Properties

 Label 287.2.e.d Level $287$ Weight $2$ Character orbit 287.e Analytic conductor $2.292$ Analytic rank $0$ Dimension $34$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$287 = 7 \cdot 41$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 287.e (of order $$3$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$2.29170653801$$ Analytic rank: $$0$$ Dimension: $$34$$ Relative dimension: $$17$$ over $$\Q(\zeta_{3})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$34q - 3q^{2} - q^{3} - 25q^{4} + q^{5} + 4q^{6} - 2q^{7} + 18q^{8} - 26q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$34q - 3q^{2} - q^{3} - 25q^{4} + q^{5} + 4q^{6} - 2q^{7} + 18q^{8} - 26q^{9} + 2q^{10} - 15q^{11} - 4q^{12} - 10q^{13} + 21q^{14} + 48q^{15} - 33q^{16} - 4q^{17} - 10q^{18} - 5q^{19} - 52q^{20} + 12q^{21} + 32q^{22} - 12q^{23} - 16q^{24} - 24q^{25} - 31q^{26} - 22q^{27} + 60q^{28} + 28q^{29} + 33q^{30} + 3q^{31} - 16q^{32} - 4q^{33} - 48q^{34} + 45q^{35} + 114q^{36} - 24q^{37} - 45q^{39} - 36q^{40} + 34q^{41} + 65q^{42} + 28q^{43} + 9q^{44} + 21q^{45} - 44q^{46} - 19q^{47} - 120q^{48} - 10q^{49} - 8q^{50} - 2q^{51} + 25q^{52} - 4q^{53} - 68q^{54} + 18q^{55} + 25q^{56} - 24q^{57} + q^{58} + 27q^{59} - 66q^{60} + q^{61} - 46q^{62} + 37q^{63} + 150q^{64} - 22q^{65} + 16q^{66} - 49q^{67} - 45q^{68} + 24q^{69} + 73q^{70} + 80q^{71} + 23q^{72} + 14q^{73} - 33q^{74} - 27q^{75} - 18q^{76} - 20q^{77} - 24q^{78} - 61q^{79} + 82q^{80} - 53q^{81} - 3q^{82} - 36q^{83} + 188q^{84} - 26q^{85} + 4q^{86} + 17q^{87} - 74q^{88} - 18q^{89} - 40q^{90} + 7q^{91} + 56q^{92} + 36q^{93} + 5q^{94} - 20q^{95} - 148q^{96} + 52q^{97} + 142q^{98} + 76q^{99} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
165.1 −1.36755 2.36867i −1.26123 + 2.18452i −2.74039 + 4.74650i 0.813928 + 1.40976i 6.89919 −1.98316 1.75131i 9.52032 −1.68141 2.91228i 2.22618 3.85585i
165.2 −1.28712 2.22936i −0.234126 + 0.405518i −2.31337 + 4.00687i −1.12112 1.94184i 1.20539 1.72083 + 2.00966i 6.76185 1.39037 + 2.40819i −2.88604 + 4.99876i
165.3 −1.26475 2.19061i 1.34808 2.33495i −2.19919 + 3.80911i −0.322251 0.558155i −6.81996 −0.460755 2.60532i 6.06670 −2.13465 3.69733i −0.815135 + 1.41186i
165.4 −1.18475 2.05204i 0.823159 1.42575i −1.80725 + 3.13025i 2.01488 + 3.48988i −3.90094 0.327045 + 2.62546i 3.82554 0.144819 + 0.250833i 4.77425 8.26924i
165.5 −0.878715 1.52198i −1.61064 + 2.78971i −0.544281 + 0.942722i −0.960818 1.66419i 5.66118 −0.778115 + 2.52874i −1.60179 −3.68832 6.38836i −1.68857 + 2.92469i
165.6 −0.604772 1.04749i −0.0752022 + 0.130254i 0.268503 0.465061i −1.66371 2.88164i 0.181921 −1.99521 1.73756i −3.06862 1.48869 + 2.57849i −2.01233 + 3.48546i
