# Properties

 Label 287.3.y.a Level 287 Weight 3 Character orbit 287.y Analytic conductor 7.820 Analytic rank 0 Dimension 432 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$287 = 7 \cdot 41$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 287.y (of order $$30$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.82018358714$$ Analytic rank: $$0$$ Dimension: $$432$$ Relative dimension: $$54$$ over $$\Q(\zeta_{30})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$432q - 3q^{2} - 24q^{3} + 101q^{4} - 9q^{5} - 6q^{7} - 16q^{8} + 576q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$432q - 3q^{2} - 24q^{3} + 101q^{4} - 9q^{5} - 6q^{7} - 16q^{8} + 576q^{9} + 72q^{10} - 11q^{11} - 33q^{12} + 182q^{14} - 54q^{15} + 197q^{16} - 63q^{17} + 48q^{18} + 63q^{19} - 26q^{21} - 52q^{22} - 24q^{23} - 510q^{24} - 253q^{25} - 159q^{26} - 65q^{28} + 152q^{29} - 131q^{30} - 45q^{31} + 94q^{32} + 36q^{33} + 84q^{35} + 474q^{36} - 46q^{37} - 6q^{38} + 74q^{39} + 258q^{40} - 220q^{42} - 88q^{43} + 128q^{44} - 156q^{45} - 82q^{46} - 309q^{47} - 338q^{49} + 704q^{50} + 66q^{51} + 291q^{52} + 68q^{53} + 483q^{54} - 182q^{56} + 114q^{57} + 159q^{58} - 207q^{59} + 430q^{60} + 423q^{61} - 172q^{63} - 896q^{64} + 204q^{65} - 1560q^{66} + 33q^{67} - 1242q^{68} + 707q^{70} - 162q^{71} - 41q^{72} - 78q^{73} - 439q^{74} - 1452q^{75} + 164q^{77} - 222q^{78} - 138q^{79} - 27q^{80} - 928q^{81} + 165q^{82} - 543q^{84} + 156q^{85} + 609q^{86} - 588q^{87} + 394q^{88} - 1161q^{89} - 950q^{91} + 482q^{92} - 45q^{93} + 1779q^{94} - 475q^{95} + 2412q^{96} - 1100q^{98} + 932q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1 −0.407277 + 3.87498i −4.43919 2.56297i −10.9370 2.32473i −1.15474 1.03974i 11.7394 16.1579i −5.75629 3.98310i 8.64655 26.6114i 8.63759 + 14.9607i 4.49926 4.05115i
10.2 −0.395467 + 3.76261i 1.42023 + 0.819968i −10.0883 2.14433i −2.55057 2.29654i −3.64688 + 5.01950i −2.63679 6.48439i 7.38140 22.7176i −3.15530 5.46515i 9.64968 8.68861i
10.3 −0.377653 + 3.59313i 1.08377 + 0.625718i −8.85534 1.88226i −2.01323 1.81272i −2.65757 + 3.65783i −0.128124 + 6.99883i 5.64163 17.3631i −3.71695 6.43795i 7.27365 6.54922i
10.4 −0.372260 + 3.54182i 2.27091 + 1.31111i −8.49333 1.80531i 5.49683 + 4.94937i −5.48909 + 7.55509i 6.95621 0.781768i 5.15378 15.8617i −1.06198 1.83940i −19.5760 + 17.6264i
10.5 −0.352334 + 3.35223i 4.51935 + 2.60925i −7.20072 1.53056i 2.89418 + 2.60593i −10.3391 + 14.2306i −6.73483 + 1.90841i 3.50144 10.7763i 9.11637 + 15.7900i −9.75540 + 8.78380i
10.6 −0.350757 + 3.33723i −2.35294 1.35847i −7.10146 1.50946i −4.71627 4.24655i 5.35883 7.37579i 6.89642 + 1.19976i 3.38054 10.4042i −0.809124 1.40144i 15.8260 14.2498i
10.7 −0.342604 + 3.25966i −2.07689 1.19909i −6.59542 1.40190i 4.83955 + 4.35755i 4.62020 6.35915i 3.55259 6.03151i 2.77798 8.54975i −1.62434 2.81344i −15.8622 + 14.2824i
