# Properties

 Label 4410.2.b.e 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: $$\Q(\zeta_{24})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 630) 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 $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\zeta_{24}^{6} q^{2} - q^{4} + q^{5} + \zeta_{24}^{6} q^{8} +O(q^{10})$$ $$q -\zeta_{24}^{6} q^{2} - q^{4} + q^{5} + \zeta_{24}^{6} q^{8} -\zeta_{24}^{6} q^{10} + ( -2 - \zeta_{24} + \zeta_{24}^{3} + 4 \zeta_{24}^{4} - \zeta_{24}^{7} ) q^{11} + ( \zeta_{24}^{5} + 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{13} + q^{16} + ( -2 \zeta_{24} + 4 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{17} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{19} - q^{20} + ( 4 \zeta_{24}^{2} - \zeta_{24}^{5} - 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{22} + ( 1 - 2 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{23} + q^{25} + ( 2 + \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{7} ) q^{26} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{29} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{31} -\zeta_{24}^{6} q^{32} + ( 2 - 4 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{34} + ( -6 - \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{37} + ( -1 + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{38} + \zeta_{24}^{6} q^{40} + ( -4 - 2 \zeta_{24} - 2 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{41} + ( -2 - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{43} + ( 2 + \zeta_{24} - \zeta_{24}^{3} - 4 \zeta_{24}^{4} + \zeta_{24}^{7} ) q^{44} + ( -2 - 2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{46} + ( -2 - 2 \zeta_{24}^{2} - 4 \zeta_{24}^{5} + \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{47} -\zeta_{24}^{6} q^{50} + ( -\zeta_{24}^{5} - 2 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{52} + ( -2 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 4 \zeta_{24}^{5} + \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{53} + ( -2 - \zeta_{24} + \zeta_{24}^{3} + 4 \zeta_{24}^{4} - \zeta_{24}^{7} ) q^{55} + ( 4 + 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{58} + ( -6 + 5 \zeta_{24} + 4 \zeta_{24}^{2} + 5 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{59} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{61} + ( 2 - 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{62} - q^{64} + ( \zeta_{24}^{5} + 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{65} + ( 6 - 6 \zeta_{24} - 8 \zeta_{24}^{2} - 6 \zeta_{24}^{3} + 4 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{67} + ( 2 \zeta_{24} - 4 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{68} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{71} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 6 \zeta_{24}^{6} ) q^{73} + ( \zeta_{24}^{5} + 6 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{74} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{76} + ( 6 - 4 \zeta_{24} + 8 \zeta_{24}^{2} - 4 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 4 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{79} + q^{80} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{82} + ( 2 - 8 \zeta_{24} - 8 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{83} + ( -2 \zeta_{24} + 4 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{85} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{86} + ( -4 \zeta_{24}^{2} + \zeta_{24}^{5} + 2 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{88} + ( -4 - 3 \zeta_{24} - 3 \zeta_{24}^{3} + 7 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{89} + ( -1 + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{92} + ( -1 + 4 \zeta_{24} - 4 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{94} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{95} + ( -6 + 6 \zeta_{24} - 6 \zeta_{24}^{3} + 12 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 6 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{4} + 8q^{5} + O(q^{10})$$ $$8q - 8q^{4} + 8q^{5} + 8q^{16} - 8q^{20} + 8q^{25} + 16q^{26} - 48q^{37} - 8q^{38} - 32q^{41} - 16q^{43} - 16q^{46} - 16q^{47} + 32q^{58} - 48q^{59} + 16q^{62} - 8q^{64} + 48q^{67} + 48q^{79} + 8q^{80} + 16q^{83} - 32q^{89} + O(q^{100})$$

## 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
 0.258819 + 0.965926i −0.258819 − 0.965926i 0.965926 + 0.258819i −0.965926 − 0.258819i −0.965926 + 0.258819i 0.965926 − 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i
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.e 8
3.b odd 2 1 4410.2.b.b 8
7.b odd 2 1 4410.2.b.b 8
7.c even 3 1 630.2.be.a 8
7.d odd 6 1 630.2.be.b yes 8
21.c even 2 1 inner 4410.2.b.e 8
21.g even 6 1 630.2.be.a 8
21.h odd 6 1 630.2.be.b yes 8
35.i odd 6 1 3150.2.bf.c 8
35.j even 6 1 3150.2.bf.b 8
35.k even 12 1 3150.2.bp.c 8
35.k even 12 1 3150.2.bp.f 8
35.l odd 12 1 3150.2.bp.a 8
35.l odd 12 1 3150.2.bp.d 8
105.o odd 6 1 3150.2.bf.c 8
105.p even 6 1 3150.2.bf.b 8
105.w odd 12 1 3150.2.bp.a 8
105.w odd 12 1 3150.2.bp.d 8
105.x even 12 1 3150.2.bp.c 8
105.x even 12 1 3150.2.bp.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.be.a 8 7.c even 3 1
630.2.be.a 8 21.g even 6 1
630.2.be.b yes 8 7.d odd 6 1
630.2.be.b yes 8 21.h odd 6 1
3150.2.bf.b 8 35.j even 6 1
3150.2.bf.b 8 105.p even 6 1
3150.2.bf.c 8 35.i odd 6 1
3150.2.bf.c 8 105.o odd 6 1
3150.2.bp.a 8 35.l odd 12 1
3150.2.bp.a 8 105.w odd 12 1
3150.2.bp.c 8 35.k even 12 1
3150.2.bp.c 8 105.x even 12 1
3150.2.bp.d 8 35.l odd 12 1
3150.2.bp.d 8 105.w odd 12 1
3150.2.bp.f 8 35.k even 12 1
3150.2.bp.f 8 105.x even 12 1
4410.2.b.b 8 3.b odd 2 1
4410.2.b.b 8 7.b odd 2 1
4410.2.b.e 8 1.a even 1 1 trivial
4410.2.b.e 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 $$S_{2}^{\mathrm{new}}(4410, [\chi])$$:

