# Properties

 Label 4410.2.b.f Level $4410$ Weight $2$ Character orbit 4410.b Analytic conductor $35.214$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4410.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$35.2140272914$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{16})$$ Defining polynomial: $$x^{8} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{16}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{16}^{4} q^{2} - q^{4} + q^{5} -\zeta_{16}^{4} q^{8} +O(q^{10})$$ $$q + \zeta_{16}^{4} q^{2} - q^{4} + q^{5} -\zeta_{16}^{4} q^{8} + \zeta_{16}^{4} q^{10} + ( -2 \zeta_{16} - \zeta_{16}^{3} - 2 \zeta_{16}^{4} - \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{11} + ( \zeta_{16} - \zeta_{16}^{2} + \zeta_{16}^{3} + \zeta_{16}^{5} - \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{13} + q^{16} + ( -\zeta_{16}^{2} - \zeta_{16}^{3} + \zeta_{16}^{5} + \zeta_{16}^{6} ) q^{17} + ( -2 \zeta_{16} + 3 \zeta_{16}^{2} + 3 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{19} - q^{20} + ( 2 + \zeta_{16} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{22} + ( -\zeta_{16} + \zeta_{16}^{2} + \zeta_{16}^{3} - 2 \zeta_{16}^{4} + \zeta_{16}^{5} + \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{23} + q^{25} + ( -\zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{3} + \zeta_{16}^{5} - \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{26} + ( 3 \zeta_{16} + 2 \zeta_{16}^{2} + 2 \zeta_{16}^{4} + 2 \zeta_{16}^{6} + 3 \zeta_{16}^{7} ) q^{29} + ( 2 \zeta_{16} - 2 \zeta_{16}^{2} + \zeta_{16}^{3} + \zeta_{16}^{5} - 2 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{31} + \zeta_{16}^{4} q^{32} + ( -\zeta_{16} - \zeta_{16}^{2} - \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{34} + ( 2 - 2 \zeta_{16} + \zeta_{16}^{2} - 5 \zeta_{16}^{3} + 5 \zeta_{16}^{5} - \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{37} + ( -3 \zeta_{16}^{2} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} + 3 \zeta_{16}^{6} ) q^{38} -\zeta_{16}^{4} q^{40} + ( -2 - \zeta_{16} + 4 \zeta_{16}^{2} - \zeta_{16}^{3} + \zeta_{16}^{5} - 4 \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{41} + ( -2 + 4 \zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{3} + \zeta_{16}^{5} - \zeta_{16}^{6} - 4 \zeta_{16}^{7} ) q^{43} + ( 2 \zeta_{16} + \zeta_{16}^{3} + 2 \zeta_{16}^{4} + \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{44} + ( 2 - \zeta_{16} - \zeta_{16}^{2} + \zeta_{16}^{3} - \zeta_{16}^{5} + \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{46} + ( 2 + 3 \zeta_{16} + \zeta_{16}^{2} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - \zeta_{16}^{6} - 3 \zeta_{16}^{7} ) q^{47} + \zeta_{16}^{4} q^{50} + ( -\zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{3} - \zeta_{16}^{5} + \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{52} + ( -\zeta_{16} - 2 \zeta_{16}^{2} - 3 \zeta_{16}^{3} + 4 \zeta_{16}^{4} - 3 \zeta_{16}^{5} - 2 \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{53} + ( -2 \zeta_{16} - \zeta_{16}^{3} - 2 \zeta_{16}^{4} - \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{55} + ( -2 - 2 \zeta_{16}^{2} - 3 \zeta_{16}^{3} + 3 \zeta_{16}^{5} + 2 \zeta_{16}^{6} ) q^{58} + ( 6 + 2 \zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{59} + ( 2 \zeta_{16} - \zeta_{16}^{2} - 2 \zeta_{16}^{4} - \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{61} + ( -\zeta_{16} + 