		5	23.d	odd	22	1
9522.2.a.bz		5	23.c	even	11	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{10} - 4T_{5}^{9} + 16T_{5}^{8} - 53T_{5}^{7} + 102T_{5}^{6} - 111T_{5}^{5} + 70T_{5}^{4} - 27T_{5}^{3} + 9T_{5}^{2} - 3T_{5} + 1 $ T5^10 - 4*T5^9 + 16*T5^8 - 53*T5^7 + 102*T5^6 - 111*T5^5 + 70*T5^4 - 27*T5^3 + 9*T5^2 - 3*T5 + 1 acting on $S_{2}^{\mathrm{new}}(414, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + \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^10
$5$	$ T^{10} - 4 T^{9} + 16 T^{8} - 53 T^{7} + \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} + 16 T^{8} + \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} + 26 T^{8} - 74 T^{7} + \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} + 48 T^{8} + \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} + 81 T^{8} - 454 T^{7} + \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} + 15 T^{8} + \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} + 210 T^{8} + \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} + 48 T^{8} + 25 T^{7} + \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} + 110 T^{8} + \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} + 225 T^{8} + \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} + 25 T^{8} + \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} + 232 T^{8} + \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} - 66 T^{3} + 726 T^{2} + \cdots - 3883)^{2} $ (T^5 - 11T^4 - 66T^3 + 726T^2 + 385T - 3883)^2
$53$	$ T^{10} - 7 T^{9} + 16 T^{8} + \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} + 859 T^{8} + \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} + 53 T^{8} + 104 T^{7} + \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} - 54 T^{8} + \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} + 110 T^{8} + \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} + 366 T^{8} + \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} - 34 T^{9} + \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} - 3 T^{9} - 277 T^{8} + \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} + 1191 T^{8} + \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} - 16 T^{9} + \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:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	414.i (of order \(11\), degree \(10\), minimal)

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

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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	414.2.i.c

