# Properties

 Label 354.3.f.a Level 354 Weight 3 Character orbit 354.f Analytic conductor 9.646 Analytic rank 0 Dimension 560 CM No

# Related objects

## Newspace parameters

 Level: $$N$$ = $$354 = 2 \cdot 3 \cdot 59$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 354.f (of order $$58$$ and degree $$28$$)

## Newform invariants

 Self dual: No Analytic conductor: $$9.64580135835$$ Analytic rank: $$0$$ Dimension: $$560$$ Relative dimension: $$20$$ over $$\Q(\zeta_{58})$$ Sato-Tate group: $\mathrm{SU}(2)[C_{58}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$560q + 40q^{4} - 8q^{7} - 60q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$560q + 40q^{4} - 8q^{7} - 60q^{9} + 24q^{15} - 80q^{16} - 72q^{19} - 16q^{22} - 140q^{25} - 64q^{26} + 16q^{28} - 56q^{29} + 80q^{35} + 120q^{36} + 8q^{41} + 1376q^{46} + 1276q^{47} + 2036q^{49} + 1856q^{50} + 696q^{52} + 1128q^{53} + 1044q^{55} + 48q^{57} - 424q^{59} - 48q^{60} - 696q^{61} - 448q^{62} - 24q^{63} + 160q^{64} - 2436q^{65} - 96q^{66} - 2088q^{67} - 1160q^{68} - 2784q^{70} - 2448q^{71} - 1740q^{73} - 1568q^{74} + 96q^{75} + 144q^{76} - 192q^{78} - 528q^{79} - 180q^{81} - 568q^{85} + 416q^{86} + 216q^{87} + 32q^{88} + 480q^{94} + 456q^{95} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1 −1.07786 + 0.915542i −1.37887 + 1.04819i 0.323564 1.97365i −4.07963 + 7.69500i 0.526568 2.39222i −11.4872 + 1.24931i 1.45821 + 2.42356i 0.802585 2.89065i −2.64783 12.0292i
13.2 −1.07786 + 0.915542i −1.37887 + 1.04819i 0.323564 1.97365i −2.10279 + 3.96628i 0.526568 2.39222i 8.08885 0.879715i 1.45821 + 2.42356i 0.802585 2.89065i −1.36479 6.20029i
13.3 −1.07786 + 0.915542i −1.37887 + 1.04819i 0.323564 1.97365i 0.0942749 0.177821i 0.526568 2.39222i −5.93063 + 0.644994i 1.45821 + 2.42356i 0.802585 2.89065i 0.0611878 + 0.277979i
13.4 −1.07786 + 0.915542i −1.37887 + 1.04819i 0.323564 1.97365i 1.26748 2.39072i 0.526568 2.39222i −3.28554 + 0.357324i 1.45821 + 2.42356i 0.802585 2.89065i 0.822641 + 3.73729i
13.5 −1.07786 + 0.915542i −1.37887 + 1.04819i 0.323564 1.97365i 3.35081 6.32029i 0.526568 2.39222i 13.6879 1.48864i 1.45821 + 2.42356i 0.802585 2.89065i 2.17480 + 9.88019i
13.6 −1.07786 + 0.915542i 1.37887 1.04819i 0.323564 1.97365i −3.73483 + 7.04464i −0.526568 + 2.39222i −0.463903 + 0.0504525i 1.45821 + 2.42356i 0.802585 2.89065i −2.42404 11.0125i
13.7 −1.07786 + 0.915542i 1.37887 1.04819i 0.323564 1.97365i −1.59599 + 3.01035i −0.526568 + 2.39222i 8.94827 0.973183i 1.45821 + 2.42356i 0.802585 2.89065i −1.03585 4.70593i
13.8 −1.07786 + 0.915542i 1.37887 1.04819i 0.323564 1.97365i 0.0757215 0.142826i −0.526568 + 2.39222i −7.17117 + 0.779911i 1.45821 + 2.42356i 0.802585 2.89065i 0.0491460 + 0.223272i
13.9 −1.07786 + 0.915542i 1.37887 1.04819i 0.323564 1.97365i 2.81638 5.31225i −0.526568 + 2.39222i 8.73242 0.949708i 1.45821 + 2.42356i 0.802585 2.89065i 1.82793 + 8.30438i
13.10 −1.07786 + 0.915542i 1.37887 1.04819i 0.323564 1.97365i 4.19506 7.91273i −0.526568 + 2.39222i −6.36367 + 0.692091i 1.45821 + 2.42356i 0.802585 2.89065i 2.72275 + 12.3696i
13.11 1.07786 0.915542i −1.37887 + 1.04819i 0.323564 1.97365i −2.82525 + 5.32898i −0.526568 + 2.39222i −3.51596 + 0.382383i −1.45821 2.42356i 0.802585 2.89065i 1.83369 + 8.33052i
13.12 1.07786 0.915542i −1.37887 + 1.04819i 0.323564 1.97365i −1.81071 + 3.41535i −0.526568 + 2.39222i 0.390182 0.0424348i −1.45821 2.42356i 0.802585 2.89065i 1.17521 + 5.33905i
13.13 1.07786 0.915542i −1.37887 + 1.04819i 0.323564 1.97365i −0.726244 + 1.36984i −0.526568 + 2.39222i 9.02581 0.981616i −1.45821 2.42356i 0.802585 2.89065i 0.471359 + 2.14140i
13.14 1.07786 0.915542i −1.37887 + 1.04819i 0.323564 1.97365i −0.150737 + 0.284320i −0.526568 + 2.39222i −7.95376 + 0.865024i −1.45821 2.42356i 0.802585 2.89065i 0.0978339 + 0.444464i
13.15 1.07786 0.915542i −1.37887 + 1.04819i 0.323564 1.97365i 3.75659 7.08568i −0.526568 + 2.39222i 3.86735 0.420600i −1.45821 2.42356i 0.802585 2.89065i −2.43816 11.0767i
13.16 1.07786 0.915542i 1.37887 1.04819i 0.323564 1.97365i −2.69168 + 5.07704i 0.526568 2.39222i −13.1254 + 1.42747i −1.45821 2.42356i 0.802585 2.89065i 1.74700 + 7.93668i
13.17 1.07786 0.915542i 1.37887 1.04819i 0.323564 1.97365i −2.22471 + 4.19624i 0.526568 2.39222i 7.39400 0.804146i −1.45821 2.42356i 0.802585 2.89065i 1.44392 + 6.55978i
13.18 1.07786 0.915542i 1.37887 1.04819i 0.323564 1.97365i 0.383356 0.723087i 0.526568 2.39222i 0.622882 0.0677425i −1.45821 2.42356i 0.802585 2.89065i −0.248812 1.13037i
13.19 1.07786 0.915542i 1.37887 1.04819i 0.323564 1.97365i 2.48920 4.69513i 0.526568 2.39222i −5.51808 + 0.600128i −1.45821 2.42356i 0.802585 2.89065i −1.61558 7.33966i
13.20 1.07786 0.915542i 1.37887 1.04819i 0.323564 1.97365i 3.51369 6.62752i 0.526568 2.39222i 9.83160 1.06925i −1.45821 2.42356i 0.802585 2.89065i −2.28051 10.3605i
See next 80 embeddings (of 560 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 349.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not have CM; other inner twists have not been computed.

## Hecke kernels

There are no other newforms in $$S_{3}^{\mathrm{new}}(354, [\chi])$$.