LMFDB - Newform orbit 414.2.i.d

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 138)
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

−0.455922

−

3.17101i

0.628663

−

1.37658i

0.654861

−

0.755750i

−1.33083

−

2.91411i

73.1

−0.415415

−

0.909632i

−0.654861

0.755750i

−1.27310

−

0.818172i

0.369215

2.56794i

0.959493

0.281733i

−0.215370

1.49793i

127.1

−0.841254

0.540641i

0.415415

−

0.909632i

−1.18639

0.348356i

0.968468

1.11767i

0.142315

0.989821i

0.809721

−

0.934468i

163.1

−0.841254

−

0.540641i

0.415415

0.909632i

−1.18639

−

0.348356i

0.968468

−

1.11767i

0.142315

−

0.989821i

0.809721

0.934468i

271.1

0.959493

0.281733i

0.841254

0.540641i

−0.455922

3.17101i

0.628663

1.37658i

0.654861

0.755750i

−1.33083

2.91411i

289.1

0.654861

0.755750i

−0.142315

0.989821i

0.614354

−

1.34525i

3.07385

0.902563i

−0.841254

0.540641i

1.41899

−

0.416652i

307.1

0.142315

−

0.989821i

−0.959493

−

0.281733i

−1.69894

−

1.96068i

−1.04019

−

0.668491i

−0.415415

0.909632i

−2.18251

1.40261i

325.1

0.142315

0.989821i

−0.959493

0.281733i

−1.69894

1.96068i

−1.04019

0.668491i

−0.415415

−

0.909632i

−2.18251

−

1.40261i

361.1

0.654861

−

0.755750i

−0.142315

−

0.989821i

0.614354

1.34525i

3.07385

−

0.902563i

−0.841254

−

0.540641i

1.41899

0.416652i

397.1

−0.415415

0.909632i

−0.654861

−

0.755750i

−1.27310

0.818172i

0.369215

−

2.56794i

0.959493

−

0.281733i

−0.215370

−

1.49793i

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.d		10
3.b	odd	2	1		138.2.e.a	✓	10
23.c	even	11	1	inner	414.2.i.d		10
23.c	even	11	1		9522.2.a.bt		5
23.d	odd	22	1		9522.2.a.bq		5
69.g	even	22	1		3174.2.a.bd		5
69.h	odd	22	1		138.2.e.a	✓	10
69.h	odd	22	1		3174.2.a.bc		5

