# Properties

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

# Related objects

## 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.

 $$\operatorname{Tr}(f)(q) =$$ $$340q + 34q^{3} - 2q^{4} - 11q^{6} + 4q^{7} + 27q^{8} - 34q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$340q + 34q^{3} - 2q^{4} - 11q^{6} + 4q^{7} + 27q^{8} - 34q^{9} - 27q^{10} - 9q^{12} + 4q^{14} - 2q^{16} - 12q^{20} - 4q^{21} - 3q^{22} + 6q^{24} + 34q^{25} - 10q^{26} + 34q^{27} + 47q^{28} + 16q^{29} - 6q^{30} - 4q^{31} - 55q^{32} + 9q^{36} - 12q^{37} + 26q^{38} - 37q^{40} - 4q^{42} - 4q^{43} - 51q^{44} - 103q^{46} + 2q^{48} - 46q^{49} - 51q^{50} + 32q^{52} - 14q^{56} + 66q^{57} + 92q^{58} + 67q^{60} + 2q^{62} - 18q^{63} + 7q^{64} - 8q^{66} - 18q^{67} - 208q^{68} + 56q^{70} - 6q^{72} + 54q^{73} + 22q^{74} - 34q^{75} - 56q^{76} + 8q^{77} + 10q^{78} + 10q^{79} - 68q^{80} - 34q^{81} + 84q^{82} + 8q^{84} + 104q^{86} - 16q^{87} - 82q^{88} - 27q^{90} + 134q^{92} + 4q^{93} + 32q^{94} - 40q^{95} - 7q^{98} + O(q^{100})$$

## 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 −1.41407 + 0.0203147i −0.841254 + 0.540641i 1.99917 0.0574526i −3.35521 0.482407i 1.17861 0.781592i −1.56720 + 3.43169i −2.82580 + 0.121854i 0.415415 0.909632i 4.75430 + 0.613996i
43.2 −1.40503 + 0.160944i −0.841254 + 0.540641i 1.94819 0.452262i 3.02422 + 0.434817i 1.09497 0.895009i 1.21277 2.65559i −2.66447 + 0.948990i 0.415415 0.909632i −4.31908 0.124198i
43.3 −1.39651 0.223063i −0.841254 + 0.540641i 1.90049 + 0.623021i 1.76201 + 0.253339i 1.29542 0.567358i −0.821285 + 1.79836i −2.51508 1.29398i 0.415415 0.909632i −2.40416 0.746831i
43.4 −1.37823 + 0.316994i −0.841254 + 0.540641i 1.79903 0.873781i −0.438381 0.0630297i 0.988060 1.01180i −0.0714523 + 0.156459i −2.20249 + 1.77455i 0.415415 0.909632i 0.624169 0.0520948i
43.5 −1.34334 0.442069i −0.841254 + 0.540641i 1.60915 + 1.18770i −1.70218 0.244737i 1.36909 0.354375i 1.89274 4.14453i −1.63660 2.30685i 0.415415 0.909632i 2.17843 + 1.08125i
43.6 −1.17113 + 0.792755i −0.841254 + 0.540641i 0.743078 1.85683i −0.379042 0.0544979i 0.556619 1.30007i −0.215316 + 0.471476i 0.601776 + 2.76367i 0.415415 0.909632i 0.487109 0.236663i
43.7 −1.13018 0.850112i −0.841254 + 0.540641i 0.554618 + 1.92156i −2.84397 0.408902i 1.41037 + 0.104138i −0.447300 + 0.979450i 1.00673 2.64320i 0.415415 0.909632i 2.86659 + 2.87983i
43.8 −1.07982 + 0.913229i −0.841254 + 0.540641i 0.332026 1.97225i 4.34031 + 0.624043i 0.414674 1.35205i −2.08213 + 4.55923i 1.44258 + 2.43289i 0.415415 0.909632i −5.25666 + 3.28985i
43.9 −0.983467 1.01626i −0.841254 + 0.540641i −0.0655858 + 1.99892i 3.71883 + 0.534687i 1.37678 + 0.323233i 1.13953 2.49522i 2.09594 1.89922i 0.415415 0.909632i −3.11397 4.30516i
43.10 −0.970424 + 1.02873i −0.841254 + 0.540641i −0.116555 1.99660i 0.394110 + 0.0566645i 0.260201 1.39007i 1.35700 2.97142i 2.16706 + 1.81765i 0.415415 0.909632i −0.440746 + 0.350443i
43.11 −0.654349 1.25373i −0.841254 + 0.540641i −1.14366 + 1.64075i 0.977951 + 0.140608i 1.22829 + 0.700934i −2.04888 + 4.48642i 2.80540 + 0.360211i 0.415415 0.909632i −0.463637 1.31809i
43.12 −0.481489 + 1.32972i −0.841254 + 0.540641i −1.53634 1.28050i −3.58365 0.515251i −0.313849 1.37895i 0.509125 1.11483i 2.44244 1.42636i 0.415415 0.909632i 2.41063 4.51718i
43.13 −0.413712 + 1.35235i −0.841254 + 0.540641i −1.65768 1.11896i 0.112828 + 0.0162222i −0.383097 1.36134i −0.733664 + 1.60650i 2.19903 1.77884i 0.415415 0.909632i −0.0686163 + 0.145871i
43.14 −0.396640 1.35745i −0.841254 + 0.540641i −1.68535 + 1.07684i −0.935143 0.134453i 1.06757 + 0.927522i 0.374614 0.820291i 2.13024 + 1.86067i 0.415415 0.909632i 0.188401 + 1.32274i
43.15 −0.385682 1.36061i −0.841254 + 0.540641i −1.70250 + 1.04952i 0.627365 + 0.0902015i 1.06006 + 0.936100i 1.21188 2.65366i 2.08461 + 1.91165i 0.415415 0.909632i −0.119235 0.888386i
43.16 0.0430767 + 1.41356i −0.841254 + 0.540641i −1.99629 + 0.121783i 1.24980 + 0.179694i −0.800465 1.16587i −0.832575 + 1.82308i −0.258141 2.81662i 0.415415 0.909632i −0.200171 + 1.77440i
43.17 0.208996 + 1.39869i −0.841254 + 0.540641i −1.91264 + 0.584640i 4.20210 + 0.604171i −0.932005 1.06366i 0.989284 2.16623i −1.21746 2.55300i 0.415415 0.909632i 0.0331777 + 6.00369i
43.18 0.243897 1.39302i −0.841254 + 0.540641i −1.88103 0.679507i −2.48843 0.357782i 0.547946 + 1.30375i −1.70447 + 3.73227i −1.40535 + 2.45459i 0.415415 0.909632i −1.10532 + 3.37917i
43.19 0.302140 + 1.38156i −0.841254 + 0.540641i −1.81742 + 0.834849i −1.61949 0.232847i −1.00110 0.998894i 1.26738 2.77517i −1.70251 2.25864i 0.415415 0.909632i −0.167619 2.30778i
43.20 0.324531 1.37647i −0.841254 + 0.540641i −1.78936 0.893416i 2.62053 + 0.376775i 0.471165 + 1.33342i −0.207219 + 0.453747i −1.81047 + 2.17307i 0.415415 0.909632i 1.36906 3.48481i
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: 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

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