Properties

Label 287.2.z.a
Level 287
Weight 2
Character orbit 287.z
Analytic conductor 2.292
Analytic rank 0
Dimension 208
CM No

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Newspace parameters

Level: \( N \) = \( 287 = 7 \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 287.z (of order \(30\) and degree \(8\))

Newform invariants

Self dual: No
Analytic conductor: \(2.29170653801\)
Analytic rank: \(0\)
Dimension: \(208\)
Relative dimension: \(26\) over \(\Q(\zeta_{30})\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 208q - 3q^{2} + 21q^{4} + q^{5} - 40q^{6} - 10q^{7} - 8q^{8} + 84q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 208q - 3q^{2} + 21q^{4} + q^{5} - 40q^{6} - 10q^{7} - 8q^{8} + 84q^{9} - 6q^{10} - 5q^{11} - 35q^{12} - 20q^{13} - 50q^{15} + 21q^{16} - 5q^{17} + 18q^{18} - 5q^{19} - 96q^{20} + 8q^{21} + 20q^{22} + 40q^{24} + 27q^{25} - 5q^{26} + 5q^{28} + 20q^{29} - 45q^{30} - 11q^{31} - 30q^{32} - 10q^{33} + 100q^{34} - 106q^{36} - 16q^{37} - 4q^{39} + 6q^{40} - 14q^{41} - 8q^{42} - 8q^{43} + 34q^{45} - 32q^{46} + 25q^{47} - 50q^{48} + 14q^{49} - 120q^{50} + 2q^{51} - 105q^{52} + 20q^{53} - 35q^{54} + 100q^{56} - 98q^{57} - 5q^{58} - 37q^{59} - 100q^{60} + 51q^{61} - 70q^{62} - 30q^{63} - 100q^{64} + 40q^{65} - 176q^{66} + 15q^{67} - 30q^{69} + 105q^{70} - 10q^{71} - 33q^{72} - 34q^{73} - 23q^{74} - 120q^{75} + 110q^{76} + 72q^{77} - 18q^{78} + 127q^{80} - 24q^{81} - 63q^{82} - 128q^{83} + 33q^{84} - 61q^{86} + 28q^{87} + 150q^{88} + 35q^{89} - 34q^{90} + 14q^{91} + 102q^{92} - 55q^{93} - 155q^{94} + 55q^{95} + 20q^{97} + 88q^{98} + 120q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.53193 1.12729i −0.633945 0.366008i 3.80165 + 4.22216i −2.99462 + 0.636527i 1.19251 + 1.64135i −1.33751 2.28278i −3.15302 9.70400i −1.23208 2.13402i 8.29974 + 1.76416i
4.2 −2.33544 1.03981i 2.55694 + 1.47625i 3.03484 + 3.37053i −1.62101 + 0.344555i −4.43657 6.10641i 0.110850 + 2.64343i −2.00302 6.16466i 2.85861 + 4.95126i 4.14404 + 0.880843i
4.3 −2.12964 0.948175i 1.26625 + 0.731069i 2.29805 + 2.55224i 3.03336 0.644761i −2.00347 2.75753i 2.17812 1.50193i −1.03329 3.18014i −0.431077 0.746647i −7.07130 1.50305i
4.4 −1.89191 0.842332i 0.0223335 + 0.0128943i 1.53153 + 1.70094i 0.705370 0.149931i −0.0313917 0.0432070i −2.31716 + 1.27702i −0.184847 0.568899i −1.49967 2.59750i −1.46079 0.310500i
4.5 −1.85683 0.826713i −2.83125 1.63462i 1.42609 + 1.58384i 2.48571 0.528355i 3.90578 + 5.37584i −2.56260 0.658072i −0.0824439 0.253736i 3.84399 + 6.65798i −5.05234 1.07391i
4.6 −1.78806 0.796097i −2.45203 1.41568i 1.22514 + 1.36066i −3.36959 + 0.716229i 3.25737 + 4.48338i 2.05104 + 1.67130i 0.102251 + 0.314698i 2.50830 + 4.34450i 6.59524 + 1.40186i
