# Properties

 Label 804.2.bf.a Level 804 Weight 2 Character orbit 804.bf Analytic conductor 6.420 Analytic rank 0 Dimension 680 CM No

# Related objects

## Newspace parameters

 Level: $$N$$ = $$804 = 2^{2} \cdot 3 \cdot 67$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 804.bf (of order $$66$$ and degree $$20$$)

## Newform invariants

 Self dual: No Analytic conductor: $$6.41997232251$$ Analytic rank: $$0$$ Dimension: $$680$$ Relative dimension: $$34$$ over $$\Q(\zeta_{66})$$ Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$680q - 68q^{3} + 2q^{4} - 11q^{6} + 4q^{7} - 39q^{8} - 68q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$680q - 68q^{3} + 2q^{4} - 11q^{6} + 4q^{7} - 39q^{8} - 68q^{9} + 39q^{10} - 9q^{12} - 6q^{13} - 10q^{14} + 2q^{16} + 12q^{20} + 4q^{21} + 33q^{22} - 6q^{24} + 68q^{25} + 19q^{26} - 68q^{27} - 92q^{28} + 8q^{29} + 6q^{30} + 2q^{31} + 40q^{32} - 9q^{36} - 12q^{37} - 4q^{38} - 6q^{39} + 37q^{40} - 10q^{42} - 4q^{43} + 159q^{44} - 93q^{46} + 2q^{48} + 46q^{49} + 6q^{50} - 28q^{52} + 17q^{56} - 66q^{57} + 92q^{58} - 98q^{60} - 6q^{61} + 34q^{62} - 18q^{63} - 49q^{64} + 22q^{66} - 18q^{67} + 208q^{68} + 56q^{70} - 36q^{71} - 6q^{72} - 72q^{73} - 11q^{74} + 68q^{75} + 162q^{76} + 4q^{77} + 19q^{78} + 28q^{79} - 41q^{80} - 68q^{81} - 84q^{82} + 12q^{83} + 7q^{84} - 77q^{86} + 8q^{87} - 132q^{88} + 39q^{90} - 186q^{92} + 2q^{93} - 16q^{94} + 20q^{95} - 15q^{96} - 18q^{97} + 65q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −1.41094 0.0961208i −0.959493 0.281733i 1.98152 + 0.271242i 2.15142 + 1.86421i 1.32671 + 0.489736i −0.0281804 + 0.591578i −2.76974 0.573173i 0.841254 + 0.540641i −2.85634 2.83710i
7.2 −1.36544 + 0.368200i −0.959493 0.281733i 1.72886 1.00551i −0.923471 0.800192i 1.41386 + 0.0314034i −0.158881 + 3.33532i −1.99042 + 2.00953i 0.841254 + 0.540641i 1.55558 + 0.752593i
7.3 −1.34464 0.438103i −0.959493 0.281733i 1.61613 + 1.17819i −0.767783 0.665288i 1.16675 + 0.799187i 0.217582 4.56760i −1.65695 2.29227i 0.841254 + 0.540641i 0.740929 + 1.23094i
7.4 −1.33247 0.473847i −0.959493 0.281733i 1.55094 + 1.26277i −2.82961 2.45187i 1.14500 + 0.830052i −0.0304265 + 0.638731i −1.46822 2.41751i 0.841254 + 0.540641i 2.60855 + 4.60783i
7.5 −1.32744 + 0.487752i −0.959493 0.281733i 1.52420 1.29492i 2.75137 + 2.38407i 1.41109 0.0940111i 0.196492 4.12487i −1.39168 + 2.46236i 0.841254 + 0.540641i −4.81511 1.82273i
7.6 −1.30759 0.538708i −0.959493 0.281733i 1.41959 + 1.40882i 2.42058 + 2.09744i 1.10285 + 0.885278i −0.217948 + 4.57528i −1.09729 2.60690i 0.841254 + 0.540641i −2.03521 4.04658i
7.7 −1.16345 + 0.803974i −0.959493 0.281733i 0.707250 1.87077i 0.406963 + 0.352635i 1.34283 0.443625i 0.0731812 1.53626i 0.681202 + 2.74517i 0.841254 + 0.540641i −0.756992 0.0830872i