2 \zeta_{16}^{2} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{62} - q^{64} + ( \zeta_{16} - \zeta_{16}^{2} + \zeta_{16}^{3} + \zeta_{16}^{5} - \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{65} + ( -4 + 3 \zeta_{16} + 3 \zeta_{16}^{2} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - 3 \zeta_{16}^{6} - 3 \zeta_{16}^{7} ) q^{67} + ( \zeta_{16}^{2} + \zeta_{16}^{3} - \zeta_{16}^{5} - \zeta_{16}^{6} ) q^{68} + ( -\zeta_{16}^{2} - 4 \zeta_{16}^{3} - 2 \zeta_{16}^{4} - 4 \zeta_{16}^{5} - \zeta_{16}^{6} ) q^{71} + ( -3 \zeta_{16} - 5 \zeta_{16}^{3} - 4 \zeta_{16}^{4} - 5 \zeta_{16}^{5} - 3 \zeta_{16}^{7} ) q^{73} + ( -5 \zeta_{16} + \zeta_{16}^{2} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{4} - 2 \zeta_{16}^{5} + \zeta_{16}^{6} - 5 \zeta_{16}^{7} ) q^{74} + ( 2 \zeta_{16} - 3 \zeta_{16}^{2} - 3 \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{76} + ( -4 \zeta_{16} - 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} + 4 \zeta_{16}^{7} ) q^{79} + q^{80} + ( -\zeta_{16} + 4 \zeta_{16}^{2} - \zeta_{16}^{3} - 2 \zeta_{16}^{4} - \zeta_{16}^{5} + 4 \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{82} + ( -2 + 2 \zeta_{16} + 8 \zeta_{16}^{2} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} - 8 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{83} + ( -\zeta_{16}^{2} - \zeta_{16}^{3} + \zeta_{16}^{5} + \zeta_{16}^{6} ) q^{85} + ( -\zeta_{16} + \zeta_{16}^{2} + 4 \zeta_{16}^{3} - 2 \zeta_{16}^{4} + 4 \zeta_{16}^{5} + \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{86} + ( -2 - \zeta_{16} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{88} + ( 10 + 2 \zeta_{16}^{2} + 4 \zeta_{16}^{3} - 4 \zeta_{16}^{5} - 2 \zeta_{16}^{6} ) q^{89} + ( \zeta_{16} - \zeta_{16}^{2} - \zeta_{16}^{3} + 2 \zeta_{16}^{4} - \zeta_{16}^{5} - \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{92} + ( -2 \zeta_{16} + \zeta_{16}^{2} + 3 \zeta_{16}^{3} + 2 \zeta_{16}^{4} + 3 \zeta_{16}^{5} + \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{94} + ( -2 \zeta_{16} + 3 \zeta_{16}^{2} + 3 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{95} + ( -4 \zeta_{16}^{2} - 2 \zeta_{16}^{4} - 4 \zeta_{16}^{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{4} + 8q^{5} + O(q^{10})$$ $$8q - 8q^{4} + 8q^{5} + 8q^{16} - 8q^{20} + 16q^{22} + 8q^{25} + 16q^{37} - 16q^{41} - 16q^{43} + 16q^{46} + 16q^{47} - 16q^{58} + 48q^{59} - 8q^{64} - 32q^{67} + 8q^{80} - 16q^{83} - 16q^{88} + 80q^{89} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4410\mathbb{Z}\right)^\times$$.

 $$n$$ $$1081$$ $$2647$$ $$3431$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
881.1
 −0.923880 + 0.382683i 0.382683 + 0.923880i −0.382683 − 0.923880i 0.923880 − 0.382683i 0.923880 + 0.382683i −0.382683 + 0.923880i 0.382683 − 0.923880i −0.923880 − 0.382683i
1.00000i 0 −1.00000 1.00000 0 0 1.00000i 0 1.00000i
881.2 1.00000i 0 −1.00000 1.00000 0 0 1.00000i 0 1.00000i
881.3 1.00000i 0 −1.00000 1.00000 0 0 1.00000i 0 1.00000i
881.4 1.00000i 0 −1.00000 1.00000 0 0 1.00000i 0 1.00000i
881.5 1.00000i 0 −1.00000 1.00000 0 0 1.00000i 0 1.00000i
881.6 1.00000i 0 −1.00000 1.00000 0 0 1.00000i 0 1.00000i
881.7 1.00000i 0 −1.00000 1.00000 0 0 1.00000i 0 1.00000i
881.8 1.00000i 0 −1.00000 1.00000 0 0 1.00000i 0 1.00000i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 881.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4410.2.b.f yes 8
