Properties

Label 349.2.g.a
Level 349
Weight 2
Character orbit 349.g
Analytic conductor 2.787
Analytic rank 0
Dimension 756
CM No

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Newspace parameters

Level: \( N \) = \( 349 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 349.g (of order \(29\) and degree \(28\))

Newform invariants

Self dual: No
Analytic conductor: \(2.78677903054\)
Analytic rank: \(0\)
Dimension: \(756\)
Relative dimension: \(27\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 756q - 26q^{2} - 25q^{3} - 46q^{4} - 19q^{5} - 9q^{6} - 19q^{7} - 95q^{8} - 44q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 756q - 26q^{2} - 25q^{3} - 46q^{4} - 19q^{5} - 9q^{6} - 19q^{7} - 95q^{8} - 44q^{9} - 7q^{10} - 25q^{11} - 5q^{12} - 5q^{13} - 9q^{14} - 5q^{15} - 26q^{16} - 11q^{17} - 106q^{18} - 57q^{19} + 15q^{20} - 95q^{21} - 9q^{22} + 11q^{23} + 26q^{24} - 22q^{25} - 35q^{26} + 17q^{27} + 47q^{28} + 9q^{29} + 61q^{30} + 15q^{31} + 16q^{32} + 13q^{33} + 35q^{34} - 27q^{35} + 56q^{36} - 186q^{37} + 37q^{38} + 19q^{39} + 79q^{40} - 33q^{41} - 107q^{42} + 19q^{43} - 93q^{44} + 73q^{45} + 21q^{46} - q^{47} + 39q^{48} + 8q^{49} - 305q^{50} - 83q^{51} - 238q^{52} + q^{53} + 99q^{54} - 81q^{55} + 69q^{56} + 61q^{57} + 97q^{58} + 9q^{59} - 203q^{60} - 227q^{61} + 43q^{62} + 49q^{63} - 151q^{64} + 61q^{65} - 226q^{66} + 53q^{67} + 133q^{68} - 51q^{69} + 165q^{70} + 3q^{71} - 150q^{72} + 71q^{73} - 87q^{74} + 39q^{75} + 159q^{76} + 99q^{77} - 59q^{78} + 33q^{79} - 7q^{80} - 384q^{81} + 107q^{82} - 95q^{83} - 165q^{84} - 36q^{85} + 49q^{86} - 81q^{87} + 101q^{88} - 349q^{89} + 231q^{90} + 107q^{91} + 121q^{92} + 117q^{93} + 157q^{94} - 123q^{95} - 379q^{96} + 57q^{97} - 304q^{98} + 89q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −0.409995 2.50086i −1.52790 + 2.88192i −4.19089 + 1.41208i 2.44011 2.31140i 7.83371 + 2.63949i −2.74627 0.298674i 2.87552 + 5.42381i −4.28745 6.32351i −6.78090 5.15471i
31.2 −0.409444 2.49750i −0.175952 + 0.331880i −4.17455 + 1.40657i −1.14986 + 1.08920i 0.900913 + 0.303553i −0.0267563 0.00290993i 2.85122 + 5.37796i 1.60438 + 2.36628i 3.19108 + 2.42580i
31.3 −0.397225 2.42297i −0.263553 + 0.497114i −3.81768 + 1.28632i 1.72972 1.63847i 1.30918 + 0.441115i 3.06696 + 0.333552i 2.33302 + 4.40055i 1.50590 + 2.22103i −4.65705 3.54020i
31.4 −0.329871 2.01212i 0.865058 1.63167i −2.04453 + 0.688881i −2.66306 + 2.52258i −3.56849 1.20236i −3.28966 0.357772i 0.150388 + 0.283663i −0.230467 0.339913i 5.95421 + 4.52627i
31.5 −0.328287 2.00246i 1.20481 2.27251i −2.00677 + 0.676161i 0.881952 0.835429i −4.94615 1.66655i 1.30077 + 0.141467i 0.111802 + 0.210881i −2.02919 2.99283i −1.96245 1.49181i
31.6 −0.260307 1.58780i −1.02525 + 1.93382i −0.558048 + 0.188028i −0.675040 + 0.639432i 3.33740 + 1.12450i −2.74814 0.298878i −1.06352 2.00601i −1.00497 1.48221i 1.19101 + 0.905382i
