# Properties

 Label 349.2.i.a Level 349 Weight 2 Character orbit 349.i Analytic conductor 2.787 Analytic rank 0 Dimension 1568 CM No

# Related objects

## Newspace parameters

 Level: $$N$$ = $$349$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 349.i (of order $$87$$ and degree $$56$$)

## Newform invariants

 Self dual: No Analytic conductor: $$2.78677903054$$ Analytic rank: $$0$$ Dimension: $$1568$$ Relative dimension: $$28$$ over $$\Q(\zeta_{87})$$ Sato-Tate group: $\mathrm{SU}(2)[C_{87}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$1568q - 58q^{2} - 58q^{3} - 28q^{4} - 62q^{5} - 72q^{6} - 60q^{7} + 23q^{8} - 30q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$1568q - 58q^{2} - 58q^{3} - 28q^{4} - 62q^{5} - 72q^{6} - 60q^{7} + 23q^{8} - 30q^{9} - 56q^{10} - 56q^{11} - 69q^{12} - 56q^{13} - 60q^{14} - 67q^{15} - 24q^{16} - 58q^{17} + 40q^{18} - 169q^{19} - 72q^{20} + 46q^{21} - 51q^{22} - 47q^{23} + q^{24} - 52q^{25} + 11q^{26} - 28q^{27} - 6q^{28} - 66q^{29} - 37q^{30} - 10q^{31} - 52q^{32} - 70q^{33} - 44q^{34} - 42q^{35} - 124q^{36} - 159q^{37} - 118q^{38} - 32q^{39} - 82q^{40} - 403q^{42} - 35q^{43} + 78q^{44} - 76q^{45} - 63q^{46} - 32q^{47} + 22q^{48} - 32q^{49} + 308q^{50} + 44q^{51} + 255q^{52} - 46q^{53} - 51q^{54} + 48q^{55} - 39q^{56} - 83q^{57} - 46q^{58} - 42q^{59} - 742q^{60} + 190q^{61} - 31q^{62} - 89q^{63} + 149q^{64} - 130q^{65} + 385q^{66} - 24q^{67} - q^{68} + 48q^{69} - 6q^{70} - 48q^{71} + 243q^{72} - 81q^{73} + 75q^{74} - 32q^{75} - 67q^{76} - 48q^{77} - 547q^{78} - 106q^{79} + 148q^{80} + 418q^{81} - 50q^{82} - 304q^{83} + 338q^{84} + 90q^{85} - 46q^{86} + 102q^{87} - 116q^{88} + 340q^{89} - 126q^{90} - 112q^{91} - 106q^{92} - q^{93} - 91q^{94} + 120q^{95} - 914q^{96} - 58q^{97} + 316q^{98} - 131q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1 −2.46125 1.24848i 0.387351 0.146288i 3.31766 + 4.53187i 1.25689 0.324652i −1.13601 0.123548i 4.59338 + 0.165940i −1.61470 9.84922i −2.12241 + 1.86979i −3.49886 0.770156i
9.2 −2.17302 1.10227i 1.67917 0.634157i 2.32562 + 3.17675i −1.08355 + 0.279878i −4.34788 0.472861i −4.21470 0.152260i −0.763578 4.65762i 0.166393 0.146588i 2.66309 + 0.586190i
9.3 −2.14572 1.08842i −2.78217 + 1.05072i 2.23804 + 3.05713i 2.44600 0.631793i 7.11339 + 0.773627i −1.34778 0.0486897i −0.696275 4.24709i 4.38540 3.86344i −5.93608 1.30663i
9.4 −2.00904 1.01909i −1.03329 + 0.390235i 1.81630 + 2.48103i 0.540716 0.139665i 2.47361 + 0.269021i −0.925700 0.0334418i −0.391715 2.38936i −1.33564 + 1.17667i −1.22865 0.270447i
9.5 −1.95105 0.989675i 2.11329 0.798109i 1.64572 + 2.24803i −2.39358 + 0.618254i −4.91299 0.534320i 2.11340 + 0.0763486i −0.278204 1.69697i 1.57796 1.39014i 5.28186 + 1.16263i
9.6 −1.86652 0.946796i 2.80322 1.05867i 1.40605 + 1.92064i 2.94974 0.761907i −6.23459 0.678053i −0.347829 0.0125657i −0.128769 0.785453i 4.48620 3.95224i −6.22710 1.37069i
