# Properties

 Label 429.2.s.b Level $429$ Weight $2$ Character orbit 429.s Analytic conductor $3.426$ Analytic rank $0$ Dimension $28$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$14$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$28q - 14q^{3} + 18q^{4} - 14q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q - 14q^{3} + 18q^{4} - 14q^{9} - 36q^{12} - 6q^{13} - 4q^{14} + 6q^{15} - 22q^{16} + 2q^{17} + 12q^{19} + 18q^{20} - 6q^{22} + 2q^{23} - 40q^{25} - 18q^{26} + 28q^{27} - 18q^{28} - 30q^{32} + 2q^{35} + 18q^{36} + 20q^{38} + 6q^{39} + 20q^{40} + 18q^{41} + 2q^{42} - 2q^{43} - 6q^{45} + 48q^{46} - 22q^{48} + 10q^{49} + 24q^{50} - 4q^{51} - 28q^{52} + 16q^{53} - 12q^{55} - 10q^{56} - 48q^{58} - 12q^{59} - 4q^{61} - 6q^{62} - 32q^{64} + 6q^{65} + 12q^{66} + 12q^{67} - 22q^{68} + 2q^{69} - 18q^{71} + 48q^{74} + 20q^{75} + 96q^{76} - 24q^{77} + 6q^{78} - 48q^{79} + 66q^{80} - 14q^{81} + 46q^{82} + 18q^{84} - 66q^{85} + 12q^{88} + 8q^{91} + 72q^{92} + 6q^{93} + 50q^{94} - 60q^{95} - 36q^{97} + 18q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
166.1 −2.14836 + 1.24036i −0.500000 0.866025i 2.07696 3.59741i 0.0418579i 2.14836 + 1.24036i −3.59366 2.07480i 5.34327i −0.500000 + 0.866025i 0.0519187 + 0.0899258i
166.2 −2.12238 + 1.22536i −0.500000 0.866025i 2.00301 3.46932i 4.42067i 2.12238 + 1.22536i 0.902680 + 0.521163i 4.91619i −0.500000 + 0.866025i −5.41690 9.38235i
166.3 −2.08148 + 1.20174i −0.500000 0.866025i 1.88836 3.27074i 3.20365i 2.08148 + 1.20174i 2.13033 + 1.22995i 4.27033i −0.500000 + 0.866025i 3.84996 + 6.66833i
166.4 −1.57303 + 0.908192i −0.500000 0.866025i 0.649624 1.12518i 0.705953i 1.57303 + 0.908192i 1.35880 + 0.784504i 1.27283i −0.500000 + 0.866025i 0.641141 + 1.11049i
166.5 −0.945010 + 0.545602i −0.500000 0.866025i −0.404638 + 0.700853i 1.08345i 0.945010 + 0.545602i −2.79805 1.61546i 3.06549i −0.500000 + 0.866025i −0.591129 1.02387i
166.6 −0.489701 + 0.282729i −0.500000 0.866025i −0.840128 + 1.45515i 1.14515i 0.489701 + 0.282729i 2.75529 + 1.59077i 2.08103i −0.500000 + 0.866025i 0.323767 + 0.560780i
166.7 −0.458122 + 0.264497i −0.500000 0.866025i −0.860083 + 1.48971i 3.29610i 0.458122 + 0.264497i 0.609605 + 0.351955i 1.96794i −0.500000 + 0.866025i −0.871809 1.51002i
166.8 0.252478 0.145768i −0.500000 0.866025i −0.957503 + 1.65844i 3.62473i −0.252478 0.145768i −1.17162 0.676437i 1.14137i −0.500000 + 0.866025i −0.528371 0.915165i
166.9 0.813989 0.469957i −0.500000 0.866025i −0.558281 + 0.966972i 0.163878i −0.813989 0.469957i 2.40248 + 1.38708i 2.92930i −0.500000 + 0.866025i −0.0770155 0.133395i
166.10 0.962812 0.555880i −0.500000 0.866025i −0.381995 + 0.661635i 2.79143i −0.962812 0.555880i −3.71222 2.14325i 3.07289i −0.500000 + 0.866025i 1.55170 + 2.68762i
166.11 1.56752 0.905006i −0.500000 0.866025i 0.638073 1.10517i 2.99723i −1.56752 0.905006i 3.93315 + 2.27081i 1.31019i −0.500000 + 0.866025i 2.71252 + 4.69821i
166.12 1.80882 1.04432i −0.500000 0.866025i 1.18121 2.04591i 2.92121i −1.80882 1.04432i 0.684943 + 0.395452i 0.756961i −0.500000 + 0.866025i −3.05068 5.28394i
166.13 1.96842 1.13647i −0.500000 0.866025i 1.58313 2.74206i 1.61338i −1.96842 1.13647i −2.77524 1.60229i 2.65083i −0.500000 + 0.866025i −1.83356 3.17581i
166.14 2.44405 1.41107i −0.500000 0.866025i 2.98226 5.16542i 2.29504i −2.44405 1.41107i −0.726492 0.419441i 11.1884i −0.500000 + 0.866025i 3.23847 + 5.60919i
199.1 −2.14836 1.24036i −0.500000 + 0.866025i 2.07696 + 3.59741i 0.0418579i 2.14836 1.24036i −3.59366 + 2.07480i 5.34327i −0.500000 0.866025i 0.0519187 0.0899258i
199.2 −2.12238 1.22536i −0.500000 + 0.866025i 2.00301 + 3.46932i 4.42067i 2.12238 1.22536i 0.902680 0.521163i 4.91619i −0.500000 0.866025i −5.41690 + 9.38235i
199.3 −2.08148 1.20174i −0.500000 + 0.866025i 1.88836 + 3.27074i 3.20365i 2.08148 1.20174i 2.13033 1.22995i 4.27033i −0.500000 0.866025i 3.84996 6.66833i
199.4 −1.57303 0.908192i −0.500000 + 0.866025i 0.649624 + 1.12518i 0.705953i 1.57303 0.908192i 1.35880 0.784504i 1.27283i −0.500000 0.866025i 0.641141 1.11049i
199.5 −0.945010 0.545602i −0.500000 + 0.866025i −0.404638 0.700853i 1.08345i 0.945010 0.545602i −2.79805 + 1.61546i 3.06549i −0.500000 0.866025i −0.591129 + 1.02387i
199.6 −0.489701 0.282729i −0.500000 + 0.866025i −0.840128 1.45515i 1.14515i 0.489701 0.282729i 2.75529 1.59077i 2.08103i −0.500000 0.866025i 0.323767 0.560780i
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e 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.s.b 28
13.e even 6 1 inner 429.2.s.b 28
13.f odd 12 1 5577.2.a.bf 14
13.f odd 12 1 5577.2.a.bg 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.s.b 28 1.a even 1 1 trivial
429.2.s.b 28 13.e even 6 1 inner
5577.2.a.bf 14 13.f odd 12 1
5577.2.a.bg 14 13.f odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{28} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(429, [\chi])$$.