Properties

 Label 354.3.h.a Level 354 Weight 3 Character orbit 354.h Analytic conductor 9.646 Analytic rank 0 Dimension 1120 CM No

Related objects

Newspace parameters

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

Newform invariants

 Self dual: No Analytic conductor: $$9.64580135835$$ Analytic rank: $$0$$ Dimension: $$1120$$ Relative dimension: $$40$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$1120q + 80q^{4} - 8q^{6} - 8q^{7} + 24q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$1120q + 80q^{4} - 8q^{6} - 8q^{7} + 24q^{9} + 16q^{10} - 34q^{15} - 160q^{16} - 16q^{18} - 24q^{19} + 18q^{21} + 16q^{22} + 16q^{24} + 216q^{25} + 30q^{27} + 16q^{28} + 64q^{30} - 96q^{31} - 76q^{33} - 80q^{34} - 48q^{36} + 200q^{37} + 28q^{39} - 32q^{40} - 48q^{42} + 104q^{43} + 696q^{45} - 32q^{46} - 288q^{49} + 1800q^{51} + 852q^{54} - 360q^{55} + 76q^{57} + 128q^{58} - 280q^{60} + 32q^{61} - 1318q^{63} + 320q^{64} - 1512q^{66} + 344q^{67} - 2640q^{69} - 192q^{70} + 32q^{72} - 40q^{73} - 1014q^{75} + 48q^{76} - 96q^{78} - 32q^{79} - 336q^{81} + 80q^{82} - 36q^{84} - 168q^{85} + 162q^{87} - 32q^{88} - 112q^{90} - 88q^{91} + 316q^{93} + 400q^{94} - 32q^{96} + 184q^{97} + 148q^{99} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −0.451561 + 1.34018i −2.93432 + 0.624304i −1.59219 1.21035i 1.01598 + 0.404804i 0.488341 4.21444i 0.548538 0.253781i 2.34106 1.58728i 8.22049 3.66382i −1.00129 + 1.17881i
5.2 −0.451561 + 1.34018i −2.92397 0.671113i −1.59219 1.21035i −8.23136 3.27967i 2.21976 3.61561i −7.65524 + 3.54169i 2.34106 1.58728i 8.09922 + 3.92463i 8.11232 9.55056i
5.3 −0.451561 + 1.34018i −2.91198 0.721372i −1.59219 1.21035i 8.94495 + 3.56399i 2.28171 3.57684i −3.17617 + 1.46945i 2.34106 1.58728i 7.95924 + 4.20124i −8.81560 + 10.3785i
5.4 −0.451561 + 1.34018i −2.24157 1.99383i −1.59219 1.21035i −1.72634 0.687837i 3.68430 2.10378i 6.44007 2.97949i 2.34106 1.58728i 1.04928 + 8.93862i 1.70137 2.00301i
5.5 −0.451561 + 1.34018i −1.87528 + 2.34165i −1.59219 1.21035i −7.13337 2.84219i −2.29144 3.57062i 1.14957 0.531850i 2.34106 1.58728i −1.96666 8.78250i 7.03021 8.27660i
5.6 −0.451561 + 1.34018i −1.79506 2.40370i −1.59219 1.21035i 1.34145 + 0.534483i 4.03197 1.32030i −2.67432 + 1.23727i 2.34106 1.58728i −2.55552 + 8.62956i −1.32205 + 1.55644i
5.7 −0.451561 + 1.34018i −1.59946 + 2.53806i −1.59219 1.21035i 2.85503 + 1.13755i −2.67921 3.28965i −8.37649 + 3.87538i 2.34106 1.58728i −3.88348 8.11903i −2.81375 + 3.31260i
5.8 −0.451561 + 1.34018i −0.780089 + 2.89680i −1.59219 1.21035i 2.36262 + 0.941352i −3.52999 2.35354i 10.5523 4.88202i 2.34106 1.58728i −7.78292 4.51953i −2.32845 + 2.74126i
5.9 −0.451561 + 1.34018i −0.520443 2.95451i −1.59219 1.21035i −4.82640 1.92301i 4.19460 + 0.636651i −5.47968 + 2.53517i 2.34106 1.58728i −8.45828 + 3.07531i 4.75660 5.59991i
5.10 −0.451561 + 1.34018i 0.214023 2.99236i −1.59219 1.21035i 6.91221 + 2.75408i 3.91366 + 1.63806i 12.1903 5.63984i 2.34106 1.58728i −8.90839 1.28087i −6.81225 + 8.02001i
5.11 −0.451561 + 1.34018i 0.908343 2.85918i −1.59219 1.21035i 4.52215 + 1.80179i 3.42166 + 2.50844i −8.14687 + 3.76914i 2.34106 1.58728i −7.34983 5.19423i −4.45675 + 5.24690i
5.12 −0.451561 + 1.34018i 0.924357 + 2.85404i −1.59219 1.21035i −0.606021 0.241461i −4.24235 0.0499646i 6.70675 3.10287i 2.34106 1.58728i −7.29113 + 5.27631i 0.597257 0.703146i
5.13 −0.451561 + 1.34018i 1.39003 2.65854i −1.59219 1.21035i −7.10734 2.83182i 2.93524 + 3.06339i 7.21460 3.33783i 2.34106 1.58728i −5.13563 7.39089i 7.00456 8.24641i
5.14 −0.451561 + 1.34018i 1.42759 + 2.63856i −1.59219 1.21035i 5.73580 + 2.28535i −4.18080 + 0.721757i −0.923037 + 0.427042i 2.34106 1.58728i −4.92400 + 7.53354i −5.65286 + 6.65506i
5.15 −0.451561 + 1.34018i 1.69004 + 2.47866i −1.59219 1.21035i −2.80901 1.11921i −4.08502 + 1.14570i −7.23899 + 3.34911i 2.34106 1.58728i −3.28751 + 8.37808i 2.76838 3.25919i
5.16 −0.451561 + 1.34018i 2.35308 1.86092i −1.59219 1.21035i −1.37348 0.547244i 1.43141 + 3.99388i −4.55846 + 2.10897i 2.34106 1.58728i 2.07397 8.75778i 1.35362 1.59360i
5.17 −0.451561 + 1.34018i 2.75259 + 1.19300i −1.59219 1.21035i 7.91048 + 3.15182i −2.84180 + 3.15027i −1.21658 + 0.562850i 2.34106 1.58728i 6.15350 + 6.56768i −7.79609 + 9.17826i
5.18 −0.451561 + 1.34018i 2.87597 + 0.853709i −1.59219 1.21035i −5.83416 2.32454i −2.44280 + 3.46882i 6.28996 2.91005i 2.34106 1.58728i 7.54236 + 4.91048i 5.74979 6.76917i
5.19 −0.451561 + 1.34018i 2.89671 0.780416i −1.59219 1.21035i 4.65663 + 1.85537i −0.262140 + 4.23453i 4.95113 2.29064i 2.34106 1.58728i 7.78190 4.52129i −4.58929 + 5.40293i
5.20 −0.451561 + 1.34018i 2.95183 + 0.535437i −1.59219 1.21035i −1.84046 0.733305i −2.05051 + 3.71421i −10.2065 + 4.72201i 2.34106 1.58728i 8.42661 + 3.16104i 1.81384 2.13542i
See next 80 embeddings (of 1120 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 341.40 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 $$S_{3}^{\mathrm{new}}(354, [\chi])$$.