4.7 −1.44506 0.643384i 1.73343 + 1.00080i 0.336004 + 0.373171i −3.23302 + 0.687199i −1.86102 2.56148i 0.537448 2.59059i 0.732162 + 2.25336i 0.503196 + 0.871562i 5.11405 + 1.08702i
4.8 −1.41117 0.628293i 0.158984 + 0.0917894i 0.258386 + 0.286967i −1.25090 + 0.265887i −0.166683 0.229419i 2.06082 + 1.65923i 0.770360 + 2.37093i −1.48315 2.56889i 1.93229 + 0.410720i
4.9 −0.904302 0.402621i 2.57326 + 1.48567i −0.682602 0.758107i 2.08760 0.443733i −1.72884 2.37954i −0.221798 + 2.63644i 0.923830 + 2.84326i 2.91443 + 5.04794i −2.06648 0.439243i
4.10 −0.747873 0.332974i 1.29885 + 0.749893i −0.889820 0.988245i 3.52515 0.749293i −0.721681 0.993309i −2.07667 1.63935i 0.842364 + 2.59253i −0.375322 0.650076i −2.88586 0.613408i
4.11 −0.681771 0.303544i −0.925344 0.534248i −0.965589 1.07239i −0.547576 + 0.116391i 0.468705 + 0.645117i 0.200982 2.63811i 0.794024 + 2.44376i −0.929159 1.60935i 0.408651 + 0.0868614i
4.12 −0.598911 0.266652i −1.68485 0.972747i −1.05067 1.16689i 3.52149 0.748516i 0.749688 + 1.03186i 2.63226 + 0.266882i 0.723281 + 2.22603i 0.392475 + 0.679786i −2.30865 0.490719i
4.13 −0.482956 0.215026i −1.64448 0.949441i −1.15125 1.27859i −0.483613 + 0.102795i 0.590058 + 0.812145i −1.39423 + 2.24858i 0.607804 + 1.87063i 0.302877 + 0.524598i 0.255668 + 0.0543438i
4.14 0.109542 + 0.0487713i 1.26282 + 0.729092i −1.32864 1.47560i −4.28969 + 0.911801i 0.102774 + 0.141456i −0.860452 + 2.50192i −0.147683 0.454521i −0.436849 0.756645i −0.514371 0.109333i
4.15 0.202575 + 0.0901924i 2.46443 + 1.42284i −1.30536 1.44975i −0.436054 + 0.0926862i 0.370904 + 0.510506i 2.35455 1.20668i −0.270724 0.833203i 2.54896 + 4.41492i −0.0966935 0.0205528i
4.16 0.348205 + 0.155031i 0.286740 + 0.165549i −1.24105 1.37832i −0.838738 + 0.178279i 0.0741790 + 0.102099i −2.26785 1.36266i −0.454025 1.39735i −1.44519 2.50314i −0.319692 0.0679526i
4.17 0.909063 + 0.404741i 0.864285 + 0.498995i −0.675680 0.750419i 2.62197 0.557318i 0.583726 + 0.803430i 1.31896 + 2.29354i −0.925513 2.84843i −1.00201 1.73553i 2.60911 + 0.554583i
4.18 0.964922 + 0.429611i −0.754585 0.435660i −0.591752 0.657207i −2.31912 + 0.492944i −0.540952 0.744556i 2.56690 0.641092i −0.941443 2.89746i −1.12040 1.94059i −2.44955 0.520667i
4.19 1.05975 + 0.471833i −2.06446 1.19192i −0.437810 0.486237i 2.17853 0.463060i −1.62543 2.23722i −2.30952 + 1.29079i −0.951495 2.92840i 1.34133 + 2.32325i 2.52719 + 0.537170i
4.20 1.47672 + 0.657478i 2.34477 + 1.35375i 0.410161 + 0.455530i 0.683534 0.145290i 2.57250 + 3.54075i −1.72845 2.00311i −0.692842 2.13235i 2.16529 + 3.75040i 1.10491 + 0.234856i
See next 80 embeddings (of 208 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 277.26
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(287, [\chi])\).