# Properties

 Label 287.3.i.a Level 287 Weight 3 Character orbit 287.i Analytic conductor 7.820 Analytic rank 0 Dimension 108 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$287 = 7 \cdot 41$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 287.i (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.82018358714$$ Analytic rank: $$0$$ Dimension: $$108$$ Relative dimension: $$54$$ over $$\Q(\zeta_{6})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$108q - 2q^{2} - 106q^{4} - 6q^{5} + 20q^{8} - 136q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$108q - 2q^{2} - 106q^{4} - 6q^{5} + 20q^{8} - 136q^{9} - 60q^{10} - 202q^{16} - 4q^{18} - 56q^{21} + 12q^{23} + 208q^{25} + 30q^{31} - 152q^{32} + 24q^{33} + 284q^{36} - 52q^{37} + 30q^{39} + 24q^{40} - 78q^{42} - 112q^{43} - 210q^{45} - 264q^{46} + 380q^{49} - 48q^{50} + 180q^{51} + 168q^{57} - 138q^{59} - 294q^{61} + 268q^{64} - 612q^{66} + 74q^{72} + 48q^{73} - 194q^{74} + 256q^{77} + 184q^{78} + 12q^{80} - 314q^{81} + 474q^{82} + 828q^{84} - 496q^{86} + 1122q^{87} - 786q^{91} + 160q^{92} - 176q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
40.1 −1.94902 + 3.37581i −0.674249 1.16783i −5.59740 9.69497i 4.70200 + 2.71470i 5.25651 1.91113 6.73406i 28.0457 3.59078 6.21941i −18.3286 + 10.5820i
40.2 −1.94902 + 3.37581i 0.674249 + 1.16783i −5.59740 9.69497i 4.70200 + 2.71470i −5.25651 −1.91113 + 6.73406i 28.0457 3.59078 6.21941i −18.3286 + 10.5820i
40.3 −1.79153 + 3.10301i −1.23268 2.13507i −4.41913 7.65416i −7.00807 4.04611i 8.83353 −6.98958 0.381749i 17.3357 1.46099 2.53050i 25.1103 14.4974i
40.4 −1.79153 + 3.10301i 1.23268 + 2.13507i −4.41913 7.65416i −7.00807 4.04611i −8.83353 6.98958 + 0.381749i 17.3357 1.46099 2.53050i 25.1103 14.4974i
40.5 −1.75354 + 3.03722i −2.52990 4.38192i −4.14980 7.18766i −1.03847 0.599559i 17.7451 6.93494 + 0.952149i 15.0790 −8.30079 + 14.3774i 3.64198 2.10270i
40.6 −1.75354 + 3.03722i 2.52990 + 4.38192i −4.14980 7.18766i −1.03847 0.599559i −17.7451 −6.93494 0.952149i 15.0790 −8.30079 + 14.3774i 3.64198 2.10270i
40.7 −1.55835 + 2.69914i −2.63338 4.56115i −2.85692 4.94833i 4.74687 + 2.74061i 16.4149 −6.99950 + 0.0840686i 5.34153 −9.36940 + 16.2283i −14.7946 + 8.54166i
40.8 −1.55835 + 2.69914i 2.63338 + 4.56115i −2.85692 4.94833i 4.74687 + 2.74061i −16.4149 6.99950 0.0840686i 5.34153 −9.36940 + 16.2283i −14.7946 + 8.54166i
40.9 −1.50161 + 2.60086i −0.735502 1.27393i −2.50965 4.34684i 0.0228039 + 0.0131658i 4.41774 −1.47843 + 6.84209i 3.06116 3.41807 5.92028i −0.0684849 + 0.0395398i
40.10 −1.50161 + 2.60086i 0.735502 + 1.27393i −2.50965 4.34684i 0.0228039 + 0.0131658i −4.41774 1.47843 6.84209i 3.06116 3.41807 5.92028i −0.0684849 + 0.0395398i
40.11 −1.24731 + 2.16040i −0.923338 1.59927i −1.11156 1.92528i −1.05182 0.607270i 4.60675 4.85778 + 5.04004i −4.43264 2.79489 4.84090i 2.62389 1.51491i
40.12 −1.24731 + 2.16040i 0.923338 + 1.59927i −1.11156 1.92528i −1.05182 0.607270i −4.60675 −4.85778 5.04004i −4.43264 2.79489 4.84090i 2.62389 1.51491i
40.13 −1.19334 + 2.06692i −0.991848 1.71793i −0.848107 1.46896i 7.97769 + 4.60592i 4.73444 6.85435 1.42052i −5.49839 2.53248 4.38638i −19.0401 + 10.9928i
40.14 −1.19334 + 2.06692i 0.991848 + 1.71793i −0.848107 1.46896i 7.97769 + 4.60592i −4.73444 −6.85435 + 1.42052i −5.49839 2.53248 4.38638i −19.0401 + 10.9928i
40.15 −1.08643 + 1.88175i −2.10629 3.64821i −0.360668 0.624694i −6.23743 3.60118i 9.15338 1.17747 6.90026i −7.12409 −4.37295 + 7.57418i 13.5531 7.82487i
40.16 −1.08643 + 1.88175i 2.10629 + 3.64821i −0.360668 0.624694i −6.23743 3.60118i −9.15338 −1.17747 + 6.90026i −7.12409 −4.37295 + 7.57418i 13.5531 7.82487i
40.17 −0.736544 + 1.27573i −1.76416 3.05561i 0.915005 + 1.58483i 2.30976 + 1.33354i 5.19752 −1.63087 6.80737i −8.58812 −1.72450 + 2.98691i −3.40248 + 1.96442i
40.18 −0.736544 + 1.27573i 1.76416 + 3.05561i 0.915005 + 1.58483i 2.30976 + 1.33354i −5.19752 1.63087 + 6.80737i −8.58812 −1.72450 + 2.98691i −3.40248 + 1.96442i
40.19 −0.702038 + 1.21597i −2.20646 3.82171i 1.01428 + 1.75679i 3.32560 + 1.92003i 6.19609 −5.26852 + 4.60898i −8.46457 −5.23697 + 9.07071i −4.66939 + 2.69588i
40.20 −0.702038 + 1.21597i 2.20646 + 3.82171i 1.01428 + 1.75679i 3.32560 + 1.92003i −6.19609 5.26852 4.60898i −8.46457 −5.23697 + 9.07071i −4.66939 + 2.69588i
See next 80 embeddings (of 108 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 122.54 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.b even 2 1 inner
287.i odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 287.3.i.a 108
7.d odd 6 1 inner 287.3.i.a 108
41.b even 2 1 inner 287.3.i.a 108
287.i odd 6 1 inner 287.3.i.a 108

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.3.i.a 108 1.a even 1 1 trivial
287.3.i.a 108 7.d odd 6 1 inner
287.3.i.a 108 41.b even 2 1 inner
287.3.i.a 108 287.i odd 6 1 inner

## Hecke kernels

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