# Properties

 Label 804.2.w.a Level 804 Weight 2 Character orbit 804.w Analytic conductor 6.420 Analytic rank 0 Dimension 1320 CM No

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: No Analytic conductor: $$6.41997232251$$ Analytic rank: $$0$$ Dimension: $$1320$$ Relative dimension: $$132$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$1320q - 18q^{4} - 5q^{6} - 22q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$1320q - 18q^{4} - 5q^{6} - 22q^{9} - 26q^{10} - 15q^{12} - 44q^{13} - 34q^{16} - 13q^{18} - 66q^{21} + 74q^{22} + 13q^{24} + 80q^{25} - 110q^{28} - 36q^{30} - 22q^{33} - 54q^{34} - 37q^{36} - 72q^{37} - 22q^{40} + q^{42} - 154q^{45} - 82q^{46} + 11q^{48} + 96q^{49} - 14q^{52} - 138q^{54} - 6q^{57} + 74q^{58} + 3q^{60} - 28q^{61} - 6q^{64} - 4q^{66} - 38q^{69} - 18q^{70} - 145q^{72} + 96q^{73} + 14q^{76} - 51q^{78} - 38q^{81} - 74q^{82} - 41q^{84} - 76q^{85} + 6q^{88} - 5q^{90} + 78q^{93} - 22q^{94} + 184q^{96} - 128q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
59.1 −1.41390 + 0.0296834i −1.26016 1.18828i 1.99824 0.0839387i −2.07016 + 3.22123i 1.81701 + 1.64271i −1.08648 0.156213i −2.82282 + 0.177996i 0.175983 + 2.99483i 2.83138 4.61595i
59.2 −1.41300 + 0.0585241i −1.35142 + 1.08336i 1.99315 0.165389i −0.520814 + 0.810403i 1.84615 1.60988i −3.60615 0.518486i −2.80665 + 0.350343i 0.652653 2.92815i 0.688484 1.17558i
59.3 −1.41000 + 0.109047i −1.59567 + 0.673676i 1.97622 0.307512i −1.55997 + 2.42736i 2.17644 1.12389i 3.11234 + 0.447486i −2.75294 + 0.649094i 2.09232 2.14993i 1.93486 3.59269i
59.4 −1.40973 + 0.112573i −1.72195 0.186745i 1.97465 0.317394i 1.50339 2.33932i 2.44851 + 0.0694140i 0.181474 + 0.0260921i −2.74799 + 0.669732i 2.93025 + 0.643134i −1.85603 + 3.46705i
59.5 −1.40620 0.150299i −1.00157 1.41310i 1.95482 + 0.422702i 1.15920 1.80375i 1.19603 + 2.13764i 1.19955 + 0.172470i −2.68534 0.888214i −0.993709 + 2.83064i −1.90118 + 2.36222i
59.6 −1.40156 0.188750i −0.286791 + 1.70814i 1.92875 + 0.529089i −0.123966 + 0.192895i 0.724367 2.33993i 2.29054 + 0.329330i −2.60339 1.10560i −2.83550 0.979759i 0.210155 0.246956i
59.7 −1.39797 0.213761i 0.286791 1.70814i 1.90861 + 0.597661i −0.123966 + 0.192895i −0.766058 + 2.32662i −2.29054 0.329330i −2.54042 1.24350i −2.83550 0.979759i 0.214534 0.243162i
59.8 −1.39159 0.251963i 1.00157 + 1.41310i 1.87303 + 0.701256i 1.15920 1.80375i −1.03773 2.21881i −1.19955 0.172470i −2.42979 1.44779i −0.993709 + 2.83064i −2.06761 + 2.21800i
59.9 −1.37889 + 0.314084i 1.66550 + 0.475496i 1.80270 0.866178i 0.120449 0.187422i −2.44590 0.132550i −4.49296 0.645990i −2.21368 + 1.76057i 2.54781 + 1.58388i −0.107220 + 0.296266i
59.10 −1.37660 + 0.323993i 1.49919 0.867419i 1.79006 0.892017i 1.25921 1.95938i −1.78275 + 1.67982i 1.82139 + 0.261876i −2.17519 + 1.80792i 1.49517 2.60086i −1.09861 + 3.10525i
59.11 −1.34827 0.426823i 1.26016 + 1.18828i 1.63564 + 1.15094i −2.07016 + 3.22123i −1.19184 2.13998i 1.08648 + 0.156213i −1.71403 2.24991i 0.175983 + 2.99483i 4.16602 3.45948i
59.12 −1.33982 + 0.452629i −0.379728 + 1.68991i 1.59025 1.21289i 2.40444 3.74138i −0.256136 2.43606i −1.84812 0.265720i −1.58167 + 2.34485i −2.71161 1.28341i −1.52806 + 6.10110i
59.13 −1.33939 + 0.453903i 1.49716 + 0.870933i 1.58794 1.21591i −0.946301 + 1.47247i −2.40060 0.486956i 2.15537 + 0.309895i −1.57497 + 2.34935i 1.48295 + 2.60784i 0.599108 2.40175i
59.14 −1.33928 0.454242i 1.35142 1.08336i 1.58733 + 1.21671i −0.520814 + 0.810403i −2.30203 + 0.837052i 3.60615 + 0.518486i −1.57319 2.35055i 0.652653 2.92815i 1.06563 0.848778i
59.15 −1.32217 0.501873i 1.59567 0.673676i 1.49625 + 1.32712i −1.55997 + 2.42736i −2.44784 + 0.0898878i −3.11234 0.447486i −1.31224 2.50560i 2.09232 2.14993i 3.28076 2.42646i
59.16 −1.32091 0.505179i 1.72195 + 0.186745i 1.48959 + 1.33459i 1.50339 2.33932i −2.18020 1.11657i −0.181474 0.0260921i −1.29340 2.51537i 2.93025 + 0.643134i −3.16762 + 2.33055i
59.17 −1.30562 + 0.543469i −1.02027 1.39966i 1.40928 1.41913i 0.375021 0.583544i 2.09275 + 1.27294i 1.49818 + 0.215405i −1.06874 + 2.61874i −0.918112 + 2.85606i −0.172497 + 0.965699i
59.18 −1.26563 + 0.631011i 1.52145 0.827758i 1.20365 1.59726i −1.73166 + 2.69452i −1.40327 + 2.00769i −2.30452 0.331340i −0.515490 + 2.78106i 1.62963 2.51879i 0.491374 4.50296i
59.19 −1.25478 + 0.652334i 0.744060 + 1.56409i 1.14892 1.63706i 0.555604 0.864536i −1.95394 1.47721i 3.61918 + 0.520359i −0.373727 + 2.80363i −1.89275 + 2.32755i −0.133192 + 1.44724i
59.20 −1.23455 0.689841i −1.66550 0.475496i 1.04824 + 1.70329i 0.120449 0.187422i 1.72814 + 1.73596i 4.49296 + 0.645990i −0.119106 2.82592i 2.54781 + 1.58388i −0.277992 + 0.148292i
See next 80 embeddings (of 1320 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 695.132 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 $$S_{2}^{\mathrm{new}}(804, [\chi])$$.