31.7 −0.234020 1.42746i −0.923973 + 1.74280i −0.0875691 + 0.0295055i −1.27207 + 1.20497i 2.70400 + 0.911085i 0.523145 + 0.0568954i −1.29251 2.43793i −0.500058 0.737531i 2.01773 + 1.53384i
31.8 −0.222646 1.35808i 0.0623339 0.117574i 0.100492 0.0338599i 2.59545 2.45854i −0.173554 0.0584770i −4.60163 0.500458i −1.35762 2.56074i 1.67362 + 2.46841i −3.91676 2.97744i
31.9 −0.195197 1.19065i 0.226589 0.427392i 0.515764 0.173781i −2.64101 + 2.50170i −0.553103 0.186362i 4.84655 + 0.527094i −1.43790 2.71216i 1.55224 + 2.28938i 3.49416 + 2.65619i
31.10 −0.168433 1.02740i 0.294114 0.554758i 0.868128 0.292507i 1.98905 1.88412i −0.619496 0.208733i 1.60551 + 0.174610i −1.42208 2.68232i 1.46231 + 2.15674i −2.27077 1.72619i
31.11 −0.0855472 0.521815i 1.06052 2.00036i 1.63033 0.549323i −0.329114 + 0.311753i −1.13454 0.382272i −2.71502 0.295277i −0.921486 1.73811i −1.19317 1.75979i 0.190832 + 0.145067i
31.12 −0.0843976 0.514802i −1.24097 + 2.34071i 1.63741 0.551707i 1.82811 1.73168i 1.30974 + 0.441302i 2.27661 + 0.247596i −0.910927 1.71819i −2.25537 3.32642i −1.04576 0.794966i
31.13 −0.0384208 0.234357i −0.0824196 + 0.155460i 1.84186 0.620595i −0.451145 + 0.427347i 0.0395997 + 0.0133427i 1.45469 + 0.158207i −0.438686 0.827450i 1.66619 + 2.45744i 0.117485 + 0.0893098i
31.14 −0.000709636 0.00432859i 1.55115 2.92579i 1.89529 0.638597i −1.92168 + 1.82032i −0.0137653 0.00463806i 3.38702 + 0.368360i −0.00821841 0.0155016i −4.47059 6.59363i 0.00924309 + 0.00702642i
31.15 0.0663976 + 0.405008i −1.58682 + 2.99305i 1.73568 0.584820i −2.67584 + 2.53469i −1.31757 0.443941i 1.71607 + 0.186634i 0.736585 + 1.38935i −4.75683 7.01579i −1.20424 0.915438i
31.16 0.0860743 + 0.525030i −0.766700 + 1.44615i 1.62706 0.548220i 0.781296 0.740083i −0.825265 0.278064i −2.19262 0.238461i 0.926303 + 1.74719i 0.180042 + 0.265543i 0.455815 + 0.346502i
31.17 0.0982508 + 0.599303i 0.399218 0.753005i 1.54580 0.520839i 0.826225 0.782642i 0.490502 + 0.165269i −0.615438 0.0669329i 1.03295 + 1.94835i 1.27592 + 1.88184i 0.550217 + 0.418264i
31.18 0.183528 + 1.11947i 1.39174 2.62510i 0.675772 0.227694i 2.41800 2.29045i 3.19415 + 1.07623i −3.83030 0.416570i 1.44166 + 2.71926i −3.27065 4.82385i 3.00787 + 2.28652i
31.19 0.205773 + 1.25516i 0.489692 0.923656i 0.362216 0.122045i −2.57149 + 2.43584i 1.26010 + 0.424579i −1.26854 0.137963i 1.41927 + 2.67704i 1.07022 + 1.57845i −3.58652 2.72640i
31.20 0.226756 + 1.38315i −1.17622 + 2.21859i 0.0336242 0.0113293i −0.238431 + 0.225854i −3.33535 1.12381i −4.90800 0.533778i 1.33635 + 2.52062i −1.85508 2.73603i −0.366455 0.278572i
See next 80 embeddings (of 756 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 332.27
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 \(S_{2}^{\mathrm{new}}(349, [\chi])\).