Properties

Label 349.2.h.a
Level 349
Weight 2
Character orbit 349.h
Analytic conductor 2.787
Analytic rank 0
Dimension 784
CM No

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Newspace parameters

Level: \( N \) = \( 349 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 349.h (of order \(58\) and degree \(28\))

Newform invariants

Self dual: No
Analytic conductor: \(2.78677903054\)
Analytic rank: \(0\)
Dimension: \(784\)
Relative dimension: \(28\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 784q - 29q^{2} - 25q^{3} + 3q^{4} - 21q^{5} - 29q^{6} - 29q^{7} + 58q^{8} - 57q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 784q - 29q^{2} - 25q^{3} + 3q^{4} - 21q^{5} - 29q^{6} - 29q^{7} + 58q^{8} - 57q^{9} - 29q^{10} - 29q^{11} - 37q^{12} - 29q^{13} - 33q^{14} - 37q^{15} - 57q^{16} - 21q^{17} - 145q^{18} + 43q^{19} - 35q^{20} - 145q^{21} - 17q^{22} - 13q^{23} - 45q^{25} + 7q^{26} - 43q^{27} - 29q^{28} - 13q^{29} - 29q^{30} - 53q^{31} - 29q^{32} - 29q^{33} - 29q^{34} - 87q^{35} + 33q^{36} + 150q^{37} - 29q^{38} - 29q^{39} - 29q^{40} - 91q^{41} + 145q^{42} - 29q^{43} - 145q^{44} + 23q^{45} - 29q^{46} - 29q^{47} + 111q^{48} + 19q^{49} + 348q^{50} + 61q^{51} - 145q^{52} - 29q^{53} - 29q^{54} - 145q^{55} + 9q^{56} - 23q^{57} - 29q^{58} - 29q^{59} + 405q^{60} - 174q^{61} - 29q^{62} - 29q^{63} - 210q^{64} - 29q^{65} + 244q^{66} - 75q^{67} - 77q^{68} - 103q^{69} - 115q^{70} - 29q^{71} - 203q^{72} - 79q^{73} + 87q^{74} + 7q^{75} - 105q^{76} + 39q^{77} + 233q^{78} - 29q^{79} + 99q^{80} + 207q^{81} - 29q^{82} + 93q^{83} - 435q^{84} + 126q^{85} - 147q^{86} + 115q^{87} - 139q^{88} - 435q^{89} - 29q^{90} - 57q^{91} - 47q^{92} - 99q^{93} - 51q^{94} + 233q^{95} + 609q^{96} - 29q^{97} + 377q^{98} - 29q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −1.88811 + 1.99326i −1.86742 2.19850i −0.299824 5.52994i −1.32650 + 1.00838i 7.90807 + 0.428763i 1.94394 3.23086i 7.40358 + 6.28866i −0.860783 + 5.25055i 0.494625 4.54800i
17.2 −1.78707 + 1.88659i 2.03611 + 2.39709i −0.257313 4.74586i −1.71722 + 1.30539i −8.16098 0.442476i 0.764584 1.27075i 5.45218 + 4.63113i −1.11496 + 6.80097i 0.606047 5.57251i
17.3 −1.63108 + 1.72191i −1.05923 1.24702i −0.196276 3.62009i 3.08829 2.34766i 3.87495 + 0.210094i −1.40558 + 2.33609i 2.93823 + 2.49576i 0.0522511 0.318718i −0.994794 + 9.14698i
17.4 −1.53864 + 1.62432i −0.352185 0.414624i −0.162732 3.00141i −2.61649 + 1.98901i 1.21537 + 0.0658952i −1.47986 + 2.45955i 1.71517 + 1.45688i 0.437467 2.66843i 0.795052 7.31038i
17.5 −1.53663 + 1.62220i 0.865190 + 1.01858i −0.162026 2.98840i 1.35343 1.02885i −2.98183 0.161670i 0.922190 1.53269i 1.69073 + 1.43612i 0.196394 1.19795i −0.410720 + 3.77650i
17.6 −1.21030 + 1.27769i −1.49451 1.75947i −0.0594073 1.09570i −0.198580 + 0.150957i 4.05686 + 0.219956i −0.511468 + 0.850067i −1.21081 1.02847i −0.376837 + 2.29860i 0.0474642 0.436427i
