# Properties

 Label 429.2.n.c Level $429$ Weight $2$ Character orbit 429.n 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.n (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$7$$ over $$\Q(\zeta_{5})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$28q + q^{2} + 7q^{3} - 5q^{4} - 4q^{5} - q^{6} + q^{7} - 7q^{8} - 7q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q + q^{2} + 7q^{3} - 5q^{4} - 4q^{5} - q^{6} + q^{7} - 7q^{8} - 7q^{9} - 2q^{10} + 14q^{11} - 30q^{12} - 7q^{13} - 9q^{14} + 4q^{15} + q^{16} - 12q^{17} - 4q^{18} + 10q^{19} - 41q^{20} - 6q^{21} + 5q^{22} + 30q^{23} + 2q^{24} + 3q^{25} - 4q^{26} + 7q^{27} - 12q^{28} - 4q^{29} + 7q^{30} - 4q^{31} + 22q^{32} + q^{33} - 24q^{34} - 6q^{35} - 5q^{36} - 8q^{37} + 73q^{38} + 7q^{39} - 28q^{40} + 10q^{41} + 9q^{42} - 12q^{43} - 22q^{44} + 16q^{45} + 35q^{46} + 12q^{47} + 14q^{48} + 16q^{49} - 57q^{50} - 13q^{51} - 5q^{52} + q^{53} - 6q^{54} - 28q^{55} + 48q^{56} - 30q^{58} - 15q^{59} + 41q^{60} - 22q^{61} - 40q^{62} + q^{63} - 19q^{64} + 16q^{65} + 20q^{66} - 88q^{67} + 39q^{68} + 14q^{70} + 34q^{71} - 2q^{72} - 59q^{73} + 79q^{74} + 27q^{75} - 124q^{76} - 42q^{77} - 6q^{78} - 3q^{79} + 37q^{80} - 7q^{81} + 82q^{82} - 8q^{83} - 8q^{84} + 70q^{85} - 35q^{86} - 36q^{87} + 59q^{88} + 126q^{89} + 8q^{90} + q^{91} - 82q^{92} + 4q^{93} + 23q^{94} - 77q^{95} + 73q^{96} - 18q^{97} - 66q^{98} - 6q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
157.1 −2.05353 + 1.49197i −0.309017 0.951057i 1.37295 4.22550i −3.43019 2.49218i 2.05353 + 1.49197i 0.899200 2.76745i 1.91620 + 5.89746i −0.809017 + 0.587785i 10.7623
157.2 −1.44143 + 1.04726i −0.309017 0.951057i 0.362933 1.11699i 2.24980 + 1.63458i 1.44143 + 1.04726i −0.0806900 + 0.248338i −0.454515 1.39885i −0.809017 + 0.587785i −4.95476
157.3 −1.35790 + 0.986569i −0.309017 0.951057i 0.252528 0.777201i 0.627776 + 0.456106i 1.35790 + 0.986569i 0.0948896 0.292040i −0.613484 1.88811i −0.809017 + 0.587785i −1.30243
157.4 0.470172 0.341600i −0.309017 0.951057i −0.513663 + 1.58089i 2.42118 + 1.75909i −0.470172 0.341600i 0.122991 0.378529i 0.657702 + 2.02420i −0.809017 + 0.587785i 1.73927
157.5 0.788916 0.573181i −0.309017 0.951057i −0.324182 + 0.997729i −2.91208 2.11575i −0.788916 0.573181i −1.06635 + 3.28188i 0.918806 + 2.82779i −0.809017 + 0.587785i −3.51009
157.6 1.18037 0.857589i −0.309017 0.951057i 0.0397803 0.122431i −0.326657 0.237330i −1.18037 0.857589i 1.48593 4.57323i 0.843682 + 2.59659i −0.809017 + 0.587785i −0.589108
157.7 2.10438 1.52892i −0.309017 0.951057i 1.47277 4.53273i −1.86590 1.35565i −2.10438 1.52892i −0.646959 + 1.99114i −2.22331 6.84263i −0.809017 + 0.587785i −5.99924