 $$T_{11}^{8} + 56 T_{11}^{6} + 978 T_{11}^{4} + 6008 T_{11}^{2} + 9409$$ $$T_{17}^{4} - 40 T_{17}^{2} - 96 T_{17} - 32$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{4}$$
$3$ 1
$5$ $$( 1 - T )^{8}$$
$7$ 1
$11$ $$1 - 32 T^{2} + 670 T^{4} - 9920 T^{6} + 121411 T^{8} - 1200320 T^{10} + 9809470 T^{12} - 56689952 T^{14} + 214358881 T^{16}$$
$13$ $$1 - 80 T^{2} + 3006 T^{4} - 69568 T^{6} + 1087139 T^{8} - 11756992 T^{10} + 85854366 T^{12} - 386144720 T^{14} + 815730721 T^{16}$$
$17$ $$( 1 + 28 T^{2} - 96 T^{3} + 342 T^{4} - 1632 T^{5} + 8092 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$1 - 116 T^{2} + 6330 T^{4} - 213136 T^{6} + 4859555 T^{8} - 76942096 T^{10} + 824931930 T^{12} - 5457322196 T^{14} + 16983563041 T^{16}$$
$23$ $$( 1 - 78 T^{2} + 2531 T^{4} - 41262 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 36 T^{2} + 470 T^{4} - 30276 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 - 100 T^{2} + 4294 T^{4} - 96100 T^{6} + 923521 T^{8} )^{2}$$
$37$ $$( 1 + 24 T + 360 T^{2} + 3480 T^{3} + 25055 T^{4} + 128760 T^{5} + 492840 T^{6} + 1215672 T^{7} + 1874161 T^{8} )^{2}$$
$41$ $$( 1 + 16 T + 232 T^{2} + 2000 T^{3} + 15591 T^{4} + 82000 T^{5} + 389992 T^{6} + 1102736 T^{7} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 + 8 T + 180 T^{2} + 1000 T^{3} + 11750 T^{4} + 43000 T^{5} + 332820 T^{6} + 636056 T^{7} + 3418801 T^{8} )^{2}$$
$47$ $$( 1 + 8 T + 142 T^{2} + 688 T^{3} + 8355 T^{4} + 32336 T^{5} + 313678 T^{6} + 830584 T^{7} + 4879681 T^{8} )^{2}$$
$53$ $$1 - 148 T^{2} + 10938 T^{4} - 528080 T^{6} + 25559075 T^{8} - 1483376720 T^{10} + 86306081178 T^{12} - 3280325447092 T^{14} + 62259690411361 T^{16}$$
$59$ $$( 1 + 24 T + 352 T^{2} + 3528 T^{3} + 29874 T^{4} + 208152 T^{5} + 1225312 T^{6} + 4929096 T^{7} + 12117361 T^{8} )^{2}$$
$61$ $$1 - 248 T^{2} + 33756 T^{4} - 3125128 T^{6} + 217206374 T^{8} - 11628601288 T^{10} + 467380208796 T^{12} - 12777052841528 T^{14} + 191707312997281 T^{16}$$
$67$ $$( 1 - 24 T + 244 T^{2} - 1080 T^{3} + 2694 T^{4} - 72360 T^{5} + 1095316 T^{6} - 7218312 T^{7} + 20151121 T^{8} )^{2}$$
$71$ $$1 - 280 T^{2} + 41532 T^{4} - 4371368 T^{6} + 352979654 T^{8} - 22036066088 T^{10} + 1055397935292 T^{12} - 35868079497880 T^{14} + 645753531245761 T^{16}$$
$73$ $$( 1 - 156 T^{2} + 12134 T^{4} - 831324 T^{6} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 - 24 T + 324 T^{2} - 3096 T^{3} + 25622 T^{4} - 244584 T^{5} + 2022084 T^{6} - 11832936 T^{7} + 38950081 T^{8} )^{2}$$
$83$ $$( 1 - 8 T + 148 T^{2} - 1192 T^{3} + 18438 T^{4} - 98936 T^{5} + 1019572 T^{6} - 4574296 T^{7} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 + 16 T + 304 T^{2} + 3344 T^{3} + 36834 T^{4} + 297616 T^{5} + 2407984 T^{6} + 11279504 T^{7} + 62742241 T^{8} )^{2}$$
$97$ $$1 + 24 T^{2} + 19036 T^{4} - 653400 T^{6} + 157960902 T^{8} - 6147840600 T^{10} + 1685243393116 T^{12} + 19991328118296 T^{14} + 7837433594376961 T^{16}$$