Properties

Label 354.2.g.a
Level 354
Weight 2
Character orbit 354.g
Analytic conductor 2.827
Analytic rank 0
Dimension 280
CM No

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Newspace parameters

Level: \( N \) = \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 354.g (of order \(58\) and degree \(28\))

Newform invariants

Self dual: No
Analytic conductor: \(2.82670423155\)
Analytic rank: \(0\)
Dimension: \(280\)
Relative dimension: \(10\) over \(\Q(\zeta_{58})\)
Sato-Tate group: $\mathrm{SU}(2)[C_{58}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 280q - 10q^{2} + q^{3} - 10q^{4} + q^{6} + 2q^{7} - 10q^{8} - 3q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 280q - 10q^{2} + q^{3} - 10q^{4} + q^{6} + 2q^{7} - 10q^{8} - 3q^{9} + 4q^{11} + q^{12} - 27q^{14} + 7q^{15} - 10q^{16} - 3q^{18} + 6q^{19} + 3q^{21} + 4q^{22} - 8q^{23} + q^{24} - 4q^{25} - 77q^{27} + 2q^{28} + 7q^{30} - 10q^{32} - 3q^{36} + 6q^{38} - 4q^{39} + 3q^{42} + 4q^{44} - 18q^{45} - 8q^{46} + 232q^{47} + q^{48} - 8q^{49} - 4q^{50} - 118q^{51} - 58q^{53} - 77q^{54} + 2q^{56} - 316q^{57} + 20q^{59} - 80q^{60} + 77q^{63} - 10q^{64} + 16q^{65} - 116q^{66} - 102q^{69} - 3q^{72} - 25q^{75} + 6q^{76} + 106q^{77} - 4q^{78} - 6q^{79} - 7q^{81} - 104q^{83} + 3q^{84} + 4q^{85} + 15q^{87} + 4q^{88} - 16q^{89} + 11q^{90} - 8q^{92} + 122q^{93} + q^{96} - 8q^{98} + 2q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 0.976621 0.214970i −1.68205 + 0.413161i 0.907575 0.419889i −2.23908 0.621678i −1.55391 + 0.765093i 1.22747 1.16272i 0.796093 0.605174i 2.65860 1.38992i −2.32038 0.125807i
11.2 0.976621 0.214970i −1.66116 0.490442i 0.907575 0.419889i 3.54823 + 0.985161i −1.72776 0.121874i −3.12490 + 2.96006i 0.796093 0.605174i 2.51893 + 1.62941i 3.67705 + 0.199364i
11.3 0.976621 0.214970i −1.36107 + 1.07121i 0.907575 0.419889i 1.88995 + 0.524743i −1.09897 + 1.33876i 0.743378 0.704165i 0.796093 0.605174i 0.705020 2.91598i 1.95857 + 0.106191i
11.4 0.976621 0.214970i −0.326586 + 1.70098i 0.907575 0.419889i −2.85285 0.792090i 0.0467107 + 1.73142i −3.45977 + 3.27726i 0.796093 0.605174i −2.78668 1.11103i −2.95643 0.160293i
11.5 0.976621 0.214970i −0.207186 1.71961i 0.907575 0.419889i 1.05087 + 0.291772i −0.572009 1.63487i 1.62439 1.53870i 0.796093 0.605174i −2.91415 + 0.712562i 1.08902 + 0.0590451i
11.6 0.976621 0.214970i 0.308899 + 1.70428i 0.907575 0.419889i −0.897171 0.249098i 0.668047 + 1.59803i 3.72971 3.53297i 0.796093 0.605174i −2.80916 + 1.05290i −0.929745 0.0504093i
11.7 0.976621 0.214970i 0.566341 + 1.63684i 0.907575 0.419889i 3.87317 + 1.07538i 0.904973 + 1.47683i −0.391370 + 0.370725i 0.796093 0.605174i −2.35852 + 1.85402i 4.01379 + 0.217621i
