# Properties

 Label 287.3.q.a Level 287 Weight 3 Character orbit 287.q Analytic conductor 7.820 Analytic rank 0 Dimension 216 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$287 = 7 \cdot 41$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 287.q (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.82018358714$$ Analytic rank: $$0$$ Dimension: $$216$$ Relative dimension: $$54$$ over $$\Q(\zeta_{12})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$216q - 6q^{3} + 204q^{4} - 4q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$216q - 6q^{3} + 204q^{4} - 4q^{7} - 12q^{10} - 10q^{11} - 30q^{12} - 74q^{14} + 16q^{15} - 372q^{16} + 48q^{17} - 36q^{18} - 78q^{19} - 80q^{22} - 4q^{23} + 108q^{24} - 464q^{25} + 36q^{26} + 56q^{28} - 120q^{29} - 188q^{30} - 84q^{31} + 22q^{35} - 104q^{37} - 24q^{38} + 240q^{40} + 320q^{42} + 118q^{44} + 180q^{45} - 282q^{47} + 112q^{51} - 306q^{52} - 244q^{53} + 54q^{54} + 510q^{56} - 344q^{57} - 116q^{58} + 252q^{59} + 236q^{60} - 30q^{63} - 840q^{64} - 52q^{65} + 828q^{66} - 294q^{67} + 78q^{68} - 282q^{70} + 336q^{71} + 548q^{72} + 42q^{75} - 1528q^{78} + 8q^{79} + 792q^{81} - 342q^{82} + 4q^{85} - 212q^{86} + 252q^{88} + 396q^{89} - 352q^{92} + 118q^{93} + 576q^{94} + 278q^{95} + 138q^{96} + 780q^{98} + 40q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
73.1 −3.32140 + 1.91761i 1.15997 0.310813i 5.35446 9.27420i −2.42895 4.20707i −3.25670 + 3.25670i 6.43721 2.74996i 25.7302i −6.54530 + 3.77893i 16.1351 + 9.31558i
73.2 −3.29726 + 1.90368i 2.58181 0.691795i 5.24797 9.08975i 3.40762 + 5.90217i −7.19597 + 7.19597i 0.200258 + 6.99713i 24.7323i −1.60705 + 0.927831i −22.4717 12.9740i
73.3 −3.26748 + 1.88648i −4.81897 + 1.29124i 5.11761 8.86396i −2.61661 4.53210i 13.3100 13.3100i 2.60313 + 6.49798i 23.5252i 13.7610 7.94489i 17.0994 + 9.87236i
73.4 −3.02057 + 1.74393i 4.33367 1.16120i 4.08257 7.07121i −3.12206 5.40756i −11.0651 + 11.0651i −6.98293 + 0.488488i 14.5274i 9.63808 5.56455i 18.8608 + 10.8893i
73.5 −2.99250 + 1.72772i −3.76632 + 1.00918i 3.97002 6.87628i 0.907514 + 1.57186i 9.52711 9.52711i −0.422895 6.98721i 13.6146i 5.37248 3.10180i −5.43146 3.13586i
73.6 −2.86475 + 1.65396i −1.20386 + 0.322573i 3.47118 6.01226i −0.352767 0.611010i 2.91523 2.91523i −6.65179 + 2.18029i 9.73311i −6.44901 + 3.72334i 2.02117 + 1.16693i
73.7 −2.81671 + 1.62623i 1.49616 0.400895i 3.28926 5.69716i 4.01378 + 6.95206i −3.56231 + 3.56231i −0.176070 6.99779i 8.38651i −5.71645 + 3.30039i −22.6113 13.0547i
73.8 −2.67372 + 1.54367i −3.71258 + 0.994784i 2.76586 4.79061i 3.73945 + 6.47692i 8.39080 8.39080i 6.61930 + 2.27703i 4.72895i 4.99946 2.88644i −19.9965 11.5450i
73.9 −2.60389 + 1.50336i 5.11320 1.37008i 2.52016 4.36505i 1.80353 + 3.12380i −11.2545 + 11.2545i 1.87307 6.74475i 3.12796i 16.4735 9.51099i −9.39238 5.42269i
73.10 −2.27032 + 1.31077i 4.22026 1.13082i 1.43624 2.48764i −0.688501 1.19252i −8.09911 + 8.09911i 4.55185 + 5.31796i 2.95585i 8.73764 5.04468i 3.12624 + 1.80493i
73.11 −2.27028 + 1.31075i −0.713090 + 0.191072i 1.43611 2.48742i −3.48098 6.02923i 1.36847 1.36847i 3.27191 + 6.18826i 2.95646i −7.32224 + 4.22750i 15.8056 + 9.12535i
73.12 −2.20801 + 1.27480i 0.503298 0.134858i 1.25022 2.16544i −0.660961 1.14482i −0.939373 + 0.939373i 6.75058 1.85194i 3.82329i −7.55911 + 4.36425i 2.91882 + 1.68518i
73.13 −2.14335 + 1.23746i −3.28848 + 0.881145i 1.06263 1.84052i −4.85635 8.41145i 5.95797 5.95797i −2.38873 6.57982i 4.63986i 2.24345 1.29525i 20.8177 + 12.0191i
73.14 −2.04653 + 1.18156i −2.47138 + 0.662204i 0.792187 1.37211i 3.06597 + 5.31041i 4.27531 4.27531i −1.68831 + 6.79335i 5.70843i −2.12504 + 1.22689i −12.5492 7.24528i
73.15 −1.69267 + 0.977266i 1.83287 0.491117i −0.0899034 + 0.155717i −2.23127 3.86467i −2.62250 + 2.62250i −4.95227 4.94722i 8.16956i −4.67600 + 2.69969i 7.55362 + 4.36108i
73.16 −1.65656 + 0.956418i 1.46388 0.392246i −0.170530 + 0.295367i 2.30350 + 3.98978i −2.04986 + 2.04986i −6.96672 0.681733i 8.30373i −5.80513 + 3.35159i −7.63179 4.40622i
73.17 −1.63115 + 0.941746i −5.53126 + 1.48210i −0.226228 + 0.391838i 1.72421 + 2.98642i 7.62657 7.62657i −6.86836 1.35117i 8.38617i 20.6040 11.8957i −5.62490 3.24754i
73.18 −1.21135 + 0.699374i −5.22513 + 1.40007i −1.02175 + 1.76973i −1.85825 3.21859i 5.35029 5.35029i 6.57733 + 2.39557i 8.45334i 17.5475 10.1311i 4.50199 + 2.59923i
73.19 −1.04789 + 0.604999i 1.82152 0.488075i −1.26795 + 2.19616i 3.38546 + 5.86378i −1.61347 + 1.61347i 6.99425 0.283563i 7.90844i −4.71451 + 2.72192i −7.09517 4.09640i
73.20 −1.04052 + 0.600745i −2.35139 + 0.630054i −1.27821 + 2.21393i 0.621112 + 1.07580i 2.06817 2.06817i 4.10040 5.67333i 7.87747i −2.66215 + 1.53699i −1.29256 0.746260i
See next 80 embeddings (of 216 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 278.54 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
41.c even 4 1 inner
287.q odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 287.3.q.a 216
7.d odd 6 1 inner 287.3.q.a 216
41.c even 4 1 inner 287.3.q.a 216
287.q odd 12 1 inner 287.3.q.a 216

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.3.q.a 216 1.a even 1 1 trivial
287.3.q.a 216 7.d odd 6 1 inner
287.3.q.a 216 41.c even 4 1 inner
287.3.q.a 216 287.q odd 12 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(287, [\chi])$$.