# Properties

 Label 349.2.e.a Level 349 Weight 2 Character orbit 349.e Analytic conductor 2.787 Analytic rank 0 Dimension 58 CM no Inner twists 2

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## Newspace parameters

 Level: $$N$$ = $$349$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 349.e (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.78677903054$$ Analytic rank: $$0$$ Dimension: $$58$$ Relative dimension: $$29$$ over $$\Q(\zeta_{6})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$58q - 3q^{2} + 27q^{4} + 2q^{5} - 29q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$58q - 3q^{2} + 27q^{4} + 2q^{5} - 29q^{9} - q^{12} - 3q^{13} - 10q^{14} + q^{15} - 29q^{16} - 10q^{17} + 15q^{18} - 9q^{19} - 3q^{22} - 17q^{23} - 48q^{24} - 29q^{25} - 4q^{26} + 18q^{27} - 2q^{29} + 9q^{30} + 32q^{31} + 9q^{32} - 12q^{33} - 63q^{34} + 24q^{36} - 16q^{37} + 54q^{40} - 10q^{41} - 15q^{42} - 45q^{43} + 18q^{44} - 2q^{45} + 27q^{46} + 6q^{48} + 35q^{49} + 6q^{50} - 14q^{51} + 27q^{54} + 24q^{55} + 11q^{56} - 29q^{57} - 18q^{59} + 116q^{60} - 9q^{62} - 21q^{63} - 132q^{64} + 130q^{66} + 58q^{67} + 42q^{69} + 40q^{70} - 24q^{71} + 72q^{72} - 6q^{73} + 30q^{74} - 58q^{75} + 37q^{76} - 4q^{77} - 33q^{78} - 40q^{80} - 81q^{81} + 21q^{82} + 12q^{83} + 18q^{84} - 11q^{85} - 126q^{86} - 42q^{87} - 50q^{88} + 3q^{89} - 12q^{90} - 28q^{91} - 120q^{92} + 31q^{93} + 29q^{94} + 60q^{95} - 120q^{96} - 15q^{97} + 39q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
123.1 −2.39209 + 1.38107i −0.0982202 + 0.170122i 2.81473 4.87525i 1.45632 2.52242i 0.542597i −2.31421 + 1.33611i 10.0251i 1.48071 + 2.56466i 8.04515i
123.2 −2.28298 + 1.31808i −1.40419 + 2.43213i 2.47467 4.28626i 0.108819 0.188480i 7.40335i 3.00446 1.73462i 7.77496i −2.44351 4.23228i 0.573727i
123.3 −2.10033 + 1.21263i 0.180949 0.313412i 1.94093 3.36178i −1.51394 + 2.62221i 0.877693i 0.636711 0.367605i 4.56397i 1.43452 + 2.48465i 7.34336i
123.4 −2.00748 + 1.15902i 1.04619 1.81206i 1.68665 2.92136i −0.198863 + 0.344441i 4.85022i −2.47763 + 1.43046i 3.18335i −0.689035 1.19344i 0.921944i
123.5 −1.85370 + 1.07024i 1.18706 2.05605i 1.29081 2.23575i 1.34675 2.33263i 5.08174i 2.87980 1.66266i 1.24494i −1.31823 2.28323i 5.76535i
123.6 −1.71754 + 0.991624i −1.29117 + 2.23637i 0.966636 1.67426i −0.796466 + 1.37952i 5.12141i −1.81826 + 1.04977i 0.132337i −1.83422 3.17696i 3.15918i
123.7 −1.34862 + 0.778626i 1.56432 2.70948i 0.212518 0.368092i −1.71886 + 2.97716i 4.87208i 0.0378395 0.0218466i 2.45262i −3.39418 5.87890i 5.35340i
123.8 −1.33595 + 0.771310i −0.677977 + 1.17429i 0.189837 0.328808i 1.91694 3.32023i 2.09172i −1.98739 + 1.14742i 2.49955i 0.580694 + 1.00579i 5.91420i
123.9 −1.06897 + 0.617172i 0.625735 1.08381i −0.238198 + 0.412572i 0.733047 1.26968i 1.54474i −0.833145 + 0.481016i 3.05672i 0.716910 + 1.24172i 1.80966i
123.10 −1.03662 + 0.598491i 0.00513702 0.00889758i −0.283617 + 0.491238i −0.193021 + 0.334323i 0.0122978i 3.23720 1.86900i 3.07293i 1.49995 + 2.59798i 0.462086i
123.11 −0.941972 + 0.543848i −0.443096 + 0.767465i −0.408460 + 0.707473i −1.75571 + 3.04098i 0.963907i 2.73929 1.58153i 3.06395i 1.10733 + 1.91796i 3.81936i
123.12 −0.788972 + 0.455513i −1.46240 + 2.53295i −0.585016 + 1.01328i 1.45267 2.51610i 2.66457i 3.29342 1.90146i 2.88798i −2.77724 4.81032i 2.64684i
123.13 −0.529073 + 0.305460i 0.484492 0.839165i −0.813388 + 1.40883i −0.0329412 + 0.0570558i 0.591972i −3.18284 + 1.83761i 2.21567i 1.03054 + 1.78494i 0.0402489i
123.14 −0.327556 + 0.189114i −1.06274 + 1.84073i −0.928472 + 1.60816i −1.07891 + 1.86873i 0.803920i −2.19494 + 1.26725i 1.45881i −0.758847 1.31436i 0.816152i
123.15 −0.0377450 + 0.0217921i 1.71622 2.97259i −0.999050 + 1.73041i 1.73771 3.00980i 0.149601i −2.79927 + 1.61616i 0.174254i −4.39085 7.60517i 0.151473i
123.16 0.228167 0.131732i 0.150552 0.260763i −0.965293 + 1.67194i 0.807178 1.39807i 0.0793300i 1.23534 0.713223i 1.03557i 1.45467 + 2.51956i 0.425325i
123.17 0.499056 0.288130i −0.844268 + 1.46231i −0.833962 + 1.44447i 0.707762 1.22588i 0.973035i 1.68926 0.975293i 2.11368i 0.0744233 + 0.128905i 0.815710i
123.18 0.531309 0.306751i 0.869046 1.50523i −0.811807 + 1.40609i −1.66390 + 2.88196i 1.06632i −1.62203 + 0.936479i 2.22310i −0.0104823 0.0181558i 2.04162i
123.19 0.544820 0.314552i 1.24137 2.15012i −0.802114 + 1.38930i −0.0260007 + 0.0450346i 1.56191i 3.86429 2.23105i 2.26743i −1.58202 2.74014i 0.0327144i
123.20 0.831968 0.480337i −1.62465 + 2.81397i −0.538552 + 0.932800i −1.69786 + 2.94079i 3.12152i 3.20576 1.85085i 2.95610i −3.77897 6.54537i 3.26219i
See all 58 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 227.29 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
349.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 349.2.e.a 58
349.e even 6 1 inner 349.2.e.a 58

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
349.2.e.a 58 1.a even 1 1 trivial
349.2.e.a 58 349.e even 6 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(349, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database