Properties

Label 354.2.e.a
Level 354
Weight 2
Character orbit 354.e
Analytic conductor 2.827
Analytic rank 0
Dimension 56
CM No

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Newspace parameters

Level: \( N \) = \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 354.e (of order \(29\) and degree \(28\))

Newform invariants

Self dual: No
Analytic conductor: \(2.82670423155\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(2\) over \(\Q(\zeta_{29})\)
Sato-Tate group: $\mathrm{SU}(2)[C_{29}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56q - 2q^{2} - 2q^{3} - 2q^{4} + 4q^{5} - 2q^{6} + 9q^{7} - 2q^{8} - 2q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 56q - 2q^{2} - 2q^{3} - 2q^{4} + 4q^{5} - 2q^{6} + 9q^{7} - 2q^{8} - 2q^{9} + 4q^{10} - 18q^{11} - 2q^{12} + 13q^{13} - 20q^{14} + 4q^{15} - 2q^{16} + 21q^{17} - 2q^{18} + 20q^{19} + 4q^{20} - 20q^{21} + 11q^{22} + 34q^{23} - 2q^{24} - 28q^{25} - 16q^{26} - 2q^{27} + 9q^{28} + 38q^{29} + 4q^{30} + 34q^{31} - 2q^{32} + 11q^{33} - 8q^{34} - 19q^{35} - 2q^{36} - 29q^{37} + 20q^{38} + 13q^{39} + 4q^{40} + 43q^{41} + 9q^{42} - 65q^{43} + 11q^{44} + 4q^{45} + 5q^{46} - 60q^{47} - 2q^{48} - 13q^{49} - 86q^{50} + 21q^{51} - 16q^{52} - 15q^{53} - 2q^{54} - 15q^{55} + 9q^{56} + 20q^{57} - 78q^{58} - 2q^{59} - 54q^{60} + 4q^{61} + 5q^{62} + 9q^{63} - 2q^{64} - 57q^{65} + 11q^{66} - 56q^{67} - 8q^{68} - 53q^{69} + 10q^{70} + 23q^{71} - 2q^{72} - 21q^{73} + 30q^{75} - 9q^{76} - 57q^{77} + 13q^{78} + 99q^{79} + 4q^{80} - 2q^{81} + 43q^{82} + 33q^{83} + 9q^{84} - 45q^{85} - 7q^{86} + 9q^{87} - 18q^{88} + 32q^{89} + 4q^{90} - 75q^{91} + 34q^{92} + 5q^{93} + 27q^{94} + 82q^{95} - 2q^{96} + 70q^{97} + 45q^{98} + 11q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −0.561187 0.827689i 0.0541389 0.998533i −0.370138 + 0.928977i −3.00541 + 1.39045i −0.856857 + 0.515554i 0.371841 1.33925i 0.976621 0.214970i −0.994138 0.108119i 2.83746 + 1.70724i
7.2 −0.561187 0.827689i 0.0541389 0.998533i −0.370138 + 0.928977i 0.704657 0.326009i −0.856857 + 0.515554i 0.375788 1.35347i 0.976621 0.214970i −0.994138 0.108119i −0.665279 0.400285i
19.1 0.468408 0.883512i −0.725995 + 0.687699i −0.561187 0.827689i −1.90859 + 0.420113i 0.267528 + 0.963550i 1.36060 1.03430i −0.994138 + 0.108119i 0.0541389 0.998533i −0.522826 + 1.88305i
19.2 0.468408 0.883512i −0.725995 + 0.687699i −0.561187 0.827689i −1.59961 + 0.352101i 0.267528 + 0.963550i −1.70658 + 1.29731i −0.994138 + 0.108119i 0.0541389 0.998533i −0.438186 + 1.57820i
25.1 0.796093 + 0.605174i 0.468408 + 0.883512i 0.267528 + 0.963550i −0.300267 0.284428i −0.161782 + 0.986827i −2.60590 + 3.06790i −0.370138 + 0.928977i −0.561187 + 0.827689i −0.0669119 0.408144i
25.2 0.796093 + 0.605174i 0.468408 + 0.883512i 0.267528 + 0.963550i 2.69226 + 2.55024i −0.161782 + 0.986827i 1.09972 1.29469i −0.370138 + 0.928977i −0.561187 + 0.827689i 0.599947 + 3.65951i