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

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{10} + 8 T_{5}^{9} + 42 T_{5}^{8} + 149 T_{5}^{7} + 389 T_{5}^{6} + 736 T_{5}^{5} + 1092 T_{5}^{4} + 1465 T_{5}^{3} + 1754 T_{5}^{2} + 1426 T_{5} + 529 $ T5^10 + 8*T5^9 + 42*T5^8 + 149*T5^7 + 389*T5^6 + 736*T5^5 + 1092*T5^4 + 1465*T5^3 + 1754*T5^2 + 1426*T5 + 529 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} + 8 T^{9} + 42 T^{8} + 149 T^{7} + \cdots + 529 $ T^10 + 8T^9 + 42T^8 + 149T^7 + 389T^6 + 736T^5 + 1092T^4 + 1465T^3 + 1754T^2 + 1426*T + 529
$7$	$ T^{10} - 8 T^{9} + 31 T^{8} - 83 T^{7} + \cdots + 529 $ T^10 - 8T^9 + 31T^8 - 83T^7 + 147T^6 - 120T^5 + 36T^4 + 64T^3 + 236T^2 - 414*T + 529
$11$	$ T^{10} + 7 T^{9} + 5 T^{8} + 123 T^{7} + \cdots + 1 $ T^10 + 7T^9 + 5T^8 + 123T^7 + 894T^6 - 1354T^5 + 1016T^4 - 500T^3 + 361T^2 - 25*T + 1
$13$	$ T^{10} - 3 T^{9} + 31 T^{8} - 38 T^{7} + \cdots + 1 $ T^10 - 3T^9 + 31T^8 - 38T^7 + 114T^6 + 164T^5 - 63T^4 - 328T^3 + 192T^2 + 18*T + 1
$17$	$ T^{10} + 4 T^{9} + 5 T^{8} + 97 T^{7} + \cdots + 1849 $ T^10 + 4T^9 + 5T^8 + 97T^7 + 300T^6 - 219T^5 + 1599T^4 - 1381T^3 + 262T^2 + 344*T + 1849
$19$	$ T^{10} + 22 T^{8} + 165 T^{7} + \cdots + 64009 $ T^10 + 22T^8 + 165T^7 + 1309T^6 + 7040T^5 + 20328T^4 + 63767T^3 + 134552T^2 - 166980T + 64009
$23$	$ T^{10} - 12 T^{9} - 10 T^{8} + \cdots + 6436343 $ T^10 - 12T^9 - 10T^8 + 527T^7 - 32T^6 - 14103T^5 - 736T^4 + 278783T^3 - 121670T^2 - 3358092*T + 6436343
$29$	$ T^{10} - 25 T^{9} + 372 T^{8} + \cdots + 2866249 $ T^10 - 25T^9 + 372T^8 - 3613T^7 + 25590T^6 - 121177T^5 + 398830T^4 - 376913T^3 - 509097T^2 + 2762976*T + 2866249
$31$	$ T^{10} - 6 T^{9} + 58 T^{8} + \cdots + 896809 $ T^10 - 6T^9 + 58T^8 + 389T^7 + 449T^6 - 20305T^5 + 310898T^4 - 793207T^3 + 1387445T^2 - 1689448*T + 896809
$37$	$ T^{10} - 9 T^{9} + 26 T^{8} - 69 T^{7} + \cdots + 529 $ T^10 - 9T^9 + 26T^8 - 69T^7 + 346T^6 - 507T^5 + 911T^4 - 235T^3 + 443T^2 + 897*T + 529
$41$	$ T^{10} + 24 T^{9} + 301 T^{8} + \cdots + 529 $ T^10 + 24T^9 + 301T^8 + 1669T^7 + 4636T^6 + 5950T^5 + 17785T^4 + 4759T^3 + 29032T^2 + 7475*T + 529