165.7 −0.587784 1.01807i −1.36430 + 2.36304i 0.309020 0.535238i 1.41426 + 2.44958i 3.20766 2.04575 1.67777i −3.07768 −2.22264 3.84973i 1.66256 2.87965i
165.8 −0.557713 0.965987i 1.37705 2.38511i 0.377913 0.654564i 0.672309 + 1.16447i −3.07199 −2.64555 + 0.0328867i −3.07392 −2.29251 3.97075i 0.749910 1.29888i
165.9 −0.190962 0.330755i 0.175243 0.303530i 0.927067 1.60573i 1.39512 + 2.41642i −0.133859 2.22712 + 1.42827i −1.47198 1.43858 + 2.49169i 0.532830 0.922889i
165.10 0.145387 + 0.251818i 0.633158 1.09666i 0.957725 1.65883i −1.61895 2.80410i 0.368212 −1.14416 + 2.38556i 1.13851 0.698223 + 1.20936i 0.470748 0.815360i
165.11 0.451240 + 0.781570i 1.66259 2.87969i 0.592766 1.02670i 0.469181 + 0.812645i 3.00091 1.14720 + 2.38410i 2.87488 −4.02843 6.97744i −0.423426 + 0.733395i
165.12 0.486022 + 0.841815i −0.753325 + 1.30480i 0.527565 0.913769i −1.41815 2.45631i −1.46453 2.62766 0.308901i 2.96972 0.365002 + 0.632201i 1.37851 2.38765i
165.13 0.834451 + 1.44531i 0.106774 0.184938i −0.392616 + 0.680030i −0.501668 0.868914i 0.356391 2.44503 1.01086i 2.02733 1.47720 + 2.55858i 0.837234 1.45013i
165.14 0.889478 + 1.54062i −0.603897 + 1.04598i −0.582342 + 1.00865i 1.26642 + 2.19350i −2.14861 −0.862133 2.50135i 1.48599 0.770616 + 1.33475i −2.25290 + 3.90213i
165.15 0.992766 + 1.71952i −1.44392 + 2.50094i −0.971167 + 1.68211i −1.95720 3.38997i −5.73388 −2.26620 + 1.36540i 0.114499 −2.66979 4.62421i 3.88608 6.73089i
165.16 1.24077 + 2.14907i −0.704660 + 1.22051i −2.07901 + 3.60095i 0.507237 + 0.878561i −3.49728 −2.53048 0.772431i −5.35521 0.506908 + 0.877991i −1.25873 + 2.18018i
165.17 1.38401 + 2.39717i 1.42525 2.46860i −2.83094 + 4.90334i 1.51053 + 2.61632i 7.89019 1.12514 2.39459i −10.1361 −2.56265 4.43864i −4.18117 + 7.24200i
247.1 −1.36755 + 2.36867i −1.26123 2.18452i −2.74039 4.74650i 0.813928 1.40976i 6.89919 −1.98316 + 1.75131i 9.52032 −1.68141 + 2.91228i 2.22618 + 3.85585i
247.2 −1.28712 + 2.22936i −0.234126 0.405518i −2.31337 4.00687i −1.12112 + 1.94184i 1.20539 1.72083 2.00966i 6.76185 1.39037 2.40819i −2.88604 4.99876i
247.3 −1.26475 + 2.19061i 1.34808 + 2.33495i −2.19919 3.80911i −0.322251 + 0.558155i −6.81996 −0.460755 + 2.60532i 6.06670 −2.13465 + 3.69733i −0.815135 1.41186i
See all 34 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 247.17 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.c even 3 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 287.2.e.d 34
7.c even 3 1 inner 287.2.e.d 34
7.c even 3 1 2009.2.a.s 17
7.d odd 6 1 2009.2.a.r 17

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.2.e.d 34 1.a even 1 1 trivial
287.2.e.d 34 7.c even 3 1 inner
2009.2.a.r 17 7.d odd 6 1
2009.2.a.s 17 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{34} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(287, [\chi])$$.