# Properties

 Label 429.2.f.a Level $429$ Weight $2$ Character orbit 429.f Analytic conductor $3.426$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$429 = 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 429.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$48$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48q - 6q^{3} + 48q^{4} + 14q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q - 6q^{3} + 48q^{4} + 14q^{9} - 20q^{12} - 6q^{15} + 8q^{16} - 4q^{22} - 28q^{25} - 12q^{31} + 2q^{33} - 16q^{34} + 26q^{36} + 20q^{37} - 2q^{42} - 50q^{45} - 82q^{48} - 56q^{49} - 20q^{55} - 80q^{58} + 104q^{60} + 16q^{64} - 50q^{66} + 60q^{67} + 6q^{69} + 80q^{70} + 12q^{75} + 10q^{78} + 6q^{81} - 8q^{82} - 52q^{88} - 8q^{91} + 42q^{93} - 92q^{97} + 18q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
131.1 −2.70555 −1.68970 0.380676i 5.32000 2.40031i 4.57157 + 1.02994i 4.64879i −8.98243 2.71017 + 1.28646i 6.49415i
131.2 −2.70555 −1.68970 + 0.380676i 5.32000 2.40031i 4.57157 1.02994i 4.64879i −8.98243 2.71017 1.28646i 6.49415i
131.3 −2.48573 0.809059 1.53148i 4.17886 2.08606i −2.01110 + 3.80684i 0.856739i −5.41606 −1.69085 2.47811i 5.18539i
131.4 −2.48573 0.809059 + 1.53148i 4.17886 2.08606i −2.01110 3.80684i 0.856739i −5.41606 −1.69085 + 2.47811i 5.18539i
131.5 −2.23364 −0.681784 1.59222i 2.98915 0.707248i 1.52286 + 3.55645i 2.10340i −2.20940 −2.07034 + 2.17110i 1.57974i
131.6 −2.23364 −0.681784 + 1.59222i 2.98915 0.707248i 1.52286 3.55645i 2.10340i −2.20940 −2.07034 2.17110i 1.57974i
131.7 −1.96237 −1.16712 1.27978i 1.85089 3.33889i 2.29031 + 2.51140i 4.49805i 0.292612 −0.275676 + 2.98731i 6.55213i
131.8 −1.96237 −1.16712 + 1.27978i 1.85089 3.33889i 2.29031 2.51140i 4.49805i 0.292612 −0.275676 2.98731i 6.55213i
131.9 −1.95255 1.45999 0.931894i 1.81247 0.144939i −2.85071 + 1.81957i 1.00140i 0.366168 1.26315 2.72111i 0.283002i
131.10 −1.95255 1.45999 + 0.931894i 1.81247 0.144939i −2.85071 1.81957i 1.00140i 0.366168 1.26315 + 2.72111i 0.283002i
131.11 −1.88424 1.68512 0.400473i 1.55036 4.10471i −3.17517 + 0.754586i 3.01872i 0.847231 2.67924 1.34969i 7.73425i
131.12 −1.88424 1.68512 + 0.400473i 1.55036 4.10471i −3.17517 0.754586i 3.01872i 0.847231 2.67924 + 1.34969i 7.73425i
131.13 −1.67217 −1.56232 0.747762i 0.796163 2.44281i 2.61247 + 1.25039i 0.956855i 2.01302 1.88170 + 2.33649i 4.08480i
131.14 −1.67217 −1.56232 + 0.747762i 0.796163 2.44281i 2.61247 1.25039i 0.956855i 2.01302 1.88170 2.33649i 4.08480i
131.15 −1.25502 0.750108 1.56120i −0.424934 0.276238i −0.941398 + 1.95933i 4.01346i 3.04333 −1.87468 2.34213i 0.346684i
131.16 −1.25502 0.750108 + 1.56120i −0.424934 0.276238i −0.941398 1.95933i 4.01346i 3.04333 −1.87468 + 2.34213i 0.346684i
131.17 −0.822598 −0.0778779 1.73030i −1.32333 3.64046i 0.0640622 + 1.42334i 1.97039i 2.73377 −2.98787 + 0.269504i 2.99464i
131.18 −0.822598 −0.0778779 + 1.73030i −1.32333 3.64046i 0.0640622 1.42334i 1.97039i 2.73377 −2.98787 0.269504i 2.99464i
131.19 −0.791616 −0.917209 1.46926i −1.37334 1.17632i 0.726077 + 1.16309i 0.422249i 2.67039 −1.31745 + 2.69524i 0.931196i
131.20 −0.791616 −0.917209 + 1.46926i −1.37334 1.17632i 0.726077 1.16309i 0.422249i 2.67039 −1.31745 2.69524i 0.931196i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 131.48 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
33.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.f.a 48
3.b odd 2 1 inner 429.2.f.a 48
11.b odd 2 1 inner 429.2.f.a 48
33.d even 2 1 inner 429.2.f.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.f.a 48 1.a even 1 1 trivial
429.2.f.a 48 3.b odd 2 1 inner
429.2.f.a 48 11.b odd 2 1 inner
429.2.f.a 48 33.d even 2 1 inner

## Hecke kernels

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