# Properties

 Label 4410.2.b.a 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: $$\Z[a_1, \ldots, a_{11}]$$ 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} + ( \zeta_{16}^{3} - 2 \zeta_{16}^{4} + \zeta_{16}^{5} ) q^{11} + ( \zeta_{16} - \zeta_{16}^{2} + \zeta_{16}^{3} + 4 \zeta_{16}^{4} + \zeta_{16}^{5} - \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{13} + q^{16} + ( -\zeta_{16}^{2} + 3 \zeta_{16}^{3} - 3 \zeta_{16}^{5} + \zeta_{16}^{6} ) q^{17} + ( -2 \zeta_{16} + \zeta_{16}^{2} + \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{19} + q^{20} + ( -2 + \zeta_{16} - \zeta_{16}^{7} ) q^{22} + ( \zeta_{16} - \zeta_{16}^{2} - 5 \zeta_{16}^{3} + 2 \zeta_{16}^{4} - 5 \zeta_{16}^{5} - \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{23} + q^{25} + ( 4 + \zeta_{16} - \zeta_{16}^{2} + \zeta_{16}^{3} - \zeta_{16}^{5} + \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{26} + ( \zeta_{16} - 2 \zeta_{16}^{2} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{4} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{29} + ( -2 \zeta_{16} + 2 \zeta_{16}^{2} - \zeta_{16}^{3} - 4 \zeta_{16}^{4} - \zeta_{16}^{5} + 2 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{31} -\zeta_{16}^{4} q^{32} + ( -3 \zeta_{16} + \zeta_{16}^{2} + \zeta_{16}^{6} - 3 \zeta_{16}^{7} ) q^{34} + ( -2 + 3 \zeta_{16}^{2} - \zeta_{16}^{3} + \zeta_{16}^{5} - 3 \zeta_{16}^{6} ) q^{37} + ( \zeta_{16}^{2} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - \zeta_{16}^{6} ) q^{38} -\zeta_{16}^{4} q^{40} + ( -6 + \zeta_{16} + \zeta_{16}^{3} - \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{41} + ( -2 + 2 \zeta_{16} - \zeta_{16}^{2} - 5 \zeta_{16}^{3} + 5 \zeta_{16}^{5} + \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{43} + ( -\zeta_{16}^{3} + 2 \zeta_{16}^{4} - \zeta_{16}^{5} ) q^{44} + ( 2 - 5 \zeta_{16} - \zeta_{16}^{2} + \zeta_{16}^{3} - \zeta_{16}^{5} + \zeta_{16}^{6} + 5 \zeta_{16}^{7} ) q^{46} + ( 6 - \zeta_{16} + \zeta_{16}^{2} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{47} -\zeta_{16}^{4} q^{50} + ( -\zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{3} - 4 \zeta_{16}^{4} - \zeta_{16}^{5} + \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{52} + ( 5 \zeta_{16} + 2 \zeta_{16}^{2} - \zeta_{16}^{3} + 4 \zeta_{16}^{4} - \zeta_{16}^{5} + 2 \zeta_{16}^{6} + 5 \zeta_{16}^{7} ) q^{53} + ( -\zeta_{16}^{3} + 2 \zeta_{16}^{4} - \zeta_{16}^{5} ) q^{55} + ( -2 + 2 \zeta_{16} - 2 \zeta_{16}^{2} + \zeta_{16}^{3} - \zeta_{16}^{5} + 2 \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{58} + ( -2 + 2 \zeta_{16} - \zeta_{16}^{2} - 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} + \zeta_{16}^{6} - 2 \zeta_{16}^{7} ) q^{59} + ( 2 \zeta_{16} + \zeta_{16}^{2} + 6 \zeta_{16}^{4} + \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{61} + ( -4 - \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} - 4 \zeta_{16}^{4} - \zeta_{16}^{5} + \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{65} + ( -5 \zeta_{16} + \zeta_{16}^{2} - \zeta_{16}^{6} + 5 \zeta_{16}^{7} ) q^{67} + ( \zeta_{16}^{2} - 3 \zeta_{16}^{3} + 3 \zeta_{16}^{5} - \zeta_{16}^{6} ) q^{68} + ( 3 \zeta_{16}^{2} + 4 \zeta_{16}^{3} + 2 \zeta_{16}^{4} + 4 \zeta_{16}^{5} + 3 \zeta_{16}^{6} ) q^{71} + ( 5 \zeta_{16} + 4 \zeta_{16}^{2} - \zeta_{16}^{3} + 4 \zeta_{16}^{4} - \zeta_{16}^{5} + 4 \zeta_{16}^{6} + 5 \zeta_{16}^{7} ) q^{73} + ( \zeta_{16} - 3 \zeta_{16}^{2} + 2 \zeta_{16}^{4} - 3 \zeta_{16}^{6} + \zeta_{16}^{7} ) q^{74} + ( 2 \zeta_{16} - \zeta_{16}^{2} - \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{76} + ( -4 \zeta_{16}^{2} + 4 \zeta_{16}^{6} ) q^{79} - q^{80} + ( -\zeta_{16} - \zeta_{16}^{3} + 6 \zeta_{16}^{4} - \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{82} + ( -2 - 2 \zeta_{16} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{83} + ( \zeta_{16}^{2} - 3 \zeta_{16}^{3} + 3 \zeta_{16}^{5} - \zeta_{16}^{6} ) q^{85} + ( 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^{86} + ( 2 - \zeta_{16} + \zeta_{16}^{7} ) q^{88} + ( 6 + 4 \zeta_{16} - 2 \zeta_{16}^{2} + 4 \zeta_{16}^{3} - 4 \zeta_{16}^{5} + 2 \zeta_{16}^{6} - 4 \zeta_{16}^{7} ) q^{89} + ( -\zeta_{16} + \zeta_{16}^{2} + 5 \zeta_{16}^{3} - 2 \zeta_{16}^{4} + 5 \zeta_{16}^{5} + \zeta_{16}^{6} - \zeta_{16}^{7} ) q^{92} + ( 2 \zeta_{16} - \zeta_{16}^{2} + \zeta_{16}^{3} - 6 \zeta_{16}^{4} + \zeta_{16}^{5} - \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{94} + ( 2 \zeta_{16} - \zeta_{16}^{2} - \zeta_{16}^{6} + 2 \zeta_{16}^{7} ) q^{95} + ( 4 \zeta_{16}^{3} + 6 \zeta_{16}^{4} + 4 \zeta_{16}^{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{4} - 8q^{5} + O(q^{10})$$ $$8q - 8q^{4} - 8q^{5} + 8q^{16} + 8q^{20} - 16q^{22} + 8q^{25} + 32q^{26} - 16q^{37} - 48q^{41} - 16q^{43} + 16q^{46} + 48q^{47} - 16q^{58} - 16q^{59} - 32q^{62} - 8q^{64} - 8q^{80} - 16q^{83} + 16q^{88} + 48q^{89} + O(q^{100})$$

