# Properties

 Label 429.2.t.a Level $429$ Weight $2$ Character orbit 429.t Analytic conductor $3.426$ Analytic rank $0$ Dimension $104$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$429 = 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 429.t (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$104$$ Relative dimension: $$52$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$104q - 2q^{3} + 44q^{4} - 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$104q - 2q^{3} + 44q^{4} - 6q^{9} - 4q^{12} - 36q^{15} - 44q^{16} - 12q^{22} + 64q^{25} + 4q^{27} + 12q^{33} + 28q^{36} - 12q^{37} + 50q^{42} - 66q^{45} + 16q^{48} - 24q^{49} - 8q^{55} - 24q^{58} - 104q^{64} - 60q^{66} - 60q^{67} - 42q^{69} - 42q^{75} + 16q^{78} - 22q^{81} - 16q^{82} + 92q^{88} - 84q^{91} + 126q^{93} + 36q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
296.1 −2.31547 + 1.33684i 0.0244717 + 1.73188i 2.57426 4.45876i 2.76446 −2.37190 3.97739i 0.565134 0.978841i 8.41814i −2.99880 + 0.0847639i −6.40102 + 3.69563i
296.2 −2.31547 + 1.33684i 1.48761 + 0.887132i 2.57426 4.45876i −2.76446 −4.63048 0.0654292i −0.565134 + 0.978841i 8.41814i 1.42599 + 2.63942i 6.40102 3.69563i
296.3 −2.17254 + 1.25431i −0.913417 1.47162i 2.14661 3.71803i 2.90929 3.83031 + 2.05144i 0.464602 0.804714i 5.75282i −1.33134 + 2.68841i −6.32053 + 3.64916i
296.4 −2.17254 + 1.25431i −0.817753 1.52685i 2.14661 3.71803i −2.90929 3.69175 + 2.29142i −0.464602 + 0.804714i 5.75282i −1.66256 + 2.49718i 6.32053 3.64916i
296.5 −2.13554 + 1.23295i −1.72013 + 0.202872i 2.04035 3.53400i −0.635450 3.42327 2.55408i −2.03776 + 3.52950i 5.13084i 2.91769 0.697932i 1.35703 0.783481i
296.6 −2.13554 + 1.23295i 1.03576 1.38824i 2.04035 3.53400i 0.635450 −0.500264 + 4.24168i 2.03776 3.52950i 5.13084i −0.854416 2.87576i −1.35703 + 0.783481i
296.7 −1.92182 + 1.10956i −1.59672 + 0.671171i 1.46227 2.53272i 1.97176 2.32391 3.06154i 0.477966 0.827861i 2.05166i 2.09906 2.14335i −3.78937 + 2.18779i
296.8 −1.92182 + 1.10956i 1.37961 1.04722i 1.46227 2.53272i −1.97176 −1.48942 + 3.54334i −0.477966 + 0.827861i 2.05166i 0.806667 2.88951i 3.78937 2.18779i
296.9 −1.63490 + 0.943908i 0.664837 + 1.59937i 0.781925 1.35433i 0.196251 −2.59660 1.98726i 1.83210 3.17329i 0.823370i −2.11598 + 2.12665i −0.320851 + 0.185243i
296.10 −1.63490 + 0.943908i 1.05268 + 1.37545i 0.781925 1.35433i −0.196251 −3.01932 1.25509i −1.83210 + 3.17329i 0.823370i −0.783738 + 2.89582i 0.320851 0.185243i
296.11 −1.41960 + 0.819608i −0.961092 + 1.44094i 0.343515 0.594985i −1.26601 0.183365 2.83328i 0.208022 0.360304i 2.15224i −1.15260 2.76975i 1.79724 1.03764i
296.12 −1.41960 + 0.819608i 1.72843 0.111861i 0.343515 0.594985i 1.26601 −2.36201 + 1.57544i −0.208022 + 0.360304i 2.15224i 2.97497 0.386690i −1.79724 + 1.03764i
296.13 −1.25571 + 0.724987i −1.55144 0.770078i 0.0512119 0.0887017i −0.906362 2.50647 0.157780i 1.12801 1.95377i 2.75144i 1.81396 + 2.38947i 1.13813 0.657100i
296.14 −1.25571 + 0.724987i 0.108816 1.72863i 0.0512119 0.0887017i 0.906362 1.11659 + 2.24955i −1.12801 + 1.95377i 2.75144i −2.97632 0.376204i −1.13813 + 0.657100i
296.15 −1.09482 + 0.632095i −1.58106 0.707293i −0.200911 + 0.347987i 4.13662 2.17805 0.225019i −0.620239 + 1.07429i 3.03636i 1.99947 + 2.23654i −4.52886 + 2.61474i
296.16 −1.09482 + 0.632095i 0.177995 1.72288i −0.200911 + 0.347987i −4.13662 0.894153 + 1.99876i 0.620239 1.07429i 3.03636i −2.93664 0.613327i 4.52886 2.61474i
296.17 −0.949310 + 0.548084i −0.500493 + 1.65816i −0.399207 + 0.691447i 3.63865 −0.433691 1.84842i −2.32754 + 4.03142i 3.06753i −2.49901 1.65980i −3.45421 + 1.99429i
296.18 −0.949310 + 0.548084i 1.68626 + 0.395642i −0.399207 + 0.691447i −3.63865 −1.81763 + 0.548625i 2.32754 4.03142i 3.06753i 2.68693 + 1.33431i 3.45421 1.99429i
296.19 −0.716153 + 0.413471i −1.68064 + 0.418881i −0.658083 + 1.13983i −3.60945 1.03040 0.994878i −1.45351 + 2.51755i 2.74228i 2.64908 1.40797i 2.58492 1.49241i
296.20 −0.716153 + 0.413471i 1.20308 1.24603i −0.658083 + 1.13983i 3.60945 −0.346391 + 1.38979i 1.45351 2.51755i 2.74228i −0.105197 2.99816i −2.58492 + 1.49241i
See next 80 embeddings (of 104 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 329.52 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
13.e even 6 1 inner
33.d even 2 1 inner
39.h odd 6 1 inner
143.i odd 6 1 inner
429.t even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.t.a 104
3.b odd 2 1 inner 429.2.t.a 104
11.b odd 2 1 inner 429.2.t.a 104
13.e even 6 1 inner 429.2.t.a 104
33.d even 2 1 inner 429.2.t.a 104
39.h odd 6 1 inner 429.2.t.a 104
143.i odd 6 1 inner 429.2.t.a 104
429.t even 6 1 inner 429.2.t.a 104

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.t.a 104 1.a even 1 1 trivial
429.2.t.a 104 3.b odd 2 1 inner
429.2.t.a 104 11.b odd 2 1 inner
429.2.t.a 104 13.e even 6 1 inner
429.2.t.a 104 33.d even 2 1 inner
429.2.t.a 104 39.h odd 6 1 inner
429.2.t.a 104 143.i odd 6 1 inner
429.2.t.a 104 429.t even 6 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(429, [\chi])$$.