17.7 −1.14082 + 1.20435i 0.0908175 + 0.106919i −0.0407092 0.750837i −2.77605 + 2.11030i −0.232374 0.0125990i 2.35778 3.91865i −1.57798 1.34035i 0.482162 2.94106i 0.625439 5.75081i
17.8 −1.12530 + 1.18796i 1.22054 + 1.43693i −0.0366819 0.676559i −0.841182 + 0.639450i −3.08050 0.167020i −2.16228 + 3.59373i −1.64928 1.40091i −0.0897083 + 0.547197i 0.186938 1.71886i
17.9 −0.998387 + 1.05398i 1.55038 + 1.82525i −0.00582860 0.107502i 2.23112 1.69606i −3.47167 0.188228i −0.345966 + 0.575000i −2.09386 1.77854i −0.442511 + 2.69920i −0.439908 + 4.04489i
17.10 −0.622305 + 0.656959i −0.150428 0.177098i 0.0639457 + 1.17941i 0.905172 0.688094i 0.209959 + 0.0113836i 1.57394 2.61591i −2.19399 1.86359i 0.476611 2.90720i −0.111243 + 1.02287i
17.11 −0.450766 + 0.475868i −0.863343 1.01641i 0.0850176 + 1.56806i −0.624584 + 0.474797i 0.872840 + 0.0473240i −0.245761 + 0.408458i −1.78366 1.51505i 0.197626 1.20547i 0.0556009 0.511242i
17.12 −0.357823 + 0.377749i 2.21687 + 2.60991i 0.0936207 + 1.72673i −0.811402 + 0.616812i −1.77914 0.0964621i 0.990354 1.64598i −1.47891 1.25619i −1.41173 + 8.61120i 0.0573381 0.527216i
17.13 −0.262216 + 0.276818i −0.127909 0.150586i 0.100407 + 1.85189i 3.35848 2.55305i 0.0752247 + 0.00407857i 0.247523 0.411386i −1.12018 0.951491i 0.479031 2.92196i −0.173917 + 1.59914i
17.14 −0.144821 + 0.152886i 0.932034 + 1.09727i 0.105877 + 1.95278i −2.28614 + 1.73788i −0.302736 0.0164139i −0.728090 + 1.21009i −0.634891 0.539282i 0.150021 0.915085i 0.0653844 0.601200i
17.15 0.100044 0.105615i −1.65632 1.94997i 0.107132 + 1.97593i −0.547752 + 0.416391i −0.371651 0.0201504i 1.05893 1.75995i 0.441159 + 0.374724i −0.573647 + 3.49909i −0.0108222 + 0.0995082i
17.16 0.176456 0.186282i −2.05396 2.41811i 0.104713 + 1.93133i 1.55529 1.18230i −0.812884 0.0440733i −2.62624 + 4.36484i 0.769373 + 0.653512i −1.14315 + 6.97291i 0.0541984 0.498347i
17.17 0.412126 0.435076i −1.43091 1.68459i 0.0888344 + 1.63845i −2.64506 + 2.01072i −1.32264 0.0717115i 1.63411 2.71592i 1.66296 + 1.41253i −0.305017 + 1.86052i −0.215281 + 1.97947i
17.18 0.489556 0.516818i 0.731930 + 0.861694i 0.0808420 + 1.49104i 0.146093 0.111057i 0.803660 + 0.0435732i −1.26681 + 2.10545i 1.89530 + 1.60989i 0.278550 1.69908i 0.0141245 0.129872i
17.19 0.504938 0.533057i 1.26280 + 1.48669i 0.0790909 + 1.45875i 0.890294 0.676784i 1.43013 + 0.0775392i 2.42550 4.03121i 1.93675 + 1.64509i −0.130219 + 0.794301i 0.0887793 0.816312i
17.20 0.875876 0.924651i −0.774622 0.911956i 0.0204571 + 0.377309i 2.88562 2.19359i −1.52171 0.0825049i 0.0984930 0.163697i 2.30822 + 1.96062i 0.253722 1.54764i 0.499138 4.58950i
See next 80 embeddings (of 784 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 318.28
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 \(S_{2}^{\mathrm{new}}(349, [\chi])\).