# Properties

 Label 287.2.bb.a Level 287 Weight 2 Character orbit 287.bb Analytic conductor 2.292 Analytic rank 0 Dimension 416 CM No

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$287 = 7 \cdot 41$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 287.bb (of order $$40$$ and degree $$16$$)

## Newform invariants

 Self dual: No Analytic conductor: $$2.29170653801$$ Analytic rank: $$0$$ Dimension: $$416$$ Relative dimension: $$26$$ over $$\Q(\zeta_{40})$$ Sato-Tate group: $\mathrm{SU}(2)[C_{40}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$416q - 32q^{2} - 40q^{4} - 16q^{7} - 48q^{8} - 48q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$416q - 32q^{2} - 40q^{4} - 16q^{7} - 48q^{8} - 48q^{9} - 32q^{11} - 12q^{14} - 8q^{15} + 56q^{16} - 24q^{18} + 4q^{21} - 64q^{22} - 40q^{23} - 40q^{25} - 32q^{28} - 24q^{29} - 8q^{30} + 32q^{32} - 16q^{35} - 96q^{36} + 48q^{37} - 32q^{39} - 192q^{42} - 8q^{43} + 128q^{44} + 48q^{46} - 48q^{49} - 120q^{50} + 48q^{51} - 32q^{53} - 124q^{56} - 8q^{57} + 56q^{58} - 152q^{60} + 112q^{63} - 40q^{64} - 120q^{65} - 96q^{67} + 32q^{70} + 64q^{71} - 40q^{72} - 72q^{74} + 76q^{77} + 128q^{78} - 40q^{79} + 304q^{84} - 48q^{85} - 40q^{86} + 24q^{88} + 132q^{91} - 144q^{92} + 24q^{93} - 32q^{95} + 88q^{98} - 48q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6.1 −1.23448 + 2.42281i −1.98219 + 0.821052i −3.17047 4.36378i 1.13770 + 0.180193i 0.457733 5.81604i 0.678495 2.55727i 9.11507 1.44369i 1.13365 1.13365i −1.84104 + 2.53397i
6.2 −1.23448 + 2.42281i 1.98219 0.821052i −3.17047 4.36378i −1.13770 0.180193i −0.457733 + 5.81604i 1.07019 + 2.41965i 9.11507 1.44369i 1.13365 1.13365i 1.84104 2.53397i
6.3 −1.01388 + 1.98985i −1.23689 + 0.512338i −1.75598 2.41690i 2.31334 + 0.366396i 0.234585 2.98068i −0.573830 + 2.58277i 2.17809 0.344976i −0.853904 + 0.853904i −3.07452 + 4.23171i
6.4 −1.01388 + 1.98985i 1.23689 0.512338i −1.75598 2.41690i −2.31334 0.366396i −0.234585 + 2.98068i −0.970800 2.46121i 2.17809 0.344976i −0.853904 + 0.853904i 3.07452 4.23171i
6.5 −0.966243 + 1.89636i −1.91386 + 0.792745i −1.48698 2.04665i −3.95080 0.625746i 0.345921 4.39534i −1.55491 + 2.14062i 1.11370 0.176393i 0.913082 0.913082i 5.00407 6.88751i
6.6 −0.966243 + 1.89636i 1.91386 0.792745i −1.48698 2.04665i 3.95080 + 0.625746i −0.345921 + 4.39534i −1.87064 1.87102i 1.11370 0.176393i 0.913082 0.913082i −5.00407 + 6.88751i
6.7 −0.645342 + 1.26655i −2.64325 + 1.09487i −0.0121231 0.0166860i −1.74535 0.276436i 0.319088 4.05439i 1.63135 2.08295i −2.77901 + 0.440153i 3.66673 3.66673i 1.47647 2.03218i
6.8 −0.645342 + 1.26655i 2.64325 1.09487i −0.0121231 0.0166860i 1.74535 + 0.276436i −0.319088 + 4.05439i 1.93711 + 1.80211i −2.77901 + 0.440153i 3.66673 3.66673i −1.47647 + 2.03218i
6.9 −0.463762 + 0.910185i −0.818378 + 0.338983i 0.562210 + 0.773816i −0.839381 0.132945i 0.0709954 0.902082i −2.27076 1.35781i −2.98294 + 0.472452i −1.56649 + 1.56649i 0.510277 0.702336i
6.10 −0.463762 + 0.910185i 0.818378 0.338983i 0.562210 + 0.773816i 0.839381 + 0.132945i −0.0709954 + 0.902082i −2.03040 + 1.69632i −2.98294 + 0.472452i −1.56649 + 1.56649i −0.510277 + 0.702336i
6.11 −0.356263 + 0.699206i −2.42868 + 1.00599i 0.813605 + 1.11983i 2.63602 + 0.417504i 0.161854 2.05655i 2.22346 + 1.43396i −2.62300 + 0.415443i 2.76515 2.76515i −1.23104 + 1.69438i
6.12 −0.356263 + 0.699206i 2.42868 1.00599i 0.813605 + 1.11983i −2.63602 0.417504i −0.161854 + 2.05655i 1.97176 1.76413i −2.62300 + 0.415443i 2.76515 2.76515i 1.23104 1.69438i
6.13 −0.0362277 + 0.0711009i −0.338380 + 0.140161i 1.17183 + 1.61288i −2.94779 0.466885i 0.00229312 0.0291368i 1.96056 + 1.77657i −0.314762 + 0.0498534i −2.02646 + 2.02646i 0.139988 0.192677i
6.14 −0.0362277 + 0.0711009i 0.338380 0.140161i 1.17183 + 1.61288i 2.94779 + 0.466885i −0.00229312 + 0.0291368i 1.65850 2.06140i −0.314762 + 0.0498534i −2.02646 + 2.02646i −0.139988 + 0.192677i
6.15 0.171065 0.335734i −2.78559 + 1.15383i 1.09212 + 1.50317i −0.146833 0.0232561i −0.0891375 + 1.13260i −2.47156 + 0.944132i 1.43582 0.227411i 4.30688 4.30688i −0.0329258 + 0.0453185i
6.16 0.171065 0.335734i 2.78559 1.15383i 1.09212 + 1.50317i 0.146833 + 0.0232561i 0.0891375 1.13260i −2.58883 0.545871i 1.43582 0.227411i 4.30688 4.30688i 0.0329258 0.0453185i
6.17 0.372174 0.730434i −1.52355 + 0.631075i 0.780551 + 1.07434i 0.154931 + 0.0245387i −0.106068 + 1.34772i −1.01262 2.44430i 2.69462 0.426785i −0.198371 + 0.198371i 0.0755853 0.104034i
6.18 0.372174 0.730434i 1.52355 0.631075i 0.780551 + 1.07434i −0.154931 0.0245387i 0.106068 1.34772i −0.617776 + 2.57262i 2.69462 0.426785i −0.198371 + 0.198371i −0.0755853 + 0.104034i
6.19 0.727703 1.42820i −1.25995 + 0.521890i −0.334625 0.460572i 1.20688 + 0.191151i −0.171510 + 2.17924i 1.62275 + 2.08966i 2.26504 0.358747i −0.806204 + 0.806204i 1.15125 1.58456i
6.20 0.727703 1.42820i 1.25995 0.521890i −0.334625 0.460572i −1.20688 0.191151i 0.171510 2.17924i 1.27588 2.31778i 2.26504 0.358747i −0.806204 + 0.806204i −1.15125 + 1.58456i
See next 80 embeddings (of 416 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 272.26 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 $$S_{2}^{\mathrm{new}}(287, [\chi])$$.