Level	$414$
Weight	$2$
Character orbit	414.i
Analytic conductor	$3.306$
Analytic rank	$0$
Dimension	$10$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(3.30580664368\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Coefficient field:	\(\Q(\zeta_{22})\)
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:	no (minimal twist has level 46)
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 397.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + \cdots + 1 \) T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$3$	\( T^{10} \) T^10
$5$	\( T^{10} - 4 T^{9} + 16 T^{8} - 53 T^{7} + \cdots + 1 \) T^10 - 4T^9 + 16T^8 - 53T^7 + 102T^6 - 111T^5 + 70T^4 - 27T^3 + 9T^2 - 3*T + 1
$7$	\( T^{10} + 7 T^{9} + 16 T^{8} + \cdots + 1849 \) T^10 + 7T^9 + 16T^8 + 35T^7 + 509T^6 + 2100T^5 + 3942T^4 + 4791T^3 + 4519T^2 + 2494*T + 1849
$11$	\( T^{10} - 2 T^{9} + 26 T^{8} - 74 T^{7} + \cdots + 1 \) T^10 - 2T^9 + 26T^8 - 74T^7 + 159T^6 - 32T^5 + 20T^4 - 40T^3 + 3T^2 + 5*T + 1
$13$	\( T^{10} - 2 T^{9} + 48 T^{8} + \cdots + 436921 \) T^10 - 2T^9 + 48T^8 - 316T^7 + 1039T^6 - 6973T^5 + 28488T^4 - 29322T^3 + 6042T^2 - 156657*T + 436921
$17$	\( T^{10} - 9 T^{9} + 81 T^{8} - 454 T^{7} + \cdots + 1 \) T^10 - 9T^9 + 81T^8 - 454T^7 + 2282T^6 - 5116T^5 + 4332T^4 - 631T^3 + 47T^2 - 5*T + 1
$19$	\( T^{10} - 2 T^{9} + 15 T^{8} + \cdots + 139129 \) T^10 - 2T^9 + 15T^8 + 146T^7 + 709T^6 + 1321T^5 + 16630T^4 + 16240T^3 + 81227T^2 + 203285*T + 139129
$23$	\( T^{10} + 21 T^{9} + 210 T^{8} + \cdots + 6436343 \) T^10 + 21T^9 + 210T^8 + 1528T^7 + 9758T^6 + 52491T^5 + 224434T^4 + 808312T^3 + 2555070T^2 + 5876661*T + 6436343
$29$	\( T^{10} - 2 T^{9} + 48 T^{8} + 25 T^{7} + \cdots + 529 \) T^10 - 2T^9 + 48T^8 + 25T^7 - 61T^6 - 109T^5 + 8688T^4 - 41653T^3 + 56477T^2 - 7222*T + 529
$31$	\( T^{10} - 11 T^{9} + 110 T^{8} + \cdots + 2076481 \) T^10 - 11T^9 + 110T^8 - 946T^7 + 6831T^6 - 37048T^5 + 191301T^4 - 775368T^3 + 2273832T^2 - 3487220*T + 2076481
$37$	\( T^{10} + 18 T^{9} + 225 T^{8} + \cdots + 14645929 \) T^10 + 18T^9 + 225T^8 + 2279T^7 + 17328T^6 + 82697T^5 + 255864T^4 + 506611T^3 + 1024714T^2 + 5315703*T + 14645929
$41$	\( T^{10} + 5 T^{9} + 25 T^{8} + \cdots + 1985281 \) T^10 + 5T^9 + 25T^8 + 70T^7 + 482T^6 - 1726T^5 + 5736T^4 - 57351T^3 + 9299T^2 + 74677*T + 1985281
$43$	\( T^{10} + 21 T^{9} + 232 T^{8} + \cdots + 53333809 \) T^10 + 21T^9 + 232T^8 + 780T^7 - 4564T^6 - 75659T^5 + 91741T^4 + 226258T^3 + 4711312T^2 + 8442268*T + 53333809
$47$	\( (T^{5} - 11 T^{4} - 66 T^{3} + 726 T^{2} + \cdots - 3883)^{2} \) (T^5 - 11T^4 - 66T^3 + 726T^2 + 385T - 3883)^2
$53$	\( T^{10} - 7 T^{9} + 16 T^{8} + \cdots + 1849 \) T^10 - 7T^9 + 16T^8 - 112T^7 + 289T^6 + 1244T^5 + 5251T^4 - 6166T^3 + 87712T^2 - 16684*T + 1849
$59$	\( T^{10} + 43 T^{9} + 859 T^{8} + \cdots + 8300161 \) T^10 + 43T^9 + 859T^8 + 9921T^7 + 71941T^6 + 383173T^5 + 1810601T^4 + 4817141T^3 + 5467155T^2 - 947849*T + 8300161
$61$	\( T^{10} + 3 T^{9} + 53 T^{8} + 104 T^{7} + \cdots + 529 \) T^10 + 3T^9 + 53T^8 + 104T^7 + 884T^6 - 32T^5 - 426T^4 - 2719T^3 + 9377T^2 + 4393*T + 529
$67$	\( T^{10} + T^{9} - 54 T^{8} + \cdots + 157609 \) T^10 + T^9 - 54T^8 + 1244T^7 + 35531T^6 - 176054T^5 + 529585T^4 - 861882T^3 + 715276T^2 - 353330T + 157609
$71$	\( T^{10} - 11 T^{9} + 110 T^{8} + \cdots + 2076481 \) T^10 - 11T^9 + 110T^8 - 946T^7 + 6831T^6 - 37048T^5 + 191301T^4 - 775368T^3 + 2273832T^2 - 3487220*T + 2076481
$73$	\( T^{10} + 28 T^{9} + 366 T^{8} + \cdots + 34774609 \) T^10 + 28T^9 + 366T^8 + 3351T^7 + 22174T^6 + 84589T^5 + 317586T^4 + 1349477T^3 + 6301101T^2 + 19017825*T + 34774609
$79$	\( T^{10} - 34 T^{9} + \cdots + 118613881 \) T^10 - 34T^9 + 903T^8 - 15478T^7 + 192248T^6 - 1679030T^5 + 10509027T^4 - 44558515T^3 + 119824563T^2 - 181868809*T + 118613881
$83$	\( T^{10} - 3 T^{9} - 277 T^{8} + \cdots + 923126689 \) T^10 - 3T^9 - 277T^8 + 3042T^7 + 46501T^6 - 174879T^5 + 3259721T^4 - 8208253T^3 + 68309180T^2 - 92698533*T + 923126689
$89$	\( T^{10} - 49 T^{9} + 1191 T^{8} + \cdots + 380689 \) T^10 - 49T^9 + 1191T^8 - 16020T^7 + 124936T^6 - 545634T^5 + 1243335T^4 - 1438440T^3 + 1566448T^2 - 772484*T + 380689
$97$	\( T^{10} - 16 T^{9} + \cdots + 788093329 \) 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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\(n\)	\(47\)	\(235\)
\(\chi(n)\)	\(1\)	\(\zeta_{22}^{4}\)