Properties

Label 804.2.u.a
Level 804
Weight 2
Character orbit 804.u
Analytic conductor 6.420
Analytic rank 0
Dimension 340
CM No

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Newspace parameters

Level: \( N \) = \( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 804.u (of order \(22\) and degree \(10\))

Newform invariants

Self dual: No
Analytic conductor: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(340\)
Relative dimension: \(34\) over \(\Q(\zeta_{22})\)
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 340q - 34q^{3} - 2q^{4} + 11q^{6} - 4q^{7} + 39q^{8} - 34q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 340q - 34q^{3} - 2q^{4} + 11q^{6} - 4q^{7} + 39q^{8} - 34q^{9} - 27q^{10} + 9q^{12} + 4q^{14} - 2q^{16} + 12q^{20} - 4q^{21} - 3q^{22} + 6q^{24} + 34q^{25} - 10q^{26} - 34q^{27} - 25q^{28} + 16q^{29} + 6q^{30} + 4q^{31} - 55q^{32} + 9q^{36} - 12q^{37} - 26q^{38} - 37q^{40} + 4q^{42} + 4q^{43} + 51q^{44} + 81q^{46} - 2q^{48} - 46q^{49} - 15q^{50} - 32q^{52} - 14q^{56} + 66q^{57} - 92q^{58} - 43q^{60} + 2q^{62} + 18q^{63} + 7q^{64} + 8q^{66} + 18q^{67} - 208q^{68} - 56q^{70} + 6q^{72} + 54q^{73} - 22q^{74} + 34q^{75} + 120q^{76} + 8q^{77} - 10q^{78} - 10q^{79} + 68q^{80} - 34q^{81} + 84q^{82} + 8q^{84} + 104q^{86} + 16q^{87} + 138q^{88} - 27q^{90} - 42q^{92} + 4q^{93} - 32q^{94} + 40q^{95} + 7q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.41208 0.0777432i 0.841254 0.540641i 1.98791 + 0.219558i 3.71883 + 0.534687i −1.22994 + 0.698024i −1.13953 + 2.49522i −2.79001 0.464580i 0.415415 0.909632i −5.20970 1.04413i
43.2 −1.38258 0.297428i 0.841254 0.540641i 1.82307 + 0.822439i −2.84397 0.408902i −1.32390 + 0.497268i 0.447300 0.979450i −2.27593 1.67932i 0.415415 0.909632i 3.81041 + 1.41122i
43.3 −1.37601 + 0.326492i 0.841254 0.540641i 1.78681 0.898513i 0.977951 + 0.140608i −0.981058 + 1.01859i 2.04888 4.48642i −2.16530 + 1.81974i 0.415415 0.909632i −1.39158 + 0.125815i
43.4 −1.28564 + 0.589182i 0.841254 0.540641i 1.30573 1.51495i −0.935143 0.134453i −0.763012 + 1.19072i −0.374614 + 0.820291i −0.786116 + 2.71699i 0.415415 0.909632i 1.28147 0.378111i
43.5 −1.28085 + 0.599529i 0.841254 0.540641i 1.28113 1.53581i 0.627365 + 0.0902015i −0.753386 + 1.19683i −1.21188 + 2.65366i −0.720170 + 2.73521i 0.415415 0.909632i −0.857636 + 0.260589i
43.6 −1.21380 0.725738i 0.841254 0.540641i 0.946608 + 1.76180i −1.70218 0.244737i −1.41348 + 0.0456983i −1.89274 + 4.14453i 0.129615 2.82546i 0.415415 0.909632i 1.88849 + 1.53240i
43.7 −1.08310 0.909337i 0.841254 0.540641i 0.346212 + 1.96981i 1.76201 + 0.253339i −1.40279 0.179415i 0.821285 1.79836i 1.41624 2.44832i 0.415415 0.909632i −1.67807 1.87666i
43.8 −0.910665 1.08198i 0.841254 0.540641i −0.341380 + 1.97065i −3.35521 0.482407i −1.35106 0.417881i 1.56720 3.43169i 2.44309 1.42523i 0.415415 0.909632i 2.53352 + 4.06960i
43.9 −0.893059 + 1.09656i 0.841254 0.540641i −0.404893 1.95859i −2.48843 0.357782i −0.158443 + 1.40531i 1.70447 3.73227i 2.50930 + 1.30514i 0.415415 0.909632i 2.61464 2.40919i
43.10 −0.827747 + 1.14666i 0.841254 0.540641i −0.629670 1.89829i 2.62053 + 0.376775i −0.0764124 + 1.41215i 0.207219 0.453747i 2.69791 + 0.849287i 0.415415 0.909632i −2.60117 + 2.69298i
43.11 −0.798462 1.16724i 0.841254 0.540641i −0.724915 + 1.86400i 3.02422 + 0.434817i −1.30277 0.550266i −1.21277 + 2.65559i 2.75456 0.642182i 0.415415 0.909632i −1.90719 3.87718i
43.12 −0.662980 1.24918i 0.841254 0.540641i −1.12092 + 1.65637i −0.438381 0.0630297i −1.23309 0.692445i 0.0714523 0.156459i 2.81225 + 0.302091i 0.415415 0.909632i 0.211902 + 0.589405i
43.13 −0.636052 + 1.26311i 0.841254 0.540641i −1.19088 1.60680i −3.60229 0.517931i 0.147806 + 1.40647i −0.431165 + 0.944119i 2.78702 0.482192i 0.415415 0.909632i 2.94544 4.22064i
43.14 −0.552643 + 1.30176i 0.841254 0.540641i −1.38917 1.43882i −0.663579 0.0954083i 0.238873 + 1.39389i −1.55646 + 3.40818i 2.64072 1.01322i 0.415415 0.909632i 0.490921 0.811096i
43.15 −0.167801 1.40422i 0.841254 0.540641i −1.94369 + 0.471260i −0.379042 0.0544979i −0.900343 1.09059i 0.215316 0.471476i 0.987906 + 2.65029i 0.415415 0.909632i −0.0129238 + 0.541404i
43.16 −0.145969 + 1.40666i 0.841254 0.540641i −1.95739 0.410658i 3.06720 + 0.440996i 0.637701 + 1.26227i 0.888803 1.94621i 0.863375 2.69343i 0.415415 0.909632i −1.06805 + 4.25013i
43.17 −0.0169599 1.41411i 0.841254 0.540641i −1.99942 + 0.0479663i 4.34031 + 0.624043i −0.778794 1.18046i 2.08213 4.55923i 0.101740 + 2.82660i 0.415415 0.909632i 0.808855 6.14827i
43.18 0.141967 1.40707i 0.841254 0.540641i −1.95969 0.399514i 0.394110 + 0.0566645i −0.641289 1.26046i −1.35700 + 2.97142i −0.840356 + 2.70070i 0.415415 0.909632i 0.135682 0.546496i
43.19 0.189932 + 1.40140i 0.841254 0.540641i −1.92785 + 0.532342i −1.42059 0.204250i 0.917436 + 1.07625i −0.342785 + 0.750594i −1.11219 2.60059i 0.415415 0.909632i 0.0164208 2.02961i
43.20 0.461083 + 1.33694i 0.841254 0.540641i −1.57480 + 1.23288i 2.74527 + 0.394710i 1.11069 + 0.875423i −1.09103 + 2.38902i −2.37440 1.53696i 0.415415 0.909632i 0.738094 + 3.85225i
See next 80 embeddings (of 340 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 715.34
Significant digits:
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Inner twists

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{7}^{340} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(804, [\chi])\).