11.8 0.976621 0.214970i 1.28026 1.16660i 0.907575 0.419889i −4.02621 1.11787i 0.999541 1.41454i 1.69268 1.60339i 0.796093 0.605174i 0.278110 2.98708i −4.17238 0.226220i
11.9 0.976621 0.214970i 1.45268 0.943248i 0.907575 0.419889i 1.11366 + 0.309205i 1.21595 1.23348i −1.48196 + 1.40379i 0.796093 0.605174i 1.22057 2.74048i 1.15409 + 0.0625729i
11.10 0.976621 0.214970i 1.57574 + 0.719053i 0.907575 0.419889i −0.135298 0.0375654i 1.69348 + 0.363504i −1.37447 + 1.30197i 0.796093 0.605174i 1.96593 + 2.26608i −0.140211 0.00760200i
23.1 −0.725995 + 0.687699i −1.73134 0.0495652i 0.0541389 0.998533i 0.271078 + 0.0444409i 1.29103 1.15466i −0.468623 + 0.883917i 0.647386 + 0.762162i 2.99509 + 0.171629i −0.227363 + 0.154156i
23.2 −0.725995 + 0.687699i −1.13722 + 1.30642i 0.0541389 0.998533i 1.48491 + 0.243438i −0.0728043 1.73052i −2.01420 + 3.79918i 0.647386 + 0.762162i −0.413452 2.97137i −1.24545 + 0.844434i
23.3 −0.725995 + 0.687699i −0.941981 1.45350i 0.0541389 0.998533i 3.62342 + 0.594030i 1.68345 + 0.407438i −0.837325 + 1.57936i 0.647386 + 0.762162i −1.22535 + 2.73834i −3.03910 + 2.06056i
23.4 −0.725995 + 0.687699i −0.713094 + 1.57845i 0.0541389 0.998533i −1.84131 0.301868i −0.567794 1.63634i 1.58049 2.98111i 0.647386 + 0.762162i −1.98299 2.25116i 1.54438 1.04712i
23.5 −0.725995 + 0.687699i −0.639916 1.60951i 0.0541389 0.998533i −1.86219 0.305290i 1.57143 + 0.728423i 1.84960 3.48871i 0.647386 + 0.762162i −2.18101 + 2.05990i 1.56189 1.05899i
23.6 −0.725995 + 0.687699i 0.422484 1.67973i 0.0541389 0.998533i −1.70784 0.279986i 0.848431 + 1.51002i −2.22532 + 4.19739i 0.647386 + 0.762162i −2.64301 1.41932i 1.43243 0.971213i
23.7 −0.725995 + 0.687699i 1.07024 + 1.36183i 0.0541389 0.998533i 2.51062 + 0.411595i −1.71352 0.252679i 0.276889 0.522268i 0.647386 + 0.762162i −0.709168 + 2.91498i −2.10575 + 1.42773i
23.8 −0.725995 + 0.687699i 1.27688 + 1.17029i 0.0541389 0.998533i −3.03432 0.497451i −1.73182 + 0.0284890i −0.0547647 + 0.103297i 0.647386 + 0.762162i 0.260860 + 2.98864i 2.54500 1.72555i
23.9 −0.725995 + 0.687699i 1.38419 1.04116i 0.0541389 0.998533i 3.46404 + 0.567901i −0.288910 + 1.70779i 1.50148 2.83210i 0.647386 + 0.762162i 0.831965 2.88233i −2.90542 + 1.96993i
23.10 −0.725995 + 0.687699i 1.57094 0.729476i 0.0541389 0.998533i −1.16466 0.190936i −0.638838 + 1.60993i 0.141142 0.266222i 0.647386 + 0.762162i 1.93573 2.29193i 0.976845 0.662318i
See next 80 embeddings (of 280 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 347.10
Significant digits:
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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 \(T_{5}^{280} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(354, [\chi])\).