196.1 −0.675177 + 2.07798i 0.809017 0.587785i −2.24411 1.63044i −0.557745 1.71656i 0.675177 + 2.07798i −0.752444 0.546682i 1.36792 0.993855i 0.309017 0.951057i 3.94356
196.2 −0.488036 + 1.50202i 0.809017 0.587785i −0.399850 0.290508i 1.09306 + 3.36409i 0.488036 + 1.50202i 3.87810 + 2.81760i −1.92390 + 1.39779i 0.309017 0.951057i −5.58638
196.3 0.0279051 0.0858830i 0.809017 0.587785i 1.61144 + 1.17078i 0.964032 + 2.96698i −0.0279051 0.0858830i −2.21929 1.61241i 0.291630 0.211882i 0.309017 0.951057i 0.281715
196.4 0.0974470 0.299911i 0.809017 0.587785i 1.53758 + 1.11712i −0.824247 2.53677i −0.0974470 0.299911i 1.23058 + 0.894067i 0.995109 0.722989i 0.309017 0.951057i −0.841126
196.5 0.400428 1.23239i 0.809017 0.587785i 0.259592 + 0.188605i −0.412722 1.27023i −0.400428 1.23239i −4.15316 3.01744i 2.43305 1.76771i 0.309017 0.951057i −1.73068
196.6 0.589929 1.81561i 0.809017 0.587785i −1.33041 0.966597i 0.697262 + 2.14595i −0.589929 1.81561i 2.71045 + 1.96926i 0.549095 0.398941i 0.309017 0.951057i 4.30756
196.7 0.856521 2.63610i 0.809017 0.587785i −4.59737 3.34018i 0.276429 + 0.850760i −0.856521 2.63610i −1.00325 0.728906i −8.25799 + 5.99978i 0.309017 0.951057i 2.47946
235.1 −2.05353 1.49197i −0.309017 + 0.951057i 1.37295 + 4.22550i −3.43019 + 2.49218i 2.05353 1.49197i 0.899200 + 2.76745i 1.91620 5.89746i −0.809017 0.587785i 10.7623
235.2 −1.44143 1.04726i −0.309017 + 0.951057i 0.362933 + 1.11699i 2.24980 1.63458i 1.44143 1.04726i −0.0806900 0.248338i −0.454515 + 1.39885i −0.809017 0.587785i −4.95476
235.3 −1.35790 0.986569i −0.309017 + 0.951057i 0.252528 + 0.777201i 0.627776 0.456106i 1.35790 0.986569i 0.0948896 + 0.292040i −0.613484 + 1.88811i −0.809017 0.587785i −1.30243
235.4 0.470172 + 0.341600i −0.309017 + 0.951057i −0.513663 1.58089i 2.42118 1.75909i −0.470172 + 0.341600i 0.122991 + 0.378529i 0.657702 2.02420i −0.809017 0.587785i 1.73927
235.5 0.788916 + 0.573181i −0.309017 + 0.951057i −0.324182 0.997729i −2.91208 + 2.11575i −0.788916 + 0.573181i −1.06635 3.28188i 0.918806 2.82779i −0.809017 0.587785i −3.51009
235.6 1.18037 + 0.857589i −0.309017 + 0.951057i 0.0397803 + 0.122431i −0.326657 + 0.237330i −1.18037 + 0.857589i 1.48593 + 4.57323i 0.843682 2.59659i −0.809017 0.587785i −0.589108
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 313.7 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.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.n.c 28
11.c even 5 1 inner 429.2.n.c 28
11.c even 5 1 4719.2.a.bp 14
11.d odd 10 1 4719.2.a.bo 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.n.c 28 1.a even 1 1 trivial
429.2.n.c 28 11.c even 5 1 inner
4719.2.a.bo 14 11.d odd 10 1
4719.2.a.bp 14 11.c even 5 1

## Hecke kernels

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