# Properties

 Label 804.2.o.d Level 804 Weight 2 Character orbit 804.o Analytic conductor 6.420 Analytic rank 0 Dimension 36 CM No

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: No Analytic conductor: $$6.41997232251$$ Analytic rank: $$0$$ Dimension: $$36$$ Relative dimension: $$18$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$36q + 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q + 4q^{9} - 36q^{13} + 18q^{15} + 16q^{21} + 76q^{25} + 6q^{31} + 4q^{33} + 42q^{37} - 21q^{39} + 2q^{49} + 18q^{51} + 20q^{55} + 18q^{57} - 24q^{61} - 12q^{63} - 8q^{67} + 3q^{69} + 14q^{73} + 72q^{79} - 12q^{81} - 18q^{85} - 21q^{87} - 68q^{91} + 9q^{93} - 48q^{97} + 15q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
365.1 0 −1.72171 0.188968i 0 3.09374 0 −1.38610 0.800262i 0 2.92858 + 0.650697i 0
365.2 0 −1.71192 + 0.263286i 0 −0.454473 0 −0.768091 0.443457i 0 2.86136 0.901449i 0
365.3 0 −1.53706 + 0.798403i 0 −3.20386 0 2.52764 + 1.45934i 0 1.72511 2.45439i 0
365.4 0 −1.41646 0.996822i 0 0.835859 0 2.58614 + 1.49311i 0 1.01269 + 2.82391i 0
365.5 0 −1.34511 1.09118i 0 −3.88149 0 −3.35972 1.93974i 0 0.618641 + 2.93552i 0
365.6 0 −0.870385 + 1.49747i 0 −2.31311 0 −0.381576 0.220303i 0 −1.48486 2.60676i 0
365.7 0 −0.869231 + 1.49814i 0 2.66956 0 0.767031 + 0.442845i 0 −1.48888 2.60447i 0
365.8 0 −0.536215 + 1.64696i 0 1.65208 0 −3.18562 1.83922i 0 −2.42495 1.76625i 0
365.9 0 −0.355178 1.69524i 0 −3.60368 0 3.20029 + 1.84769i 0 −2.74770 + 1.20423i 0
365.10 0 0.355178 1.69524i 0 3.60368 0 3.20029 + 1.84769i 0 −2.74770 1.20423i 0
365.11 0 0.536215 + 1.64696i 0 −1.65208 0 −3.18562 1.83922i 0 −2.42495 + 1.76625i 0
365.12 0 0.869231 + 1.49814i 0 −2.66956 0 0.767031 + 0.442845i 0 −1.48888 + 2.60447i 0
365.13 0 0.870385 + 1.49747i 0 2.31311 0 −0.381576 0.220303i 0 −1.48486 + 2.60676i 0
365.14 0 1.34511 1.09118i 0 3.88149 0 −3.35972 1.93974i 0 0.618641 2.93552i 0
365.15 0 1.41646 0.996822i 0 −0.835859 0 2.58614 + 1.49311i 0 1.01269 2.82391i 0
365.16 0 1.53706 + 0.798403i 0 3.20386 0 2.52764 + 1.45934i 0 1.72511 + 2.45439i 0
365.17 0 1.71192 + 0.263286i 0 0.454473 0 −0.768091 0.443457i 0 2.86136 + 0.901449i 0
365.18 0 1.72171 0.188968i 0 −3.09374 0 −1.38610 0.800262i 0 2.92858 0.650697i 0
641.1 0 −1.72171 + 0.188968i 0 3.09374 0 −1.38610 + 0.800262i 0 2.92858 0.650697i 0
641.2 0 −1.71192 0.263286i 0 −0.454473 0 −0.768091 + 0.443457i 0 2.86136 + 0.901449i 0
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 641.18 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 $$T_{5}^{18} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(804, [\chi])$$.