# Properties

 Label 287.3.m.a Level 287 Weight 3 Character orbit 287.m Analytic conductor 7.820 Analytic rank 0 Dimension 168 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$287 = 7 \cdot 41$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 287.m (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.82018358714$$ Analytic rank: $$0$$ Dimension: $$168$$ Relative dimension: $$42$$ over $$\Q(\zeta_{8})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$168q + 8q^{2} - 16q^{3} + 24q^{6} - 48q^{8} - 48q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$168q + 8q^{2} - 16q^{3} + 24q^{6} - 48q^{8} - 48q^{9} + 216q^{12} + 88q^{13} - 672q^{16} - 88q^{17} + 128q^{22} - 192q^{24} + 40q^{26} + 56q^{27} - 80q^{29} + 384q^{30} - 344q^{32} - 232q^{33} - 48q^{34} - 56q^{35} - 488q^{36} - 80q^{37} - 32q^{38} - 32q^{39} + 224q^{41} - 560q^{42} + 304q^{43} - 352q^{44} - 64q^{46} - 216q^{47} + 448q^{48} + 376q^{50} + 80q^{51} + 696q^{52} - 72q^{53} + 440q^{54} - 48q^{55} + 40q^{58} + 1152q^{59} - 824q^{60} + 768q^{61} - 56q^{62} - 96q^{65} - 688q^{67} + 128q^{68} - 424q^{69} - 176q^{71} - 368q^{73} + 248q^{74} - 864q^{75} - 352q^{76} - 760q^{78} + 48q^{79} - 80q^{80} + 648q^{82} + 960q^{83} - 128q^{85} + 1120q^{87} + 392q^{88} - 752q^{89} - 1088q^{90} + 224q^{91} + 1448q^{92} + 896q^{93} + 1576q^{94} + 648q^{95} - 1600q^{96} - 544q^{97} + 160q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
85.1 −2.66518 + 2.66518i −0.402189 + 0.970969i 10.2064i −1.88272 1.88272i −1.51590 3.65972i −1.01249 + 2.44436i 16.5412 + 16.5412i 5.58294 + 5.58294i 10.0356
85.2 −2.60935 + 2.60935i −1.18140 + 2.85214i 9.61738i 5.05187 + 5.05187i −4.35956 10.5249i 1.01249 2.44436i 14.6577 + 14.6577i −0.375070 0.375070i −26.3641
85.3 −2.53377 + 2.53377i −2.06584 + 4.98738i 8.83994i −5.28886 5.28886i −7.40250 17.8712i 1.01249 2.44436i 12.2633 + 12.2633i −14.2423 14.2423i 26.8015
85.4 −2.35635 + 2.35635i 0.594151 1.43441i 7.10479i 0.766826 + 0.766826i 1.97994 + 4.77999i 1.01249 2.44436i 7.31597 + 7.31597i 4.65945 + 4.65945i −3.61382
85.5 −2.28855 + 2.28855i −1.56996 + 3.79022i 6.47494i 1.63254 + 1.63254i −5.08118 12.2670i −1.01249 + 2.44436i 5.66404 + 5.66404i −5.53702 5.53702i −7.47229
85.6 −2.24689 + 2.24689i 1.13818 2.74781i 6.09703i 6.89996 + 6.89996i 3.61666 + 8.73139i −1.01249 + 2.44436i 4.71181 + 4.71181i 0.108960 + 0.108960i −31.0069
85.7 −2.09220 + 2.09220i 0.418806 1.01109i 4.75459i −5.05847 5.05847i 1.23917 + 2.99162i −1.01249 + 2.44436i 1.57875 + 1.57875i 5.51706 + 5.51706i 21.1667
85.8 −1.90333 + 1.90333i 1.37451 3.31835i 3.24533i −1.07431 1.07431i 3.69978 + 8.93206i −1.01249 + 2.44436i −1.43639 1.43639i −2.75823 2.75823i 4.08952
85.9 −1.74186 + 1.74186i 1.60775 3.88144i 2.06815i −6.92683 6.92683i 3.96046 + 9.56139i 1.01249 2.44436i −3.36502 3.36502i −6.11678 6.11678i 24.1311
85.10 −1.67751 + 1.67751i 0.0291431 0.0703578i 1.62807i 1.99276 + 1.99276i 0.0691379 + 0.166914i 1.01249 2.44436i −3.97893 3.97893i 6.35986 + 6.35986i −6.68576
85.11 −1.56178 + 1.56178i −1.33869 + 3.23188i 0.878306i −2.93781 2.93781i −2.95674 7.13821i 1.01249 2.44436i −4.87540 4.87540i −2.28898 2.28898i 9.17643
85.12 −1.53733 + 1.53733i −1.13549 + 2.74132i 0.726768i 4.22119 + 4.22119i −2.46869 5.95995i −1.01249 + 2.44436i −5.03204 5.03204i 0.138450 + 0.138450i −12.9787
85.13 −1.25621 + 1.25621i −2.14947 + 5.18928i 0.843870i 4.68795 + 4.68795i −3.81864 9.21902i 1.01249 2.44436i −6.08492 6.08492i −15.9445 15.9445i −11.7781
85.14 −1.20352 + 1.20352i 2.13250 5.14832i 1.10308i 0.770141 + 0.770141i 3.62959 + 8.76261i −1.01249 + 2.44436i −6.14166 6.14166i −15.5936 15.5936i −1.85376
85.15 −1.04509 + 1.04509i 1.67777 4.05049i 1.81559i 0.878824 + 0.878824i 2.47970 + 5.98653i 1.01249 2.44436i −6.07779 6.07779i −7.22761 7.22761i −1.83689
85.16 −0.593548 + 0.593548i −0.514688 + 1.24257i 3.29540i −2.56222 2.56222i −0.432031 1.04301i 1.01249 2.44436i −4.33017 4.33017i 5.08489 + 5.08489i 3.04160
85.17 −0.558735 + 0.558735i 0.988435 2.38629i 3.37563i −0.00806792 0.00806792i 0.781033 + 1.88558i −1.01249 + 2.44436i −4.12102 4.12102i 1.64657 + 1.64657i 0.00901566
85.18 −0.524382 + 0.524382i 0.375483 0.906495i 3.45005i −4.83386 4.83386i 0.278453 + 0.672246i −1.01249 + 2.44436i −3.90667 3.90667i 5.68321 + 5.68321i 5.06958
85.19 −0.472219 + 0.472219i −1.96830 + 4.75189i 3.55402i −3.06808 3.06808i −1.31446 3.17340i −1.01249 + 2.44436i −3.56715 3.56715i −12.3423 12.3423i 2.89762
85.20 −0.328376 + 0.328376i −0.752203 + 1.81598i 3.78434i 4.15285 + 4.15285i −0.349318 0.843329i −1.01249 + 2.44436i −2.55619 2.55619i 3.63199 + 3.63199i −2.72739
See next 80 embeddings (of 168 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 260.42 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.e odd 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 287.3.m.a 168
41.e odd 8 1 inner 287.3.m.a 168

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.3.m.a 168 1.a even 1 1 trivial
287.3.m.a 168 41.e odd 8 1 inner

## Hecke kernels

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