Properties

Label 349.2.c.a
Level 349
Weight 2
Character orbit 349.c
Analytic conductor 2.787
Analytic rank 0
Dimension 56
CM No

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Newspace parameters

Level: \( N \) = \( 349 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 349.c (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(2.78677903054\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(\zeta_{3})\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 56q - 30q^{4} + 4q^{5} + 14q^{6} + 2q^{7} + 6q^{8} - 28q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 56q - 30q^{4} + 4q^{5} + 14q^{6} + 2q^{7} + 6q^{8} - 28q^{9} - 2q^{10} - 2q^{11} + 11q^{12} - 2q^{13} + 2q^{14} + 9q^{15} - 34q^{16} + 18q^{18} - 5q^{19} + 14q^{20} + 12q^{21} - 7q^{22} - 11q^{23} - 30q^{24} - 6q^{25} - 11q^{26} - 30q^{27} - 52q^{28} + 8q^{29} - 21q^{30} - 48q^{31} - 6q^{32} + 12q^{33} - 14q^{34} + 42q^{35} + 66q^{36} + 14q^{37} + 60q^{38} - 26q^{39} + 24q^{40} - 3q^{42} - 23q^{43} - 20q^{44} + 18q^{45} + 5q^{46} - 26q^{47} - 22q^{48} - 26q^{49} + 11q^{50} + 14q^{51} + 6q^{52} - 12q^{53} - 7q^{54} + 10q^{55} - 19q^{56} + 25q^{57} - 12q^{58} - 16q^{59} - 12q^{60} + 42q^{61} - 27q^{62} + 31q^{63} + 54q^{64} + 72q^{65} - 66q^{66} - 34q^{67} - 57q^{68} + 10q^{69} - 52q^{70} - 10q^{71} + 47q^{72} + 23q^{73} - 17q^{74} - 26q^{75} + 9q^{76} - 10q^{77} + 25q^{78} + 48q^{79} - 32q^{80} - 12q^{81} - 8q^{82} + 14q^{83} + 10q^{84} - 3q^{85} + 46q^{86} + 14q^{87} + 58q^{88} + 8q^{89} + 68q^{90} + 54q^{91} + 48q^{92} - 57q^{93} + 33q^{94} + 54q^{95} - 72q^{96} + 32q^{98} + 73q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
122.1 −1.36429 + 2.36301i −0.158062 0.273772i −2.72255 4.71559i 1.82233 + 3.15637i 0.862568 0.454802 0.787740i 9.40019 1.45003 2.51153i −9.94473
122.2 −1.30509 + 2.26049i −1.25915 2.18091i −2.40653 4.16824i −0.430393 0.745463i 6.57322 1.54999 2.68466i 7.34263 −1.67090 + 2.89409i 2.24681
122.3 −1.21954 + 2.11230i 1.25596 + 2.17538i −1.97455 3.42003i −0.137412 0.238004i −6.12676 −1.55873 + 2.69980i 4.75403 −1.65485 + 2.86629i 0.670316
122.4 −1.19066 + 2.06228i 0.388808 + 0.673436i −1.83534 3.17890i −0.996293 1.72563i −1.85175 0.620144 1.07412i 3.97841 1.19766 2.07440i 4.74498
122.5 −1.10993 + 1.92246i −0.792401 1.37248i −1.46391 2.53556i −0.325113 0.563112i 3.51805 −2.07452 + 3.59318i 2.05963 0.244201 0.422969i 1.44342
122.6 −0.926433 + 1.60463i 1.38903 + 2.40587i −0.716556 1.24111i 1.81251 + 3.13936i −5.14738 2.31800 4.01489i −1.05037 −2.35882 + 4.08560i −6.71668
122.7 −0.822866 + 1.42525i 0.352316 + 0.610228i −0.354218 0.613523i −1.45058 2.51248i −1.15963 1.11281 1.92745i −2.12557 1.25175 2.16809i 4.77454
122.8 −0.770318 + 1.33423i −1.33797 2.31743i −0.186780 0.323513i 0.251334 + 0.435324i 4.12265 0.615403 1.06591i −2.50575 −2.08032 + 3.60322i −0.774430
122.9 −0.736999 + 1.27652i −0.677079 1.17273i −0.0863339 0.149535i 1.59482 + 2.76232i 1.99602 −0.744222 + 1.28903i −2.69348 0.583129 1.01001i −4.70153
122.10 −0.433156 + 0.750249i −0.267600 0.463497i 0.624751 + 1.08210i −1.56448 2.70976i 0.463651 −2.15779 + 3.73740i −2.81509 1.35678 2.35001i 2.71066
122.11 −0.331556 + 0.574272i 1.42732 + 2.47219i 0.780141 + 1.35124i −1.03448 1.79178i −1.89295 −0.281656 + 0.487843i −2.36087 −2.57448 + 4.45914i 1.37196
122.12 −0.296091 + 0.512844i −0.340451 0.589679i 0.824661 + 1.42835i 0.413913 + 0.716918i 0.403218 2.29989 3.98352i −2.16106 1.26819 2.19656i −0.490223
122.13 −0.226116 + 0.391644i 1.02977 + 1.78362i 0.897743 + 1.55494i 0.218923 + 0.379186i −0.931393 −0.0278656 + 0.0482646i −1.71644 −0.620869 + 1.07538i −0.198008
122.14 −0.0965034 + 0.167149i −1.64186 2.84379i 0.981374 + 1.69979i 0.242663 + 0.420304i 0.633781 −1.37278 + 2.37772i −0.764837 −3.89141 + 6.74013i −0.0936712
122.15 0.0575886 0.0997464i 0.261626 + 0.453149i 0.993367 + 1.72056i 1.39129 + 2.40979i 0.0602667 −0.729104 + 1.26285i 0.459181 1.36310 2.36097i 0.320490
122.16 0.129761 0.224752i −0.563330 0.975716i 0.966324 + 1.67372i −0.922568 1.59793i −0.292393 −1.10786 + 1.91887i 1.02061 0.865319 1.49878i −0.478853
122.17 0.404409 0.700458i −1.01083 1.75081i 0.672906 + 1.16551i −1.32166 2.28919i −1.63515 1.24448 2.15550i 2.70616 −0.543548 + 0.941452i −2.13797
122.18 0.492106 0.852353i 1.32791 + 2.30001i 0.515663 + 0.893154i 0.455093 + 0.788245i 2.61390 1.87395 3.24578i 2.98347 −2.02671 + 3.51036i 0.895817
122.19 0.569861 0.987028i 0.360004 + 0.623546i 0.350517 + 0.607113i −0.221966 0.384456i 0.820609 0.125742 0.217791i 3.07843 1.24079 2.14912i −0.505958
122.20 0.608705 1.05431i −1.29631 2.24527i 0.258956 + 0.448525i 1.83464 + 3.17768i −3.15628 1.86440 3.22924i 3.06533 −1.86082 + 3.22304i 4.46701
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 226.28
Significant digits:
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Inner twists

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

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(349, [\chi])\).