$43$	$ T^{10} + 30 T^{9} + 427 T^{8} + \cdots + 192721 $ T^10 + 30T^9 + 427T^8 + 3537T^7 + 22774T^6 + 93928T^5 + 261363T^4 + 491911T^3 + 624596T^2 + 499143*T + 192721
$47$	$ (T^{5} - 24 T^{4} + 83 T^{3} + 1520 T^{2} + \cdots + 10649)^{2} $ (T^5 - 24T^4 + 83T^3 + 1520T^2 - 10380T + 10649)^2
$53$	$ T^{10} + 15 T^{9} + \cdots + 361342081 $ T^10 + 15T^9 + 71T^8 - 453T^7 - 4023T^6 + 185681T^5 + 2703089T^4 + 10436695T^3 + 61777527T^2 + 167336227*T + 361342081
$59$	$ T^{10} + 5 T^{9} - 19 T^{8} + \cdots + 31730689 $ T^10 + 5T^9 - 19T^8 - 183T^7 + 3639T^6 + 53032T^5 + 355360T^4 + 1744702T^3 + 6484581T^2 + 16346966*T + 31730689
$61$	$ T^{10} - 12 T^{9} + \cdots + 2801902489 $ T^10 - 12T^9 + 133T^8 - 3301T^7 + 47213T^6 - 199486T^5 + 1782782T^4 - 29202790T^3 + 183906086T^2 - 731957524*T + 2801902489
$67$	$ T^{10} - 18 T^{9} + \cdots + 1804635361 $ T^10 - 18T^9 + 258T^8 - 5392T^7 + 84461T^6 - 616120T^5 + 2862072T^4 - 13909000T^3 + 40130165T^2 - 164613875*T + 1804635361
$71$	$ T^{10} + 28 T^{9} + \cdots + 1490654881 $ T^10 + 28T^9 + 388T^8 + 2152T^7 + 5432T^6 + 61346T^5 + 507127T^4 + 4717127T^3 + 56624165T^2 + 238603620*T + 1490654881
$73$	$ T^{10} - 19 T^{9} + \cdots + 236452129 $ T^10 - 19T^9 + 273T^8 - 3812T^7 + 56324T^6 - 979208T^5 + 11086045T^4 - 76020584T^3 + 511706134T^2 + 599057166*T + 236452129
$79$	$ T^{10} + 52 T^{9} + \cdots + 2417590561 $ T^10 + 52T^9 + 1516T^8 + 29563T^7 + 421601T^6 + 4507821T^5 + 36651354T^4 + 218346493T^3 + 903594279T^2 + 2231780910*T + 2417590561
$83$	$ T^{10} + 7 T^{9} + 269 T^{8} + \cdots + 659102929 $ T^10 + 7T^9 + 269T^8 + 1850T^7 + 12840T^6 + 292181T^5 + 1293395T^4 + 1004438T^3 + 89637183T^2 + 410973384*T + 659102929
$89$	$ T^{10} + 3 T^{9} + 97 T^{8} + \cdots + 92871769 $ T^10 + 3T^9 + 97T^8 + 720T^7 - 84T^6 - 6423T^5 + 311941T^4 + 3946358T^3 + 20625841T^2 + 57282328*T + 92871769
$97$	$ T^{10} - 51 T^{9} + \cdots + 22314683161 $ T^10 - 51T^9 + 1061T^8 - 12300T^7 + 127372T^6 - 1721532T^5 + 20948030T^4 - 190179985T^3 + 1422410933T^2 - 5970907951*T + 22314683161
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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)

