# Properties

 Label 287.3.d.d Level 287 Weight 3 Character orbit 287.d Analytic conductor 7.820 Analytic rank 0 Dimension 32 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$287 = 7 \cdot 41$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 287.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.82018358714$$ Analytic rank: $$0$$ Dimension: $$32$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 4q^{2} + 68q^{4} - 88q^{8} + 44q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 4q^{2} + 68q^{4} - 88q^{8} + 44q^{9} - 92q^{16} - 48q^{18} - 72q^{21} + 140q^{23} - 500q^{25} + 92q^{32} - 284q^{36} + 312q^{37} + 140q^{39} + 8q^{42} - 120q^{43} - 344q^{46} - 552q^{49} + 416q^{50} - 364q^{51} - 316q^{57} - 320q^{64} + 972q^{72} + 680q^{74} + 428q^{77} + 1144q^{78} - 240q^{81} + 640q^{84} + 260q^{86} - 160q^{91} + 676q^{92} + 532q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
286.1 −3.60835 −0.377109 9.02020 7.18702i 1.36074 −2.11657 6.67234i −18.1146 −8.85779 25.9333i
286.2 −3.60835 −0.377109 9.02020 7.18702i 1.36074 −2.11657 + 6.67234i −18.1146 −8.85779 25.9333i
286.3 −3.60835 0.377109 9.02020 7.18702i −1.36074 2.11657 + 6.67234i −18.1146 −8.85779 25.9333i
286.4 −3.60835 0.377109 9.02020 7.18702i −1.36074 2.11657 6.67234i −18.1146 −8.85779 25.9333i
286.5 −2.67431 −2.89735 3.15194 3.34396i 7.74841 4.11915 5.65974i 2.26798 −0.605380 8.94280i
286.6 −2.67431 −2.89735 3.15194 3.34396i 7.74841 4.11915 + 5.65974i 2.26798 −0.605380 8.94280i
286.7 −2.67431 2.89735 3.15194 3.34396i −7.74841 −4.11915 + 5.65974i 2.26798 −0.605380 8.94280i
286.8 −2.67431 2.89735 3.15194 3.34396i −7.74841 −4.11915 5.65974i 2.26798 −0.605380 8.94280i
286.9 −2.31240 −5.31887 1.34721 9.29617i 12.2994 −2.62328 + 6.48987i 6.13432 19.2904 21.4965i
286.10 −2.31240 −5.31887 1.34721 9.29617i 12.2994 −2.62328 6.48987i 6.13432 19.2904 21.4965i
286.11 −2.31240 5.31887 1.34721 9.29617i −12.2994 2.62328 6.48987i 6.13432 19.2904 21.4965i
286.12 −2.31240 5.31887 1.34721 9.29617i −12.2994 2.62328 + 6.48987i 6.13432 19.2904 21.4965i
286.13 0.517567 −2.41493 −3.73212 2.97759i −1.24989 2.38045 + 6.58281i −4.00189 −3.16813 1.54110i
286.14 0.517567 −2.41493 −3.73212 2.97759i −1.24989 2.38045 6.58281i −4.00189 −3.16813 1.54110i
286.15 0.517567 2.41493 −3.73212 2.97759i 1.24989 −2.38045 6.58281i −4.00189 −3.16813 1.54110i
286.16 0.517567 2.41493 −3.73212 2.97759i 1.24989 −2.38045 + 6.58281i −4.00189 −3.16813 1.54110i
286.17 1.16717 −4.43412 −2.63772 9.05049i −5.17535 6.69015 2.05957i −7.74732 10.6614 10.5634i
286.18 1.16717 −4.43412 −2.63772 9.05049i −5.17535 6.69015 + 2.05957i −7.74732 10.6614 10.5634i
286.19 1.16717 4.43412 −2.63772 9.05049i 5.17535 −6.69015 + 2.05957i −7.74732 10.6614 10.5634i
286.20 1.16717 4.43412 −2.63772 9.05049i 5.17535 −6.69015 2.05957i −7.74732 10.6614 10.5634i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 286.32 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.b odd 2 1 inner
41.b even 2 1 inner
287.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 287.3.d.d 32
7.b odd 2 1 inner 287.3.d.d 32
41.b even 2 1 inner 287.3.d.d 32
287.d odd 2 1 inner 287.3.d.d 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.3.d.d 32 1.a even 1 1 trivial
287.3.d.d 32 7.b odd 2 1 inner
287.3.d.d 32 41.b even 2 1 inner
287.3.d.d 32 287.d odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(287, [\chi])$$:

 $$T_{2}^{8} - \cdots$$ $$T_{3}^{16} - \cdots$$