10.8 −0.340799 + 3.24249i −3.64025 2.10170i −6.48498 1.37843i 4.03434 + 3.63253i 8.05533 11.0872i 2.25115 + 6.62815i 2.64959 8.15460i 4.33429 + 7.50720i −13.1533 + 11.8433i
10.9 −0.288646 + 2.74628i −0.0116820 0.00674463i −3.54617 0.753762i 4.04208 + 3.63950i 0.0218947 0.0301354i −6.18024 3.28704i −0.319659 + 0.983810i −4.49991 7.79407i −11.1618 + 10.0502i
10.10 −0.282410 + 2.68695i −1.99951 1.15441i −3.22738 0.686000i 0.0649077 + 0.0584432i 3.66654 5.04656i −6.98405 + 0.472222i −0.584859 + 1.80001i −1.83465 3.17771i −0.175365 + 0.157899i
10.11 −0.276618 + 2.63184i 1.67521 + 0.967181i −2.93749 0.624382i −5.03132 4.53022i −3.00886 + 4.14134i 0.936757 6.93704i −0.815219 + 2.50899i −2.62912 4.55377i 13.3146 11.9885i
10.12 −0.262361 + 2.49620i 3.99634 + 2.30729i −2.24959 0.478164i −1.09445 0.985445i −6.80793 + 9.37031i 6.92637 + 1.01262i −1.31867 + 4.05845i 6.14714 + 10.6472i 2.74701 2.47342i
10.13 −0.256802 + 2.44330i 3.72271 + 2.14931i −1.99119 0.423241i −6.18690 5.57071i −6.20740 + 8.54375i −6.03910 + 3.53967i −1.49128 + 4.58968i 4.73903 + 8.20823i 15.1997 13.6859i
10.14 −0.223775 + 2.12907i −4.20842 2.42973i −0.570287 0.121218i −4.07393 3.66818i 6.11481 8.41631i 0.752068 6.95948i −2.26048 + 6.95703i 7.30717 + 12.6564i 8.72148 7.85285i
10.15 −0.197672 + 1.88072i −2.25015 1.29912i 0.414549 + 0.0881151i −2.05729 1.85239i 2.88808 3.97510i 6.85731 1.40618i −2.58517 + 7.95634i −1.12456 1.94779i 3.89051 3.50303i
10.16 −0.193036 + 1.83662i 1.95836 + 1.13066i 0.576688 + 0.122579i 6.02254 + 5.42272i −2.45462 + 3.37850i −0.339639 + 6.99176i −2.61914 + 8.06089i −1.94322 3.36575i −11.1220 + 10.0143i
10.17 −0.182448 + 1.73588i −4.79904 2.77073i 0.932598 + 0.198230i −1.81832 1.63723i 5.68522 7.82504i −1.44156 + 6.84996i −2.67174 + 8.22277i 10.8538 + 18.7994i 3.17378 2.85768i
10.18 −0.181634 + 1.72813i −0.294892 0.170256i 0.959146 + 0.203873i −1.02853 0.926089i 0.347787 0.478687i −1.59260 + 6.81642i −2.67439 + 8.23091i −4.44203 7.69381i 1.78722 1.60922i
10.19 −0.153412 + 1.45962i 0.632396 + 0.365114i 1.80565 + 0.383802i 2.72179 + 2.45071i −0.629943 + 0.867043i 4.21354 5.58982i −2.65133 + 8.15997i −4.23338 7.33244i −3.99465 + 3.59680i
10.20 −0.130578 + 1.24237i 4.05852 + 2.34319i 2.38617 + 0.507196i 3.04278 + 2.73974i −3.44105 + 4.73619i −2.06147 6.68957i −2.48581 + 7.65054i 6.48104 + 11.2255i −3.80107 + 3.42250i
See next 80 embeddings (of 432 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 283.54 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.d even 5 1 inner
287.y odd 30 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 287.3.y.a 432
7.d odd 6 1 inner 287.3.y.a 432
41.d even 5 1 inner 287.3.y.a 432
287.y odd 30 1 inner 287.3.y.a 432

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.3.y.a 432 1.a even 1 1 trivial
287.3.y.a 432 7.d odd 6 1 inner
287.3.y.a 432 41.d even 5 1 inner
287.3.y.a 432 287.y odd 30 1 inner

## Hecke kernels

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