3.b odd 2 1 4410.2.b.c 8
7.b odd 2 1 4410.2.b.c 8
21.c even 2 1 inner 4410.2.b.f yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4410.2.b.c 8 3.b odd 2 1
4410.2.b.c 8 7.b odd 2 1
4410.2.b.f yes 8 1.a even 1 1 trivial
4410.2.b.f yes 8 21.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4410, [\chi])$$:

 $$T_{11}^{8} + 56 T_{11}^{6} + 852 T_{11}^{4} + 2032 T_{11}^{2} + 1156$$ $$T_{17}^{4} - 8 T_{17}^{2} - 8 T_{17} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{4}$$
$3$ 1
$5$ $$( 1 - T )^{8}$$
$7$ 1
$11$ $$1 - 32 T^{2} + 544 T^{4} - 8352 T^{6} + 109154 T^{8} - 1010592 T^{10} + 7964704 T^{12} - 56689952 T^{14} + 214358881 T^{16}$$
$13$ $$1 - 80 T^{2} + 2996 T^{4} - 69104 T^{6} + 1078470 T^{8} - 11678576 T^{10} + 85568756 T^{12} - 386144720 T^{14} + 815730721 T^{16}$$
$17$ $$( 1 + 60 T^{2} - 8 T^{3} + 1460 T^{4} - 136 T^{5} + 17340 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$1 - 48 T^{2} + 1092 T^{4} - 20496 T^{6} + 392870 T^{8} - 7399056 T^{10} + 142310532 T^{12} - 2258202288 T^{14} + 16983563041 T^{16}$$
$23$ $$1 - 144 T^{2} + 9428 T^{4} - 376368 T^{6} + 10282534 T^{8} - 199098672 T^{10} + 2638340948 T^{12} - 21317168016 T^{14} + 78310985281 T^{16}$$
$29$ $$1 - 112 T^{2} + 7488 T^{4} - 339056 T^{6} + 11412770 T^{8} - 285146096 T^{10} + 5296120128 T^{12} - 66620211952 T^{14} + 500246412961 T^{16}$$
$31$ $$1 - 176 T^{2} + 14816 T^{4} - 791216 T^{6} + 29218434 T^{8} - 760358576 T^{10} + 13682887136 T^{12} - 156200647856 T^{14} + 852891037441 T^{16}$$
$37$ $$( 1 - 8 T + 52 T^{2} - 432 T^{3} + 3764 T^{4} - 15984 T^{5} + 71188 T^{6} - 405224 T^{7} + 1874161 T^{8} )^{2}$$
$41$ $$( 1 + 8 T + 116 T^{2} + 664 T^{3} + 6526 T^{4} + 27224 T^{5} + 194996 T^{6} + 551368 T^{7} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 + 8 T + 124 T^{2} + 720 T^{3} + 7508 T^{4} + 30960 T^{5} + 229276 T^{6} + 636056 T^{7} + 3418801 T^{8} )^{2}$$
$47$ $$( 1 - 8 T + 156 T^{2} - 880 T^{3} + 10404 T^{4} - 41360 T^{5} + 344604 T^{6} - 830584 T^{7} + 4879681 T^{8} )^{2}$$
$53$ $$1 - 248 T^{2} + 34028 T^{4} - 3010504 T^{6} + 189188806 T^{8} - 8456505736 T^{10} + 268497287468 T^{12} - 5496761559992 T^{14} + 62259690411361 T^{16}$$
$59$ $$( 1 - 24 T + 432 T^{2} - 4904 T^{3} + 44786 T^{4} - 289336 T^{5} + 1503792 T^{6} - 4929096 T^{7} + 12117361 T^{8} )^{2}$$
$61$ $$1 - 432 T^{2} + 84740 T^{4} - 9833616 T^{6} + 737501350 T^{8} - 36590885136 T^{10} + 1173296566340 T^{12} - 22256801723952 T^{14} + 191707312997281 T^{16}$$
$67$ $$( 1 + 16 T + 276 T^{2} + 2936 T^{3} + 27492 T^{4} + 196712 T^{5} + 1238964 T^{6} + 4812208 T^{7} + 20151121 T^{8} )^{2}$$
$71$ $$1 - 416 T^{2} + 80900 T^{4} - 9781984 T^{6} + 820517062 T^{8} - 49310981344 T^{10} + 2055804992900 T^{12} - 53289718111136 T^{14} + 645753531245761 T^{16}$$
$73$ $$1 - 248 T^{2} + 27212 T^{4} - 1705864 T^{6} + 97521382 T^{8} - 9090549256 T^{10} + 772772934092 T^{12} - 37530888119672 T^{14} + 806460091894081 T^{16}$$
$79$ $$( 1 + 188 T^{2} + 19270 T^{4} + 1173308 T^{6} + 38950081 T^{8} )^{2}$$
$83$ $$( 1 + 8 T + 68 T^{2} + 360 T^{3} + 7766 T^{4} + 29880 T^{5} + 468452 T^{6} + 4574296 T^{7} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 - 40 T + 876 T^{2} - 12824 T^{3} + 139590 T^{4} - 1141336 T^{5} + 6938796 T^{6} - 28198760 T^{7} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 316 T^{2} + 43270 T^{4} - 2973244 T^{6} + 88529281 T^{8} )^{2}$$