Properties

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

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Newspace parameters

Level: \( N \) = \( 287 = 7 \cdot 41 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 287.x (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.

\(\operatorname{Tr}(f)(q) = \) \( 432q - 3q^{2} + 101q^{4} - 9q^{5} - 10q^{7} - 40q^{8} - 624q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 432q - 3q^{2} + 101q^{4} - 9q^{5} - 10q^{7} - 40q^{8} - 624q^{9} - 90q^{10} - 5q^{11} - 15q^{12} + 70q^{15} + 197q^{16} - 15q^{17} - 6q^{18} - 15q^{19} + 166q^{21} + 60q^{22} + 18q^{23} + 480q^{24} - 213q^{25} - 15q^{26} - 105q^{28} + 360q^{29} - 15q^{30} - 45q^{31} + 142q^{32} + 36q^{33} - 150q^{35} + 46q^{36} + 82q^{37} - 80q^{39} - 54q^{40} + 228q^{42} - 88q^{43} + 330q^{45} - 96q^{46} - 15q^{47} + 50q^{49} - 472q^{50} + 150q^{51} - 15q^{52} - 230q^{53} + 465q^{54} + 180q^{56} + 382q^{57} - 5q^{58} - 207q^{59} - 480q^{60} - 441q^{61} + 200q^{63} - 128q^{64} - 290q^{65} - 918q^{66} + 115q^{67} + 1175q^{70} - 730q^{71} - 309q^{72} - 78q^{73} + 589q^{74} + 240q^{75} + 684q^{77} - 434q^{78} - 27q^{80} - 1936q^{81} - 309q^{82} - 173q^{84} - 439q^{86} - 1002q^{87} + 1335q^{89} - 274q^{91} - 270q^{92} + 765q^{93} + 1515q^{94} + 715q^{95} - 454q^{98} + 480q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −0.404206 + 3.84576i −2.18229 + 3.77983i −10.7139 2.27731i 6.18506 + 5.56906i −13.6543 9.92039i −5.32339 4.54549i 8.30884 25.5720i −5.02476 8.70314i −23.9173 + 21.5352i
31.2 −0.382108 + 3.63551i −0.296842 + 0.514145i −9.15836 1.94667i −3.78875 3.41141i −1.75576 1.27563i 4.94220 4.95728i 6.05813 18.6450i 4.32377 + 7.48899i 13.8499 12.4705i
31.3 −0.377598 + 3.59260i 2.17214 3.76225i −8.85161 1.88147i −0.351696 0.316669i 12.6961 + 9.22425i −6.40224 + 2.83043i 5.63655 17.3475i −4.93637 8.55005i 1.27046 1.14393i
31.4 −0.374024 + 3.55860i 0.0587614 0.101778i −8.61118 1.83036i 2.71663 + 2.44606i 0.340208 + 0.247176i 2.46506 + 6.55160i 5.31143 16.3469i 4.49309 + 7.78227i −9.72066 + 8.75252i
31.5 −0.367514 + 3.49667i 1.64260 2.84507i −8.17901 1.73850i 3.05662 + 2.75219i 9.34458 + 6.78924i −1.45189 6.84777i 4.73894 14.5850i −0.896289 1.55242i −10.7469 + 9.67651i
31.6 −0.353899 + 3.36712i −2.83548 + 4.91119i −7.29966 1.55159i −3.86461 3.47971i −15.5331 11.2855i 6.16762 + 3.31066i 3.62282 11.1499i −11.5799 20.0569i 13.0843 11.7811i
31.7 −0.329922 + 3.13900i 2.50545 4.33957i −5.83189 1.23961i −5.08648 4.57989i 12.7953 + 9.29634i 6.86155 1.38531i 1.91380 5.89008i −8.05460 13.9510i 16.0544 14.4555i
31.8 −0.327450 + 3.11548i −1.62561 + 2.81565i −5.68637 1.20868i −3.32056 2.98985i −8.23977 5.98654i −6.84936 1.44437i 1.75545 5.40272i −0.785238 1.36007i 10.4021 9.36610i
