# Properties

 Label 804.2.s.b Level 804 Weight 2 Character orbit 804.s Analytic conductor 6.420 Analytic rank 0 Dimension 200 CM no Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$6.41997232251$$ Analytic rank: $$0$$ Dimension: $$200$$ Relative dimension: $$20$$ over $$\Q(\zeta_{22})$$ Coefficient ring index: multiple of None Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$200q - 10q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$200q - 10q^{9} + 2q^{15} + 6q^{19} - 10q^{21} - 20q^{25} - 44q^{31} - 5q^{33} + 78q^{39} - 22q^{43} - 22q^{45} - 16q^{49} + 36q^{55} + 66q^{57} + 176q^{61} + 132q^{63} + 46q^{67} - 26q^{73} - 165q^{75} - 44q^{79} + 42q^{81} - 66q^{87} - 20q^{91} + 84q^{93} - 55q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 0 −1.72660 0.137338i 0 2.48625 0.730028i 0 3.64531 + 3.15868i 0 2.96228 + 0.474256i 0
5.2 0 −1.58909 + 0.689041i 0 3.06595 0.900243i 0 0.591666 + 0.512682i 0 2.05045 2.18990i 0
5.3 0 −1.57919 0.711443i 0 −0.583807 + 0.171421i 0 −0.967947 0.838730i 0 1.98770 + 2.24701i 0
5.4 0 −1.56138 + 0.749732i 0 −3.06595 + 0.900243i 0 0.591666 + 0.512682i 0 1.87580 2.34123i 0
5.5 0 −1.53236 0.807386i 0 −3.45254 + 1.01376i 0 −0.403229 0.349400i 0 1.69626 + 2.47441i 0
5.6 0 −1.17979 1.26810i 0 2.83063 0.831147i 0 −2.86249 2.48036i 0 −0.216176 + 2.99220i 0
5.7 0 −1.02689 + 1.39481i 0 −2.48625 + 0.730028i 0 3.64531 + 3.15868i 0 −0.891005 2.86463i 0
5.8 0 −0.786808 1.54303i 0 0.966140 0.283684i 0 1.54930 + 1.34247i 0 −1.76187 + 2.42813i 0
5.9 0 −0.496478 + 1.65937i 0 0.583807 0.171421i 0 −0.967947 0.838730i 0 −2.50702 1.64768i 0
5.10 0 −0.393301 + 1.68681i 0 3.45254 1.01376i 0 −0.403229 0.349400i 0 −2.69063 1.32685i 0
5.11 0 0.0434100 1.73151i 0 −1.09364 + 0.321121i 0 2.09055 + 1.81147i 0 −2.99623 0.150329i 0
5.12 0 0.185769 + 1.72206i 0 −2.83063 + 0.831147i 0 −2.86249 2.48036i 0 −2.93098 + 0.639810i 0
5.13 0 0.429842 1.67787i 0 −3.03558 + 0.891328i 0 −1.25477 1.08727i 0 −2.63047 1.44244i 0
5.14 0 0.575290 1.63372i 0 2.05430 0.603197i 0 −2.91035 2.52183i 0 −2.33808 1.87973i 0
5.15 0 0.650892 + 1.60510i 0 −0.966140 + 0.283684i 0 1.54930 + 1.34247i 0 −2.15268 + 2.08949i 0
5.16 0 1.33701 + 1.10109i 0 1.09364 0.321121i 0 2.09055 + 1.81147i 0 0.575207 + 2.94434i 0
5.17 0 1.48256 0.895546i 0 −2.22357 + 0.652900i 0 2.01808 + 1.74867i 0 1.39599 2.65541i 0
5.18 0 1.54953 + 0.773916i 0 3.03558 0.891328i 0 −1.25477 1.08727i 0 1.80211 + 2.39842i 0
5.19 0 1.61142 + 0.635084i 0 −2.05430 + 0.603197i 0 −2.91035 2.52183i 0 2.19334 + 2.04677i 0
5.20 0 1.64768 0.533989i 0 2.22357 0.652900i 0 2.01808 + 1.74867i 0 2.42971 1.75969i 0
See next 80 embeddings (of 200 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 713.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
67.f odd 22 1 inner
201.j even 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 804.2.s.b 200
3.b odd 2 1 inner 804.2.s.b 200
67.f odd 22 1 inner 804.2.s.b 200
201.j even 22 1 inner 804.2.s.b 200

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
804.2.s.b 200 1.a even 1 1 trivial
804.2.s.b 200 3.b odd 2 1 inner
804.2.s.b 200 67.f odd 22 1 inner
804.2.s.b 200 201.j even 22 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{200} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(804, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database