# Properties

 Label 287.3.ba.a Level 287 Weight 3 Character orbit 287.ba Analytic conductor 7.820 Analytic rank 0 Dimension 672 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$287 = 7 \cdot 41$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 287.ba (of order $$40$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.82018358714$$ Analytic rank: $$0$$ Dimension: $$672$$ Relative dimension: $$42$$ over $$\Q(\zeta_{40})$$ Coefficient ring index: multiple of None Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$672q - 8q^{2} + 16q^{3} - 24q^{6} + 48q^{8} + 48q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$672q - 8q^{2} + 16q^{3} - 24q^{6} + 48q^{8} + 48q^{9} - 216q^{12} - 88q^{13} + 672q^{16} + 88q^{17} - 128q^{22} + 192q^{24} - 40q^{26} - 56q^{27} + 80q^{29} - 384q^{30} - 360q^{31} - 776q^{32} + 232q^{33} - 552q^{34} + 56q^{35} - 632q^{36} + 80q^{37} - 128q^{38} - 128q^{39} - 184q^{41} + 560q^{42} - 184q^{43} + 352q^{44} + 800q^{45} + 544q^{46} + 216q^{47} + 1792q^{48} + 624q^{50} - 80q^{51} + 984q^{52} + 592q^{53} - 440q^{54} + 48q^{55} - 40q^{58} - 1152q^{59} + 824q^{60} - 768q^{61} + 56q^{62} - 224q^{65} - 2400q^{66} - 992q^{67} - 128q^{68} + 424q^{69} - 1424q^{71} - 3240q^{72} - 912q^{73} - 1928q^{74} + 864q^{75} + 352q^{76} - 440q^{78} - 368q^{79} - 320q^{80} - 648q^{82} - 960q^{83} + 1488q^{85} + 2000q^{86} - 160q^{87} + 2408q^{88} + 752q^{89} + 1088q^{90} - 224q^{91} + 1192q^{92} + 1024q^{93} + 3104q^{94} + 1592q^{95} + 1600q^{96} + 544q^{97} + 2000q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
15.1 −3.78044 0.598763i −2.87761 + 1.19194i 10.1290 + 3.29111i −3.66111 7.18534i 11.5923 2.78307i −2.57265 0.617638i −22.6799 11.5560i 0.495936 0.495936i 9.53830 + 29.3559i
15.2 −3.66968 0.581219i 5.26184 2.17952i 9.32448 + 3.02971i 3.63247 + 7.12913i −20.5760 + 4.93986i 2.57265 + 0.617638i −19.2150 9.79055i 16.5726 16.5726i −9.18641 28.2728i
15.3 −3.66700 0.580796i 0.570646 0.236369i 9.30533 + 3.02349i −0.556019 1.09125i −2.22984 + 0.535337i 2.57265 + 0.617638i −19.1344 9.74948i −6.09419 + 6.09419i 1.40513 + 4.32454i
15.4 −3.58163 0.567275i −4.62584 + 1.91609i 8.70208 + 2.82748i 1.82688 + 3.58546i 17.6550 4.23860i 2.57265 + 0.617638i −16.6395 8.47827i 11.3631 11.3631i −4.50928 13.8781i
15.5 −3.43813 0.544547i 3.38252 1.40108i 7.72000 + 2.50838i −2.27644 4.46777i −12.3925 + 2.97517i −2.57265 0.617638i −12.7701 6.50670i 3.11441 3.11441i 5.39381 + 16.6004i
15.6 −3.42144 0.541903i 2.05380 0.850711i 7.60836 + 2.47211i 0.819120 + 1.60761i −7.48794 + 1.79770i −2.57265 0.617638i −12.3458 6.29050i −2.86959 + 2.86959i −1.93140 5.94424i
15.7 −2.84388 0.450427i −3.65012 + 1.51193i 4.08055 + 1.32585i −0.933634 1.83236i 11.0615 2.65563i −2.57265 0.617638i −0.745394 0.379797i 4.67346 4.67346i 1.82980 + 5.63155i
