Properties

Label 804.2.l.a
Level 804
Weight 2
Character orbit 804.l
Analytic conductor 6.420
Analytic rank 0
Dimension 264
CM No

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Newspace parameters

Level: \( N \) = \( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 804.l (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(264\)
Relative dimension: \(132\) over \(\Q(\zeta_{6})\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 264q - 2q^{4} - 3q^{6} - 12q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 264q - 2q^{4} - 3q^{6} - 12q^{9} + 2q^{10} + 2q^{12} - 18q^{16} + q^{18} - 4q^{22} - 248q^{25} + 7q^{30} - 38q^{34} + 13q^{36} + 4q^{37} + 48q^{40} - 12q^{42} + 48q^{45} - 2q^{46} - 11q^{48} + 128q^{49} + 32q^{52} - 8q^{54} - 2q^{57} - 32q^{58} + 68q^{60} - 8q^{61} + 4q^{64} + 30q^{66} - 40q^{69} - 140q^{70} + 108q^{72} - 4q^{73} - 120q^{76} + 44q^{78} - 28q^{81} - 64q^{82} + 33q^{84} + 16q^{85} - 14q^{88} + 42q^{90} + 16q^{93} - 12q^{94} - 4q^{96} + 20q^{97} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
431.1 −1.41407 0.0202093i −1.20523 1.24395i 1.99918 + 0.0571548i 3.26426i 1.67914 + 1.78339i −0.138760 + 0.0801133i −2.82583 0.121223i −0.0948278 + 2.99850i −0.0659686 + 4.61589i
431.2 −1.41390 + 0.0296860i −1.00788 + 1.40861i 1.99824 0.0839461i 0.377411i 1.38323 2.02155i −3.78586 + 2.18577i −2.82282 + 0.178011i −0.968338 2.83942i 0.0112038 + 0.533622i
431.3 −1.41324 0.0525133i −1.72604 + 0.144124i 1.99448 + 0.148428i 3.64727i 2.44688 0.113041i −0.289615 + 0.167210i −2.81089 0.314501i 2.95846 0.497529i 0.191530 5.15446i
431.4 −1.41305 + 0.0574626i 1.12603 1.31608i 1.99340 0.162395i 3.12863i −1.51551 + 1.92438i −3.36179 + 1.94093i −2.80743 + 0.344017i −0.464114 2.96388i −0.179779 4.42090i
431.5 −1.41271 + 0.0652340i 0.500155 + 1.65827i 1.99149 0.184313i 0.631752i −0.814749 2.31002i 2.68471 1.55002i −2.80137 + 0.390294i −2.49969 + 1.65878i −0.0412117 0.892481i
431.6 −1.40119 0.191474i −0.473255 1.66614i 1.92668 + 0.536583i 0.531500i 0.344098 + 2.42520i 0.910512 0.525684i −2.59690 1.12076i −2.55206 + 1.57702i −0.101768 + 0.744733i
431.7 −1.39862 + 0.209424i 1.33282 1.10616i 1.91228 0.585811i 2.33442i −1.63245 + 1.82622i 1.17857 0.680446i −2.55188 + 1.21981i 0.552819 2.94863i 0.488885 + 3.26497i
431.8 −1.39364 + 0.240373i −1.35771 + 1.07547i 1.88444 0.669985i 4.20267i 1.63364 1.82517i 3.17693 1.83420i −2.46518 + 1.38668i 0.686739 2.92034i 1.01021 + 5.85699i
431.9 −1.39349 0.241196i 1.40279 + 1.01597i 1.88365 + 0.672210i 2.69591i −1.70973 1.75409i −2.30125 + 1.32862i −2.46272 1.39105i 0.935617 + 2.85037i 0.650241 3.75673i
431.10 −1.38134 0.303157i 0.740294 + 1.56587i 1.81619 + 0.837524i 3.95379i −0.547892 2.38743i −2.00371 + 1.15684i −2.25488 1.70749i −1.90393 + 2.31842i −1.19862 + 5.46152i
431.11 −1.37942 0.311775i 0.422580 1.67971i 1.80559 + 0.860137i 2.80374i −1.10661 + 2.18527i 4.57629 2.64212i −2.22250 1.74943i −2.64285 1.41963i 0.874136 3.86753i
431.12 −1.37670 + 0.323579i 1.62001 + 0.612827i 1.79059 0.890942i 0.604920i −2.42857 0.319475i 2.48563 1.43508i −2.17681 + 1.80596i 2.24889 + 1.98558i −0.195740 0.832792i
431.13 −1.35424 0.407489i 1.72726 0.128753i 1.66791 + 1.10367i 1.48944i −2.39158 0.529476i −2.25234 + 1.30039i −1.80900 2.17428i 2.96685 0.444780i −0.606929 + 2.01705i
431.14 −1.32505 + 0.494196i −0.696446 1.58586i 1.51154 1.30967i 1.99192i 1.70656 + 1.75718i −1.23529 + 0.713198i −1.35564 + 2.48239i −2.02993 + 2.20894i −0.984400 2.63941i
431.15 −1.31532 + 0.519539i −1.71904 0.211862i 1.46016 1.36673i 0.428467i 2.37117 0.614444i −1.49589 + 0.863651i −1.21051 + 2.55630i 2.91023 + 0.728399i 0.222606 + 0.563574i
431.16 −1.31048 0.531632i −1.38818 + 1.03584i 1.43473 + 1.39339i 1.57106i 2.36987 0.619443i 0.285742 0.164973i −1.13942 2.58877i 0.854091 2.87585i 0.835229 2.05885i
431.17 −1.27761 0.606402i −0.369021 + 1.69228i 1.26455 + 1.54948i 0.628413i 1.49767 1.93830i 1.58089 0.912727i −0.675992 2.74646i −2.72765 1.24898i −0.381071 + 0.802864i
431.18 −1.25248 0.656737i −1.29766 1.14721i 1.13739 + 1.64509i 2.89696i 0.871868 + 2.28907i −1.32141 + 0.762919i −0.344163 2.80741i 0.367820 + 2.97737i 1.90254 3.62837i
431.19 −1.24697 + 0.667126i 1.63732 + 0.564973i 1.10988 1.66378i 1.17393i −2.41860 + 0.387791i −2.79224 + 1.61210i −0.274046 + 2.81512i 2.36161 + 1.85008i 0.783159 + 1.46386i
431.20 −1.23331 0.692067i −1.71207 0.262337i 1.04209 + 1.70706i 2.53500i 1.92995 + 1.50841i −3.95663 + 2.28436i −0.103813 2.82652i 2.86236 + 0.898278i −1.75439 + 3.12643i
See next 80 embeddings (of 264 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 707.132
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 \(S_{2}^{\mathrm{new}}(804, [\chi])\).