# Properties

 Label 429.2.y.a Level $429$ Weight $2$ Character orbit 429.y Analytic conductor $3.426$ Analytic rank $0$ Dimension $32$ CM discriminant -39 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$429 = 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 429.y (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{10}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 16q^{4} - 24q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 16q^{4} - 24q^{9} - 24q^{12} - 92q^{16} + 28q^{22} - 40q^{25} + 180q^{30} + 48q^{36} - 72q^{48} + 56q^{49} - 260q^{52} + 32q^{55} + 184q^{64} - 60q^{66} - 144q^{75} - 72q^{81} + 204q^{82} - 40q^{88} + 60q^{90} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
116.1 −2.60814 + 0.847436i −1.40126 1.01807i 4.46622 3.24490i −0.309156 + 0.951485i 4.51743 + 1.46780i 0 −5.67484 + 7.81075i 0.927051 + 2.85317i 2.74360i
116.2 −2.06579 + 0.671217i 1.40126 + 1.01807i 2.19893 1.59762i −0.710253 + 2.18593i −3.57806 1.16258i 0 −0.916732 + 1.26177i 0.927051 + 2.85317i 4.99242i
116.3 −1.72295 + 0.559822i −1.40126 1.01807i 1.03713 0.753522i 1.18548 3.64854i 2.98424 + 0.969639i 0 0.764591 1.05237i 0.927051 + 2.85317i 6.94993i
116.4 −0.658535 + 0.213971i 1.40126 + 1.01807i −1.23015 + 0.893756i −1.34694 + 4.14546i −1.14062 0.370608i 0 1.43285 1.97215i 0.927051 + 2.85317i 3.01814i
116.5 0.658535 0.213971i 1.40126 + 1.01807i −1.23015 + 0.893756i 1.34694 4.14546i 1.14062 + 0.370608i 0 −1.43285 + 1.97215i 0.927051 + 2.85317i 3.01814i
116.6 1.72295 0.559822i −1.40126 1.01807i 1.03713 0.753522i −1.18548 + 3.64854i −2.98424 0.969639i 0 −0.764591 + 1.05237i 0.927051 + 2.85317i 6.94993i
116.7 2.06579 0.671217i 1.40126 + 1.01807i 2.19893 1.59762i 0.710253 2.18593i 3.57806 + 1.16258i 0 0.916732 1.26177i 0.927051 + 2.85317i 4.99242i
116.8 2.60814 0.847436i −1.40126 1.01807i 4.46622 3.24490i 0.309156 0.951485i −4.51743 1.46780i 0 5.67484 7.81075i 0.927051 + 2.85317i 2.74360i
194.1 −1.65880 + 2.28314i −0.535233 + 1.64728i −1.84308 5.67240i −2.37713 + 1.72708i −2.87312 3.95451i 0 10.6402 + 3.45720i −2.42705 1.76336i 8.29218i
194.2 −1.54330 + 2.12417i 0.535233 1.64728i −1.51228 4.65433i 3.60386 2.61836i 2.67307 + 3.67916i 0 7.22625 + 2.34795i −2.42705 1.76336i 11.6961i
194.3 −0.618195 + 0.850873i −0.535233 + 1.64728i 0.276215 + 0.850102i −0.319931 + 0.232444i −1.07075 1.47376i 0 −2.89461 0.940514i −2.42705 1.76336i 0.415916i
194.4 −0.111034 + 0.152825i 0.535233 1.64728i 0.607007 + 1.86818i −2.72753 + 1.98167i 0.192316 + 0.264700i 0 −0.712214 0.231412i −2.42705 1.76336i 0.636866i
194.5 0.111034 0.152825i 0.535233 1.64728i 0.607007 + 1.86818i 2.72753 1.98167i −0.192316 0.264700i 0 0.712214 + 0.231412i −2.42705 1.76336i 0.636866i
194.6 0.618195 0.850873i −0.535233 + 1.64728i 0.276215 + 0.850102i 0.319931 0.232444i 1.07075 + 1.47376i 0 2.89461 + 0.940514i −2.42705 1.76336i 0.415916i
194.7 1.54330 2.12417i 0.535233 1.64728i −1.51228 4.65433i −3.60386 + 2.61836i −2.67307 3.67916i 0 −7.22625 2.34795i −2.42705 1.76336i 11.6961i
194.8 1.65880 2.28314i −0.535233 + 1.64728i −1.84308 5.67240i 2.37713 1.72708i 2.87312 + 3.95451i 0 −10.6402 3.45720i −2.42705 1.76336i 8.29218i
233.1 −2.60814 0.847436i −1.40126 + 1.01807i 4.46622 + 3.24490i −0.309156 0.951485i 4.51743 1.46780i 0 −5.67484 7.81075i 0.927051 2.85317i 2.74360i
233.2 −2.06579 0.671217i 1.40126 1.01807i 2.19893 + 1.59762i −0.710253 2.18593i −3.57806 + 1.16258i 0 −0.916732 1.26177i 0.927051 2.85317i 4.99242i
233.3 −1.72295 0.559822i −1.40126 + 1.01807i 1.03713 + 0.753522i 1.18548 + 3.64854i 2.98424 0.969639i 0 0.764591 + 1.05237i 0.927051 2.85317i 6.94993i
233.4 −0.658535 0.213971i 1.40126 1.01807i −1.23015 0.893756i −1.34694 4.14546i −1.14062 + 0.370608i 0 1.43285 + 1.97215i 0.927051 2.85317i 3.01814i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 272.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 CM by $$\Q(\sqrt{-39})$$
3.b odd 2 1 inner
11.d odd 10 1 inner
13.b even 2 1 inner
33.f even 10 1 inner
143.l odd 10 1 inner
429.y even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.y.a 32
3.b odd 2 1 inner 429.2.y.a 32
11.d odd 10 1 inner 429.2.y.a 32
13.b even 2 1 inner 429.2.y.a 32
33.f even 10 1 inner 429.2.y.a 32
39.d odd 2 1 CM 429.2.y.a 32
143.l odd 10 1 inner 429.2.y.a 32
429.y even 10 1 inner 429.2.y.a 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.y.a 32 1.a even 1 1 trivial
429.2.y.a 32 3.b odd 2 1 inner
429.2.y.a 32 11.d odd 10 1 inner
429.2.y.a 32 13.b even 2 1 inner
429.2.y.a 32 33.f even 10 1 inner
429.2.y.a 32 39.d odd 2 1 CM
429.2.y.a 32 143.l odd 10 1 inner
429.2.y.a 32 429.y even 10 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{32} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(429, [\chi])$$.