31.9 −0.287755 + 2.73781i 1.34904 2.33661i −3.50018 0.743987i −5.88685 5.30054i 6.00898 + 4.36578i −0.00249787 + 7.00000i −0.358667 + 1.10386i 0.860183 + 1.48988i 16.2058 14.5918i
31.10 −0.277470 + 2.63995i −0.490949 + 0.850348i −2.97975 0.633365i 3.54226 + 3.18946i −2.10865 1.53203i 6.74819 1.86061i −0.782288 + 2.40763i 4.01794 + 6.95927i −9.40289 + 8.46640i
31.11 −0.274542 + 2.61210i 2.49203 4.31632i −2.83508 0.602615i 6.74003 + 6.06875i 10.5905 + 7.69444i 6.28097 + 3.09022i −0.894076 + 2.75168i −7.92044 13.7186i −17.7026 + 15.9395i
31.12 −0.262518 + 2.49769i 0.353437 0.612171i −2.25695 0.479730i −1.88040 1.69312i 1.43623 + 1.04348i −5.52339 4.30025i −1.31362 + 4.04289i 4.25016 + 7.36150i 4.72252 4.25217i
31.13 −0.261584 + 2.48881i −1.88397 + 3.26313i −2.21315 0.470420i 3.91575 + 3.52575i −7.62849 5.54242i 5.00297 4.89595i −1.34357 + 4.13508i −2.59868 4.50104i −9.79923 + 8.82326i
31.14 −0.255628 + 2.43214i −1.68440 + 2.91747i −1.93737 0.411800i 2.75087 + 2.47690i −6.66510 4.84248i −3.50712 + 6.05806i −1.52605 + 4.69670i −1.17440 2.03413i −6.72735 + 6.05734i
31.15 −0.216725 + 2.06200i 0.981587 1.70016i −0.292279 0.0621259i 5.16597 + 4.65146i 3.29299 + 2.39250i −6.36962 + 2.90310i −2.37136 + 7.29830i 2.57297 + 4.45652i −10.7109 + 9.64414i
31.16 −0.194152 + 1.84723i −1.06893 + 1.85144i 0.538027 + 0.114361i −5.05032 4.54733i −3.21250 2.33402i 0.430771 + 6.98673i −2.61159 + 8.03764i 2.21478 + 3.83610i 9.38049 8.44623i
31.17 −0.185452 + 1.76446i 2.12795 3.68571i 0.833662 + 0.177200i −0.754233 0.679114i 6.10866 + 4.43820i 1.07386 6.91714i −2.66027 + 8.18748i −4.55631 7.89176i 1.33814 1.20487i
31.18 −0.149486 + 1.42226i −2.44356 + 4.23237i 1.91210 + 0.406429i −3.74322 3.37041i −5.65427 4.10807i 0.725236 6.96233i −2.63158 + 8.09918i −7.44197 12.8899i 5.35317 4.82002i
31.19 −0.139031 + 1.32279i 0.972780 1.68490i 2.18215 + 0.463831i −0.709834 0.639137i 2.09352 + 1.52103i 6.41681 + 2.79723i −2.56100 + 7.88194i 2.60740 + 4.51615i 0.944131 0.850099i
31.20 −0.113807 + 1.08280i −0.534489 + 0.925762i 2.75308 + 0.585184i 5.36495 + 4.83062i −0.941591 0.684106i −3.56891 6.02187i −2.29275 + 7.05637i 3.92864 + 6.80461i −5.84119 + 5.25943i
See next 80 embeddings (of 432 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 271.54
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
41.f even 10 1 inner
287.x 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.x.a 432
7.d odd 6 1 inner 287.3.x.a 432
41.f even 10 1 inner 287.3.x.a 432
287.x odd 30 1 inner 287.3.x.a 432
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.3.x.a 432 1.a even 1 1 trivial
287.3.x.a 432 7.d odd 6 1 inner
287.3.x.a 432 41.f even 10 1 inner
287.3.x.a 432 287.x odd 30 1 inner

Hecke kernels

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