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

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 138)
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
$5$	\( T^{10} + 8 T^{9} + 42 T^{8} + 149 T^{7} + \cdots + 529 \) T^10 + 8T^9 + 42T^8 + 149T^7 + 389T^6 + 736T^5 + 1092T^4 + 1465T^3 + 1754T^2 + 1426*T + 529
$7$	\( T^{10} - 8 T^{9} + 31 T^{8} - 83 T^{7} + \cdots + 529 \) T^10 - 8T^9 + 31T^8 - 83T^7 + 147T^6 - 120T^5 + 36T^4 + 64T^3 + 236T^2 - 414*T + 529
$11$	\( T^{10} + 7 T^{9} + 5 T^{8} + 123 T^{7} + \cdots + 1 \) T^10 + 7T^9 + 5T^8 + 123T^7 + 894T^6 - 1354T^5 + 1016T^4 - 500T^3 + 361T^2 - 25*T + 1
$13$	\( T^{10} - 3 T^{9} + 31 T^{8} - 38 T^{7} + \cdots + 1 \) T^10 - 3T^9 + 31T^8 - 38T^7 + 114T^6 + 164T^5 - 63T^4 - 328T^3 + 192T^2 + 18*T + 1
$17$	\( T^{10} + 4 T^{9} + 5 T^{8} + 97 T^{7} + \cdots + 1849 \) T^10 + 4T^9 + 5T^8 + 97T^7 + 300T^6 - 219T^5 + 1599T^4 - 1381T^3 + 262T^2 + 344*T + 1849
$19$	\( T^{10} + 22 T^{8} + 165 T^{7} + \cdots + 64009 \) T^10 + 22T^8 + 165T^7 + 1309T^6 + 7040T^5 + 20328T^4 + 63767T^3 + 134552T^2 - 166980T + 64009
$23$	\( T^{10} - 12 T^{9} - 10 T^{8} + \cdots + 6436343 \) T^10 - 12T^9 - 10T^8 + 527T^7 - 32T^6 - 14103T^5 - 736T^4 + 278783T^3 - 121670T^2 - 3358092*T + 6436343
$29$	\( T^{10} - 25 T^{9} + 372 T^{8} + \cdots + 2866249 \) T^10 - 25T^9 + 372T^8 - 3613T^7 + 25590T^6 - 121177T^5 + 398830T^4 - 376913T^3 - 509097T^2 + 2762976*T + 2866249
$31$	\( T^{10} - 6 T^{9} + 58 T^{8} + \cdots + 896809 \) T^10 - 6T^9 + 58T^8 + 389T^7 + 449T^6 - 20305T^5 + 310898T^4 - 793207T^3 + 1387445T^2 - 1689448*T + 896809
$37$	\( T^{10} - 9 T^{9} + 26 T^{8} - 69 T^{7} + \cdots + 529 \) T^10 - 9T^9 + 26T^8 - 69T^7 + 346T^6 - 507T^5 + 911T^4 - 235T^3 + 443T^2 + 897*T + 529
$41$	\( T^{10} + 24 T^{9} + 301 T^{8} + \cdots + 529 \) T^10 + 24T^9 + 301T^8 + 1669T^7 + 4636T^6 + 5950T^5 + 17785T^4 + 4759T^3 + 29032T^2 + 7475*T + 529
$43$	\( T^{10} + 30 T^{9} + 427 T^{8} + \cdots + 192721 \) T^10 + 30T^9 + 427T^8 + 3537T^7 + 22774T^6 + 93928T^5 + 261363T^4 + 491911T^3 + 624596T^2 + 499143*T + 192721
$47$	\( (T^{5} - 24 T^{4} + 83 T^{3} + 1520 T^{2} + \cdots + 10649)^{2} \) (T^5 - 24T^4 + 83T^3 + 1520T^2 - 10380T + 10649)^2
$53$	\( T^{10} + 15 T^{9} + \cdots + 361342081 \) T^10 + 15T^9 + 71T^8 - 453T^7 - 4023T^6 + 185681T^5 + 2703089T^4 + 10436695T^3 + 61777527T^2 + 167336227*T + 361342081
$59$	\( T^{10} + 5 T^{9} - 19 T^{8} + \cdots + 31730689 \) T^10 + 5T^9 - 19T^8 - 183T^7 + 3639T^6 + 53032T^5 + 355360T^4 + 1744702T^3 + 6484581T^2 + 16346966*T + 31730689
$61$	\( T^{10} - 12 T^{9} + \cdots + 2801902489 \) T^10 - 12T^9 + 133T^8 - 3301T^7 + 47213T^6 - 199486T^5 + 1782782T^4 - 29202790T^3 + 183906086T^2 - 731957524*T + 2801902489
$67$	\( T^{10} - 18 T^{9} + \cdots + 1804635361 \) T^10 - 18T^9 + 258T^8 - 5392T^7 + 84461T^6 - 616120T^5 + 2862072T^4 - 13909000T^3 + 40130165T^2 - 164613875*T + 1804635361
$71$	\( T^{10} + 28 T^{9} + \cdots + 1490654881 \) T^10 + 28T^9 + 388T^8 + 2152T^7 + 5432T^6 + 61346T^5 + 507127T^4 + 4717127T^3 + 56624165T^2 + 238603620*T + 1490654881
$73$	\( T^{10} - 19 T^{9} + \cdots + 236452129 \) T^10 - 19T^9 + 273T^8 - 3812T^7 + 56324T^6 - 979208T^5 + 11086045T^4 - 76020584T^3 + 511706134T^2 + 599057166*T + 236452129
$79$	\( T^{10} + 52 T^{9} + \cdots + 2417590561 \) T^10 + 52T^9 + 1516T^8 + 29563T^7 + 421601T^6 + 4507821T^5 + 36651354T^4 + 218346493T^3 + 903594279T^2 + 2231780910*T + 2417590561
$83$	\( T^{10} + 7 T^{9} + 269 T^{8} + \cdots + 659102929 \) T^10 + 7T^9 + 269T^8 + 1850T^7 + 12840T^6 + 292181T^5 + 1293395T^4 + 1004438T^3 + 89637183T^2 + 410973384*T + 659102929
$89$	\( T^{10} + 3 T^{9} + 97 T^{8} + \cdots + 92871769 \) T^10 + 3T^9 + 97T^8 + 720T^7 - 84T^6 - 6423T^5 + 311941T^4 + 3946358T^3 + 20625841T^2 + 57282328*T + 92871769
$97$	\( T^{10} - 51 T^{9} + \cdots + 22314683161 \) T^10 - 51T^9 + 1061T^8 - 12300T^7 + 127372T^6 - 1721532T^5 + 20948030T^4 - 190179985T^3 + 1422410933T^2 - 5970907951*T + 22314683161
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\(n\)	\(47\)	\(235\)
\(\chi(n)\)	\(1\)	\(\zeta_{22}^{4}\)