49.1 −0.370138 + 0.928977i −0.994138 0.108119i −0.725995 0.687699i −0.571390 + 0.672692i 0.468408 0.883512i 2.39323 + 1.43996i 0.907575 0.419889i 0.976621 + 0.214970i −0.413422 0.779797i
49.2 −0.370138 + 0.928977i −0.994138 0.108119i −0.725995 0.687699i 0.386052 0.454496i 0.468408 0.883512i −0.965403 0.580863i 0.907575 0.419889i 0.976621 + 0.214970i 0.279324 + 0.526860i
79.1 0.907575 + 0.419889i −0.947653 + 0.319302i 0.647386 + 0.762162i −0.314743 + 0.189375i −0.994138 0.108119i 0.132599 + 2.44564i 0.267528 + 0.963550i 0.796093 0.605174i −0.365169 + 0.0397146i
79.2 0.907575 + 0.419889i −0.947653 + 0.319302i 0.647386 + 0.762162i 2.12124 1.27631i −0.994138 0.108119i −0.243034 4.48249i 0.267528 + 0.963550i 0.796093 0.605174i 2.46109 0.267659i
85.1 0.796093 0.605174i 0.468408 0.883512i 0.267528 0.963550i −0.300267 + 0.284428i −0.161782 0.986827i −2.60590 3.06790i −0.370138 0.928977i −0.561187 0.827689i −0.0669119 + 0.408144i
85.2 0.796093 0.605174i 0.468408 0.883512i 0.267528 0.963550i 2.69226 2.55024i −0.161782 0.986827i 1.09972 + 1.29469i −0.370138 0.928977i −0.561187 0.827689i 0.599947 3.65951i
121.1 0.907575 0.419889i −0.947653 0.319302i 0.647386 0.762162i −0.314743 0.189375i −0.994138 + 0.108119i 0.132599 2.44564i 0.267528 0.963550i 0.796093 + 0.605174i −0.365169 0.0397146i
121.2 0.907575 0.419889i −0.947653 0.319302i 0.647386 0.762162i 2.12124 + 1.27631i −0.994138 + 0.108119i −0.243034 + 4.48249i 0.267528 0.963550i 0.796093 + 0.605174i 2.46109 + 0.267659i
127.1 0.647386 0.762162i 0.796093 + 0.605174i −0.161782 0.986827i −0.659445 1.24384i 0.976621 0.214970i 1.04625 + 0.113786i −0.856857 0.515554i 0.267528 + 0.963550i −1.37493 0.302644i
127.2 0.647386 0.762162i 0.796093 + 0.605174i −0.161782 0.986827i 0.653953 + 1.23349i 0.976621 0.214970i 3.53789 + 0.384769i −0.856857 0.515554i 0.267528 + 0.963550i 1.36348 + 0.300124i
133.1 −0.994138 + 0.108119i 0.647386 + 0.762162i 0.976621 0.214970i −2.52137 + 1.91669i −0.725995 0.687699i −1.41566 3.55305i −0.947653 + 0.319302i −0.161782 + 0.986827i 2.29936 2.17807i
133.2 −0.994138 + 0.108119i 0.647386 + 0.762162i 0.976621 0.214970i 1.51851 1.15434i −0.725995 0.687699i 0.899258 + 2.25697i −0.947653 + 0.319302i −0.161782 + 0.986827i −1.38480 + 1.31175i
139.1 −0.856857 0.515554i −0.370138 + 0.928977i 0.468408 + 0.883512i −3.00982 0.327338i 0.796093 0.605174i 0.507395 + 0.170961i 0.0541389 0.998533i −0.725995 0.687699i 2.41023 + 1.83221i
139.2 −0.856857 0.515554i −0.370138 + 0.928977i 0.468408 + 0.883512i 3.11390 + 0.338657i 0.796093 0.605174i 2.05914 + 0.693803i 0.0541389 0.998533i −0.725995 0.687699i −2.49357 1.89557i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 343.2
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{5}^{56} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(354, [\chi])\).