## 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.a 8
3.b odd 2 1 4410.2.b.d yes 8
7.b odd 2 1 4410.2.b.d yes 8
21.c even 2 1 inner 4410.2.b.a 8

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

## Hecke kernels

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

 $$T_{11}^{8} + 24 T_{11}^{6} + 148 T_{11}^{4} + 176 T_{11}^{2} + 4$$ $$T_{17}^{4} - 40 T_{17}^{2} - 72 T_{17} + 94$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{4}$$
$3$ 1
$5$ $$( 1 + T )^{8}$$
$7$ 1
$11$ $$1 - 64 T^{2} + 1952 T^{4} - 37312 T^{6} + 489570 T^{8} - 4514752 T^{10} + 28579232 T^{12} - 113379904 T^{14} + 214358881 T^{16}$$
$13$ $$1 - 16 T^{2} + 308 T^{4} - 5424 T^{6} + 50118 T^{8} - 916656 T^{10} + 8796788 T^{12} - 77228944 T^{14} + 815730721 T^{16}$$
$17$ $$( 1 + 28 T^{2} - 72 T^{3} + 468 T^{4} - 1224 T^{5} + 8092 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$1 - 112 T^{2} + 5956 T^{4} - 197328 T^{6} + 4474022 T^{8} - 71235408 T^{10} + 776191876 T^{12} - 5269138672 T^{14} + 16983563041 T^{16}$$
$23$ $$1 + 48 T^{2} + 980 T^{4} - 4592 T^{6} - 398298 T^{8} - 2429168 T^{10} + 274244180 T^{12} + 7105722672 T^{14} + 78310985281 T^{16}$$
$29$ $$1 - 144 T^{2} + 8384 T^{4} - 272784 T^{6} + 7329058 T^{8} - 229411344 T^{10} + 5929843904 T^{12} - 85654558224 T^{14} + 500246412961 T^{16}$$
$31$ $$1 - 112 T^{2} + 5984 T^{4} - 201840 T^{6} + 5984898 T^{8} - 193968240 T^{10} + 5526349664 T^{12} - 99400412272 T^{14} + 852891037441 T^{16}$$
$37$ $$( 1 + 8 T + 132 T^{2} + 784 T^{3} + 7188 T^{4} + 29008 T^{5} + 180708 T^{6} + 405224 T^{7} + 1874161 T^{8} )^{2}$$
$41$ $$( 1 + 24 T + 372 T^{2} + 3720 T^{3} + 28158 T^{4} + 152520 T^{5} + 625332 T^{6} + 1654104 T^{7} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 + 8 T + 76 T^{2} + 256 T^{3} + 1492 T^{4} + 11008 T^{5} + 140524 T^{6} + 636056 T^{7} + 3418801 T^{8} )^{2}$$
$47$ $$( 1 - 24 T + 380 T^{2} - 3968 T^{3} + 31844 T^{4} - 186496 T^{5} + 839420 T^{6} - 2491752 T^{7} + 4879681 T^{8} )^{2}$$
$53$ $$1 - 120 T^{2} + 14060 T^{4} - 973896 T^{6} + 61968070 T^{8} - 2735673864 T^{10} + 110940162860 T^{12} - 2659723335480 T^{14} + 62259690411361 T^{16}$$
$59$ $$( 1 + 8 T + 176 T^{2} + 888 T^{3} + 12914 T^{4} + 52392 T^{5} + 612656 T^{6} + 1643032 T^{7} + 12117361 T^{8} )^{2}$$
$61$ $$1 - 304 T^{2} + 47620 T^{4} - 4871952 T^{6} + 350453414 T^{8} - 18128533392 T^{10} + 659338948420 T^{12} - 15662193805744 T^{14} + 191707312997281 T^{16}$$
$67$ $$( 1 + 164 T^{2} - 200 T^{3} + 14052 T^{4} - 13400 T^{5} + 736196 T^{6} + 20151121 T^{8} )^{2}$$
$71$ $$1 - 352 T^{2} + 54788 T^{4} - 5305888 T^{6} + 402214086 T^{8} - 26746981408 T^{10} + 1392255178628 T^{12} - 45091299940192 T^{14} + 645753531245761 T^{16}$$
$73$ $$1 - 184 T^{2} + 31436 T^{4} - 3096008 T^{6} + 278954470 T^{8} - 16498626632 T^{10} + 892727104076 T^{12} - 27845497637176 T^{14} + 806460091894081 T^{16}$$
$79$ $$( 1 + 126 T^{2} + 6241 T^{4} )^{4}$$
$83$ $$( 1 + 8 T + 324 T^{2} + 1896 T^{3} + 40022 T^{4} + 157368 T^{5} + 2232036 T^{6} + 4574296 T^{7} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 - 24 T + 428 T^{2} - 5032 T^{3} + 54470 T^{4} - 447848 T^{5} + 3390188 T^{6} - 16919256 T^{7} + 62742241 T^{8} )^{2}$$
$97$ $$1 - 504 T^{2} + 122652 T^{4} - 19352520 T^{6} + 2189485382 T^{8} - 182087860680 T^{10} + 10858293373212 T^{12} - 419817890484216 T^{14} + 7837433594376961 T^{16}$$