9.7 −1.76537 0.895491i −1.48767 + 0.561835i 1.13323 + 1.54797i −3.26214 + 0.842600i 3.12940 + 0.340343i 1.24531 + 0.0449880i 0.0261182 + 0.159314i −0.353559 + 0.311477i 6.51343 + 1.43371i
9.8 −1.18360 0.600385i −0.682614 + 0.257797i −0.140957 0.192545i 3.36183 0.868350i 0.962720 + 0.104702i 1.83035 + 0.0661230i 0.480660 + 2.93189i −1.85155 + 1.63117i −4.50041 0.990615i
9.9 −1.05413 0.534709i 0.535376 0.202191i −0.356137 0.486477i 1.37934 0.356279i −0.672467 0.0731352i −3.69432 0.133461i 0.497739 + 3.03607i −2.00530 + 1.76663i −1.64451 0.361983i
9.10 −0.818321 0.415096i −2.39483 + 0.904438i −0.684062 0.934416i −1.74670 + 0.451166i 2.33517 + 0.253965i −4.80765 0.173681i 0.468806 + 2.85959i 2.66617 2.34884i 1.61663 + 0.355848i
9.11 −0.737293 0.373994i 0.783668 0.295962i −0.777676 1.06229i −2.65376 + 0.685457i −0.688481 0.0748768i 1.75900 + 0.0635456i 0.443583 + 2.70573i −1.72451 + 1.51925i 2.21296 + 0.487108i
9.12 −0.736757 0.373722i 2.21410 0.836181i −0.778263 1.06309i 0.343155 0.0886359i −1.94375 0.211396i 5.07513 + 0.183344i 0.443393 + 2.70458i 1.95198 1.71965i −0.285947 0.0629418i
9.13 −0.379345 0.192424i 2.17990 0.823266i −1.07453 1.46779i 2.33094 0.602075i −0.985352 0.107163i −2.42172 0.0874869i 0.262811 + 1.60307i 1.82315 1.60615i −1.00009 0.220136i
9.14 −0.0280982 0.0142529i −2.20107 + 0.831260i −1.18082 1.61298i 0.943566 0.243720i 0.0736939 + 0.00801469i 0.689625 + 0.0249134i 0.0203836 + 0.124335i 1.90266 1.67620i −0.0299862 0.00660047i
9.15 0.0699482 + 0.0354814i −0.760409 + 0.287178i −1.17777 1.60881i −2.29345 + 0.592390i −0.0633787 0.00689285i 2.08428 + 0.0752967i −0.0506779 0.309122i −1.75530 + 1.54638i −0.181442 0.0399383i
9.16 0.200938 + 0.101926i −1.14709 + 0.433214i −1.15142 1.57282i 0.757667 0.195703i −0.274651 0.0298701i 2.54552 + 0.0919592i −0.143955 0.878084i −1.12290 + 0.989249i 0.172191 + 0.0379022i
9.17 0.436267 + 0.221298i 1.26243 0.476772i −1.04005 1.42069i −3.45502 + 0.892420i 0.656265 + 0.0713731i −3.40184 0.122895i −0.297627 1.81544i −0.884633 + 0.779341i −1.70480 0.375255i
9.18 0.581003 + 0.294716i 2.59259 0.979122i −0.930699 1.27132i 0.258546 0.0667815i 1.79486 + 0.195203i −0.0188644 0.000681494i −0.376856 2.29872i 3.51178 3.09380i 0.169897 + 0.0373972i
9.19 1.07325 + 0.544407i −2.77045 + 1.04630i −0.325928 0.445212i 3.94990 1.02025i −3.54299 0.385323i −2.35651 0.0851310i −0.496810 3.03041i 4.32963 3.81430i 4.79464 + 1.05538i
9.20 1.18924 + 0.603247i 0.788193 0.297671i −0.131017 0.178967i 3.38595 0.874580i 1.11692 + 0.121472i −0.168277 0.00607917i −0.479320 2.92372i −1.71841 + 1.51388i 4.55430 + 1.00248i
See next 80 embeddings (of 1568 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 346.28 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Hecke kernels

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