# Properties

 Label 429.2.m.b Level $429$ Weight $2$ Character orbit 429.m Analytic conductor $3.426$ Analytic rank $0$ Dimension $28$ 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.m (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$14$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$28q + 28q^{3} - 4q^{5} + 28q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q + 28q^{3} - 4q^{5} + 28q^{9} - 4q^{15} - 20q^{16} - 16q^{20} - 8q^{22} + 12q^{26} + 28q^{27} + 8q^{31} - 32q^{34} - 12q^{37} + 36q^{44} - 4q^{45} - 40q^{47} - 20q^{48} + 8q^{53} - 16q^{55} + 16q^{58} - 44q^{59} - 16q^{60} - 8q^{66} - 20q^{67} - 36q^{70} - 60q^{71} + 12q^{78} - 8q^{80} + 28q^{81} + 48q^{86} + 32q^{89} + 4q^{91} + 64q^{92} + 8q^{93} + 8q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1 −1.92050 1.92050i 1.00000 5.37666i −0.179544 + 0.179544i −1.92050 1.92050i −1.23995 + 1.23995i 6.48489 6.48489i 1.00000 0.689629
109.2 −1.63677 1.63677i 1.00000 3.35804i 0.440340 0.440340i −1.63677 1.63677i 2.71876 2.71876i 2.22281 2.22281i 1.00000 −1.44147
109.3 −1.43677 1.43677i 1.00000 2.12863i −2.46780 + 2.46780i −1.43677 1.43677i 0.806735 0.806735i 0.184806 0.184806i 1.00000 7.09133
109.4 −1.20600 1.20600i 1.00000 0.908895i −0.200991 + 0.200991i −1.20600 1.20600i −3.33407 + 3.33407i −1.31588 + 1.31588i 1.00000 0.484791
109.5 −1.10479 1.10479i 1.00000 0.441107i 1.95513 1.95513i −1.10479 1.10479i 1.11517 1.11517i −1.72224 + 1.72224i 1.00000 −4.32001
109.6 −0.555689 0.555689i 1.00000 1.38242i −2.09993 + 2.09993i −0.555689 0.555689i −1.20106 + 1.20106i −1.87957 + 1.87957i 1.00000 2.33382
109.7 −0.290762 0.290762i 1.00000 1.83092i 1.55280 1.55280i −0.290762 0.290762i 0.786011 0.786011i −1.11388 + 1.11388i 1.00000 −0.902988
109.8 0.290762 + 0.290762i 1.00000 1.83092i 1.55280 1.55280i 0.290762 + 0.290762i −0.786011 + 0.786011i 1.11388 1.11388i 1.00000 0.902988
109.9 0.555689 + 0.555689i 1.00000 1.38242i −2.09993 + 2.09993i 0.555689 + 0.555689i 1.20106 1.20106i 1.87957 1.87957i 1.00000 −2.33382
109.10 1.10479 + 1.10479i 1.00000 0.441107i 1.95513 1.95513i 1.10479 + 1.10479i −1.11517 + 1.11517i 1.72224 1.72224i 1.00000 4.32001
109.11 1.20600 + 1.20600i 1.00000 0.908895i −0.200991 + 0.200991i 1.20600 + 1.20600i 3.33407 3.33407i 1.31588 1.31588i 1.00000 −0.484791
109.12 1.43677 + 1.43677i 1.00000 2.12863i −2.46780 + 2.46780i 1.43677 + 1.43677i −0.806735 + 0.806735i −0.184806 + 0.184806i 1.00000 −7.09133
109.13 1.63677 + 1.63677i 1.00000 3.35804i 0.440340 0.440340i 1.63677 + 1.63677i −2.71876 + 2.71876i −2.22281 + 2.22281i 1.00000 1.44147
109.14 1.92050 + 1.92050i 1.00000 5.37666i −0.179544 + 0.179544i 1.92050 + 1.92050i 1.23995 1.23995i −6.48489 + 6.48489i 1.00000 −0.689629
307.1 −1.92050 + 1.92050i 1.00000 5.37666i −0.179544 0.179544i −1.92050 + 1.92050i −1.23995 1.23995i 6.48489 + 6.48489i 1.00000 0.689629
307.2 −1.63677 + 1.63677i 1.00000 3.35804i 0.440340 + 0.440340i −1.63677 + 1.63677i 2.71876 + 2.71876i 2.22281 + 2.22281i 1.00000 −1.44147
307.3 −1.43677 + 1.43677i 1.00000 2.12863i −2.46780 2.46780i −1.43677 + 1.43677i 0.806735 + 0.806735i 0.184806 + 0.184806i 1.00000 7.09133
307.4 −1.20600 + 1.20600i 1.00000 0.908895i −0.200991 0.200991i −1.20600 + 1.20600i −3.33407 3.33407i −1.31588 1.31588i 1.00000 0.484791
307.5 −1.10479 + 1.10479i 1.00000 0.441107i 1.95513 + 1.95513i −1.10479 + 1.10479i 1.11517 + 1.11517i −1.72224 1.72224i 1.00000 −4.32001
307.6 −0.555689 + 0.555689i 1.00000 1.38242i −2.09993 2.09993i −0.555689 + 0.555689i −1.20106 1.20106i −1.87957 1.87957i 1.00000 2.33382
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 307.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
13.d odd 4 1 inner
143.g even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.m.b 28
11.b odd 2 1 inner 429.2.m.b 28
13.d odd 4 1 inner 429.2.m.b 28
143.g even 4 1 inner 429.2.m.b 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.m.b 28 1.a even 1 1 trivial
429.2.m.b 28 11.b odd 2 1 inner
429.2.m.b 28 13.d odd 4 1 inner
429.2.m.b 28 143.g even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{28} + 115 T_{2}^{24} + 4521 T_{2}^{20} + 76475 T_{2}^{16} + 564863 T_{2}^{12} + 1562521 T_{2}^{8} + 556319 T_{2}^{4} + 14641$$ acting on $$S_{2}^{\mathrm{new}}(429, [\chi])$$.