7.8 −1.15194 0.820385i −0.959493 0.281733i 0.653935 + 1.89007i −1.53250 1.32792i 0.874150 + 1.11169i −0.182174 + 3.82430i 0.797292 2.71373i 0.841254 + 0.540641i 0.675944 + 2.78693i
7.9 −1.07757 + 0.915880i −0.959493 0.281733i 0.322326 1.97386i −0.775668 0.672120i 1.29196 0.575194i −0.0588862 + 1.23617i 1.46049 + 2.42218i 0.841254 + 0.540641i 1.45142 + 0.0138391i
7.10 −0.967854 1.03114i −0.959493 0.281733i −0.126517 + 1.99599i 0.342563 + 0.296833i 0.638142 + 1.26205i 0.0123453 0.259160i 2.18061 1.80137i 0.841254 + 0.540641i −0.0254738 0.640523i
7.11 −0.774785 + 1.18309i −0.959493 0.281733i −0.799417 1.83328i 2.32348 + 2.01331i 1.07672 0.916887i −0.144795 + 3.03962i 2.78832 + 0.474616i 0.841254 + 0.540641i −4.18213 + 1.18901i
7.12 −0.687234 1.23601i −0.959493 0.281733i −1.05542 + 1.69885i −1.21985 1.05701i 0.311173 + 1.37955i 0.201271 4.22520i 2.82511 + 0.136998i 0.841254 + 0.540641i −0.468144 + 2.23415i
7.13 −0.546520 1.30434i −0.959493 0.281733i −1.40263 + 1.42570i 3.24307 + 2.81014i 0.156906 + 1.40548i 0.0543857 1.14170i 2.62617 + 1.05034i 0.841254 + 0.540641i 1.89299 5.76588i
7.14 −0.530629 + 1.31089i −0.959493 0.281733i −1.43687 1.39119i −3.07171 2.66165i 0.878455 1.10829i −0.0582013 + 1.22180i 2.58614 1.14537i 0.841254 + 0.540641i 5.11907 2.61432i
7.15 −0.518960 + 1.31555i −0.959493 0.281733i −1.46136 1.36544i −0.507778 0.439992i 0.868573 1.11606i 0.160858 3.37682i 2.55470 1.21389i 0.841254 + 0.540641i 0.842349 0.439671i
7.16 −0.289585 1.38425i −0.959493 0.281733i −1.83228 + 0.801716i −0.243036 0.210592i −0.112132 + 1.40976i −0.0786161 + 1.65036i 1.64037 + 2.30416i 0.841254 + 0.540641i −0.221132 + 0.397407i
7.17 0.0531079 + 1.41322i −0.959493 0.281733i −1.99436 + 0.150106i 1.40058 + 1.21361i 0.347192 1.37093i 0.0696355 1.46183i −0.318048 2.81049i 0.841254 + 0.540641i −1.64071 + 2.04377i
7.18 0.130801 1.40815i −0.959493 0.281733i −1.96578 0.368376i −2.24944 1.94915i −0.522225 + 1.31426i −0.0649402 + 1.36326i −0.775856 + 2.71994i 0.841254 + 0.540641i −3.03893 + 2.91260i
7.19 0.269053 1.38838i −0.959493 0.281733i −1.85522 0.747098i 1.87962 + 1.62870i −0.649308 + 1.25634i 0.129508 2.71872i −1.53641 + 2.37475i 0.841254 + 0.540641i 2.76699 2.17143i
7.20 0.335515 + 1.37384i −0.959493 0.281733i −1.77486 + 0.921885i −2.14113 1.85530i 0.0651310 1.41271i 0.0976058 2.04900i −1.86201 2.12906i 0.841254 + 0.540641i 1.83050 3.56404i
See next 80 embeddings (of 680 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 787.34 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 $$T_{7}^{680} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(804, [\chi])$$.