# Properties

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

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## 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: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ 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.b 8
3.b odd 2 1 4410.2.b.e 8
7.b odd 2 1 4410.2.b.e 8
7.c even 3 1 630.2.be.b yes 8
7.d odd 6 1 630.2.be.a 8
21.c even 2 1 inner 4410.2.b.b 8
21.g even 6 1 630.2.be.b yes 8
21.h odd 6 1 630.2.be.a 8
35.i odd 6 1 3150.2.bf.b 8
35.j even 6 1 3150.2.bf.c 8
35.k even 12 1 3150.2.bp.a 8
35.k even 12 1 3150.2.bp.d 8
35.l odd 12 1 3150.2.bp.c 8
35.l odd 12 1 3150.2.bp.f 8
105.o odd 6 1 3150.2.bf.b 8
105.p even 6 1 3150.2.bf.c 8
105.w odd 12 1 3150.2.bp.c 8
105.w odd 12 1 3150.2.bp.f 8
105.x even 12 1 3150.2.bp.a 8
105.x even 12 1 3150.2.bp.d 8

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

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