# Properties

 Label 287.2.r.c Level 287 Weight 2 Character orbit 287.r Analytic conductor 2.292 Analytic rank 0 Dimension 96 CM No

# Related objects

## Newspace parameters

 Level: $$N$$ = $$287 = 7 \cdot 41$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 287.r (of order $$12$$ and degree $$4$$)

## Newform invariants

 Self dual: No Analytic conductor: $$2.29170653801$$ Analytic rank: $$0$$ Dimension: $$96$$ Relative dimension: $$24$$ over $$\Q(\zeta_{12})$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$96q - 4q^{3} + 48q^{4} - 28q^{6} - 14q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$96q - 4q^{3} + 48q^{4} - 28q^{6} - 14q^{7} - 28q^{10} + 12q^{12} - 8q^{13} + 8q^{14} - 20q^{15} - 40q^{16} - 20q^{17} - 16q^{18} - 8q^{19} - 12q^{22} + 12q^{23} - 30q^{24} + 40q^{25} + 8q^{26} - 4q^{27} - 20q^{28} - 72q^{29} + 14q^{30} + 24q^{31} + 40q^{34} + 20q^{35} + 16q^{37} - 18q^{38} + 80q^{40} - 88q^{41} - 76q^{42} + 4q^{44} - 16q^{45} + 14q^{47} - 24q^{48} - 8q^{51} + 10q^{52} - 4q^{53} + 16q^{54} - 60q^{55} + 36q^{56} + 128q^{57} - 16q^{58} - 8q^{59} + 54q^{60} + 30q^{63} - 16q^{64} + 48q^{66} + 14q^{67} - 30q^{68} + 56q^{69} - 34q^{70} - 68q^{71} + 112q^{72} - 62q^{75} - 84q^{76} - 96q^{78} - 26q^{79} - 32q^{81} + 14q^{82} + 56q^{83} - 92q^{85} + 36q^{86} + 6q^{88} + 40q^{89} - 160q^{92} - 78q^{93} + 96q^{94} + 72q^{95} + 24q^{96} + 60q^{97} - 116q^{98} - 128q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1 −2.36599 1.36600i 0.327392 1.22184i 2.73193 + 4.73184i 0.808637 + 0.466867i −2.44364 + 2.44364i −2.58896 + 0.545252i 9.46328i 1.21236 + 0.699958i −1.27548 2.20920i
9.2 −2.01867 1.16548i 0.0710883 0.265305i 1.71670 + 2.97340i −1.99653 1.15270i −0.452712 + 0.452712i 2.25474 + 1.38425i 3.34118i 2.53274 + 1.46228i 2.68690 + 4.65385i
9.3 −1.94459 1.12271i −0.198974 + 0.742581i 1.52096 + 2.63438i −1.21146 0.699435i 1.22063 1.22063i −0.635224 2.56836i 2.33955i 2.08624 + 1.20449i 1.57053 + 2.72023i
9.4 −1.85131 1.06886i 0.785917 2.93308i 1.28491 + 2.22553i 1.72481 + 0.995818i −4.59003 + 4.59003i 2.23519 1.41560i 1.21811i −5.38724 3.11032i −2.12877 3.68714i
9.5 −1.81296 1.04671i −0.502344 + 1.87477i 1.19122 + 2.06326i 2.30381 + 1.33010i 2.87308 2.87308i −1.22422 + 2.34548i 0.800616i −0.664344 0.383559i −2.78448 4.82286i
9.6 −1.53151 0.884220i 0.753885 2.81354i 0.563689 + 0.976338i −1.29997 0.750535i −3.64237 + 3.64237i −1.57140 + 2.12855i 1.54318i −4.74957 2.74217i 1.32728 + 2.29891i
9.7 −1.21275 0.700184i 0.113489 0.423548i −0.0194839 0.0337470i 3.39793 + 1.96180i −0.434197 + 0.434197i −0.0318551 2.64556i 2.85531i 2.43156 + 1.40386i −2.74724 4.75836i
9.8 −1.07981 0.623428i −0.766789 + 2.86170i −0.222676 0.385686i −0.266103 0.153635i 2.61205 2.61205i −2.02642 1.70106i 3.04900i −5.00326 2.88863i 0.191560 + 0.331792i
9.9 −0.620778 0.358407i 0.524149 1.95615i −0.743090 1.28707i −3.52688 2.03625i −1.02648 + 1.02648i 2.16565 1.51985i 2.49894i −0.953719 0.550630i 1.45961 + 2.52811i
9.10 −0.391313 0.225925i 0.417030 1.55638i −0.897916 1.55524i −0.146422 0.0845369i −0.514813 + 0.514813i −2.59980 0.490939i 1.71514i 0.349682 + 0.201889i 0.0381979 + 0.0661608i
9.11 −0.194567 0.112333i −0.270068 + 1.00791i −0.974763 1.68834i −0.875130 0.505256i 0.165768 0.165768i −2.25178 + 1.38907i 0.887324i 1.65513 + 0.955593i 0.113514 + 0.196612i
9.12 0.160177 + 0.0924783i −0.754663 + 2.81644i −0.982896 1.70242i 3.38010 + 1.95150i −0.381339 + 0.381339i 2.61706 + 0.388572i 0.733499i −4.76474 2.75093i 0.360943 + 0.625172i
9.13 0.366500 + 0.211599i −0.581121 + 2.16877i −0.910452 1.57695i −3.23301 1.86658i −0.671891 + 0.671891i 0.706651 + 2.54964i 1.61700i −1.76780 1.02064i −0.789933 1.36820i
9.14 0.395417 + 0.228294i 0.613202 2.28850i −0.895764 1.55151i 1.59768 + 0.922421i 0.764922 0.764922i 2.63921 + 0.185981i 1.73117i −2.26315 1.30663i 0.421167 + 0.729482i
9.15 0.628687 + 0.362972i −0.172314 + 0.643084i −0.736502 1.27566i 0.0766403 + 0.0442483i −0.341753 + 0.341753i 0.877316 2.49606i 2.52121i 2.21421 + 1.27838i 0.0321218 + 0.0556367i
9.16 1.07606 + 0.621266i 0.164202 0.612812i −0.228056 0.395005i −3.15567 1.82193i 0.557412 0.557412i −1.89756 1.84371i 3.05180i 2.24950 + 1.29875i −2.26381 3.92103i
9.17 1.12292 + 0.648320i 0.752557 2.80858i −0.159362 0.276022i −1.50273 0.867599i 2.66592 2.66592i −0.701479 + 2.55106i 3.00655i −4.72371 2.72723i −1.12496 1.94850i
9.18 1.23416 + 0.712544i 0.388603 1.45029i 0.0154369 + 0.0267375i 3.19102 + 1.84234i 1.51299 1.51299i −2.63716 0.213102i 2.80618i 0.645761 + 0.372830i 2.62549 + 4.54748i
9.19 1.58851 + 0.917128i −0.0945920 + 0.353022i 0.682248 + 1.18169i −0.406421 0.234647i −0.474027 + 0.474027i 2.16453 + 1.52145i 1.16568i 2.48240 + 1.43321i −0.430403 0.745481i
9.20 1.59242 + 0.919385i −0.699204 + 2.60946i 0.690539 + 1.19605i 0.418240 + 0.241471i −3.51253 + 3.51253i −1.85609 + 1.88546i 1.13806i −3.72234 2.14909i 0.444010 + 0.769047i
See all 96 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 214.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not have CM; other inner twists have not been computed.

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(287, [\chi])$$:

 $$T_{2}^{96} - \cdots$$ $$T_{3}^{96} + \cdots$$