# Properties

 Label 429.2.s.a Level $429$ Weight $2$ Character orbit 429.s Analytic conductor $3.426$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

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## 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: $$24$$ Relative dimension: $$12$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 12q^{3} + 14q^{4} + 6q^{7} - 12q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 12q^{3} + 14q^{4} + 6q^{7} - 12q^{9} + 28q^{12} - 4q^{13} + 20q^{14} - 6q^{15} - 14q^{16} + 10q^{17} - 18q^{20} + 2q^{22} - 14q^{23} + 4q^{25} - 34q^{26} - 24q^{27} - 30q^{28} + 4q^{29} + 30q^{32} + 6q^{35} + 14q^{36} + 12q^{38} - 2q^{39} + 20q^{40} - 30q^{41} + 10q^{42} - 4q^{43} - 6q^{45} - 24q^{46} + 14q^{48} + 18q^{49} - 84q^{50} + 20q^{51} + 40q^{52} - 56q^{53} - 4q^{55} + 26q^{56} + 48q^{58} + 60q^{59} - 2q^{61} + 18q^{62} - 6q^{63} - 48q^{64} - 10q^{65} + 4q^{66} - 42q^{67} - 18q^{68} + 14q^{69} + 6q^{71} + 2q^{75} - 48q^{76} + 24q^{77} - 26q^{78} - 20q^{79} + 30q^{80} - 12q^{81} - 10q^{82} - 30q^{84} + 6q^{85} - 4q^{87} - 12q^{88} + 12q^{89} + 18q^{91} + 8q^{92} + 12q^{93} - 22q^{94} + 4q^{95} + 6q^{97} - 114q^{98} + O(q^{100})$$

## 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 −2.37717 + 1.37246i 0.500000 + 0.866025i 2.76728 4.79308i 0.261284i −2.37717 1.37246i −3.66637 2.11678i 9.70209i −0.500000 + 0.866025i −0.358601 0.621115i
166.2 −1.91936 + 1.10815i 0.500000 + 0.866025i 1.45597 2.52182i 0.692237i −1.91936 1.10815i 2.81071 + 1.62277i 2.02113i −0.500000 + 0.866025i 0.767100 + 1.32866i
166.3 −1.70879 + 0.986569i 0.500000 + 0.866025i 0.946639 1.63963i 1.03130i −1.70879 0.986569i −0.142489 0.0822660i 0.210579i −0.500000 + 0.866025i 1.01745 + 1.76227i
166.4 −1.00815 + 0.582055i 0.500000 + 0.866025i −0.322425 + 0.558456i 1.30774i −1.00815 0.582055i −2.07195 1.19624i 3.07889i −0.500000 + 0.866025i 0.761176 + 1.31840i
166.5 −0.761768 + 0.439807i 0.500000 + 0.866025i −0.613140 + 1.06199i 3.40975i −0.761768 0.439807i 4.33112 + 2.50057i 2.83788i −0.500000 + 0.866025i −1.49963 2.59744i
166.6 −0.529161 + 0.305511i 0.500000 + 0.866025i −0.813326 + 1.40872i 2.30641i −0.529161 0.305511i −2.90273 1.67589i 2.21596i −0.500000 + 0.866025i −0.704633 1.22046i
166.7 0.219134 0.126517i 0.500000 + 0.866025i −0.967987 + 1.67660i 0.770885i 0.219134 + 0.126517i 2.38762 + 1.37849i 0.995935i −0.500000 + 0.866025i 0.0975301 + 0.168927i
166.8 0.884735 0.510802i 0.500000 + 0.866025i −0.478163 + 0.828202i 0.892170i 0.884735 + 0.510802i −0.417948 0.241303i 3.02019i −0.500000 + 0.866025i 0.455722 + 0.789334i
166.9 1.30176 0.751572i 0.500000 + 0.866025i 0.129721 0.224683i 4.13566i 1.30176 + 0.751572i 3.62040 + 2.09024i 2.61631i −0.500000 + 0.866025i −3.10825 5.38364i
166.10 1.56256 0.902143i 0.500000 + 0.866025i 0.627724 1.08725i 3.28340i 1.56256 + 0.902143i −1.35933 0.784811i 1.34338i −0.500000 + 0.866025i 2.96210 + 5.13050i
166.11 2.15823 1.24606i 0.500000 + 0.866025i 2.10532 3.64652i 1.90483i 2.15823 + 1.24606i 1.37703 + 0.795028i 5.50915i −0.500000 + 0.866025i 2.37353 + 4.11108i
166.12 2.17798 1.25745i 0.500000 + 0.866025i 2.16238 3.74536i 2.19769i 2.17798 + 1.25745i −0.966057 0.557753i 5.84658i −0.500000 + 0.866025i −2.76349 4.78651i
199.1 −2.37717 1.37246i 0.500000 0.866025i 2.76728 + 4.79308i 0.261284i −2.37717 + 1.37246i −3.66637 + 2.11678i 9.70209i −0.500000 0.866025i −0.358601 + 0.621115i
199.2 −1.91936 1.10815i 0.500000 0.866025i 1.45597 + 2.52182i 0.692237i −1.91936 + 1.10815i 2.81071 1.62277i 2.02113i −0.500000 0.866025i 0.767100 1.32866i
199.3 −1.70879 0.986569i 0.500000 0.866025i 0.946639 + 1.63963i 1.03130i −1.70879 + 0.986569i −0.142489 + 0.0822660i 0.210579i −0.500000 0.866025i 1.01745 1.76227i
199.4 −1.00815 0.582055i 0.500000 0.866025i −0.322425 0.558456i 1.30774i −1.00815 + 0.582055i −2.07195 + 1.19624i 3.07889i −0.500000 0.866025i 0.761176 1.31840i
199.5 −0.761768 0.439807i 0.500000 0.866025i −0.613140 1.06199i 3.40975i −0.761768 + 0.439807i 4.33112 2.50057i 2.83788i −0.500000 0.866025i −1.49963 + 2.59744i
199.6 −0.529161 0.305511i 0.500000 0.866025i −0.813326 1.40872i 2.30641i −0.529161 + 0.305511i −2.90273 + 1.67589i 2.21596i −0.500000 0.866025i −0.704633 + 1.22046i
199.7 0.219134 + 0.126517i 0.500000 0.866025i −0.967987 1.67660i 0.770885i 0.219134 0.126517i 2.38762 1.37849i 0.995935i −0.500000 0.866025i 0.0975301 0.168927i
199.8 0.884735 + 0.510802i 0.500000 0.866025i −0.478163 0.828202i 0.892170i 0.884735 0.510802i −0.417948 + 0.241303i 3.02019i −0.500000 0.866025i 0.455722 0.789334i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.12 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.a 24
13.e even 6 1 inner 429.2.s.a 24
13.f odd 12 1 5577.2.a.z 12
13.f odd 12 1 5577.2.a.be 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.s.a 24 1.a even 1 1 trivial
429.2.s.a 24 13.e even 6 1 inner
5577.2.a.z 12 13.f odd 12 1
5577.2.a.be 12 13.f odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{24} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(429, [\chi])$$.