15.8 −2.75441 0.436256i 4.50992 1.86807i 3.59224 + 1.16719i −3.73781 7.33586i −13.2371 + 3.17796i 2.57265 + 0.617638i 0.553839 + 0.282195i 10.4857 10.4857i 7.09515 + 21.8366i
15.9 −2.70944 0.429133i −3.73900 + 1.54874i 3.35268 + 1.08935i 2.16823 + 4.25539i 10.7952 2.59170i −2.57265 0.617638i 1.16048 + 0.591295i 5.21755 5.21755i −4.04856 12.4602i
15.10 −2.68738 0.425640i −0.267448 + 0.110781i 3.23664 + 1.05165i −2.28153 4.47776i 0.765888 0.183873i 2.57265 + 0.617638i 1.44684 + 0.737200i −6.30470 + 6.30470i 4.22544 + 13.0046i
15.11 −2.42937 0.384775i 1.48197 0.613852i 1.94957 + 0.633455i 3.86932 + 7.59398i −3.83645 + 0.921050i −2.57265 0.617638i 4.27377 + 2.17760i −4.54454 + 4.54454i −6.47806 19.9374i
15.12 −2.13834 0.338680i 4.07301 1.68710i 0.653585 + 0.212363i −0.409196 0.803093i −9.28089 + 2.22815i −2.57265 0.617638i 6.39045 + 3.25609i 7.37918 7.37918i 0.603011 + 1.85588i
15.13 −1.96641 0.311449i 3.24786 1.34531i −0.0344593 0.0111965i 1.89942 + 3.72783i −6.80562 + 1.63388i 2.57265 + 0.617638i 7.15997 + 3.64819i 2.37478 2.37478i −2.57402 7.92201i
15.14 −1.83089 0.289985i −4.82616 + 1.99906i −0.536147 0.174205i −2.39266 4.69587i 9.41587 2.26055i 2.57265 + 0.617638i 7.53780 + 3.84070i 12.9316 12.9316i 3.01898 + 9.29147i
15.15 −1.43794 0.227747i −2.27018 + 0.940338i −1.78843 0.581095i 3.73773 + 7.33572i 3.47854 0.835122i 2.57265 + 0.617638i 7.62804 + 3.88668i −2.09449 + 2.09449i −3.70394 11.3996i
15.16 −1.13941 0.180465i −1.89135 + 0.783423i −2.53853 0.824819i 0.0171779 + 0.0337136i 2.29641 0.551319i 2.57265 + 0.617638i 6.85510 + 3.49285i −3.40051 + 3.40051i −0.0134886 0.0415137i
15.17 −1.05438 0.166997i −2.28946 + 0.948324i −2.72040 0.883911i 1.32214 + 2.59484i 2.57232 0.617560i −2.57265 0.617638i 6.52540 + 3.32486i −2.02167 + 2.02167i −0.960704 2.95674i
15.18 −1.00230 0.158748i 1.12918 0.467721i −2.82483 0.917843i −1.44168 2.82945i −1.20602 + 0.289540i −2.57265 0.617638i 6.30235 + 3.21121i −5.30768 + 5.30768i 0.995819 + 3.06482i
15.19 −0.313369 0.0496328i 2.18150 0.903609i −3.70849 1.20496i 0.209267 + 0.410709i −0.728465 + 0.174889i 2.57265 + 0.617638i 2.23310 + 1.13782i −2.42151 + 2.42151i −0.0451931 0.139090i
15.20 −0.140552 0.0222612i 1.74628 0.723333i −3.78497 1.22981i −4.24244 8.32626i −0.261545 + 0.0627914i 2.57265 + 0.617638i 1.01178 + 0.515528i −3.83768 + 3.83768i 0.410930 + 1.26471i
See next 80 embeddings (of 672 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 281.42 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.h odd 40 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 287.3.ba.a 672
41.h odd 40 1 inner 287.3.ba.a 672

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.3.ba.a 672 1.a even 1 1 trivial
287.3.ba.a 672 41.h odd 40 1 inner

## Hecke kernels

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