# Properties

 Label 804.2.ba.b Level 804 Weight 2 Character orbit 804.ba Analytic conductor 6.420 Analytic rank 0 Dimension 440 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.ba (of order $$66$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.41997232251$$ Analytic rank: $$0$$ Dimension: $$440$$ Relative dimension: $$22$$ over $$\Q(\zeta_{66})$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$440q - 12q^{7} + 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$440q - 12q^{7} + 4q^{9} - 2q^{15} - 10q^{19} + 22q^{21} - 68q^{25} + 50q^{31} + 11q^{33} - 22q^{37} - 45q^{39} + 22q^{43} + 22q^{45} - 18q^{49} - 6q^{51} + 126q^{55} - 183q^{57} - 56q^{61} - 141q^{63} - 12q^{67} + 33q^{69} + 356q^{73} + 165q^{75} + 228q^{79} + 24q^{81} - 6q^{85} + 75q^{87} - 4q^{91} - 75q^{93} + 12q^{97} + 88q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
41.1 0 −1.70276 + 0.317213i 0 0.152781 1.06262i 0 0.324444 + 3.39773i 0 2.79875 1.08027i 0
41.2 0 −1.69899 0.336812i 0 −0.202812 + 1.41059i 0 −0.147283 1.54242i 0 2.77312 + 1.14448i 0
41.3 0 −1.68954 0.381398i 0 0.606307 4.21696i 0 −0.295017 3.08956i 0 2.70907 + 1.28877i 0
41.4 0 −1.47255 + 0.911919i 0 −0.412679 + 2.87025i 0 −0.273979 2.86924i 0 1.33681 2.68569i 0
41.5 0 −1.32247 1.11851i 0 0.102406 0.712247i 0 0.0729329 + 0.763788i 0 0.497878 + 2.95840i 0
41.6 0 −1.19819 + 1.25074i 0 0.253886 1.76582i 0 −0.160706 1.68299i 0 −0.128702 2.99724i 0
41.7 0 −1.14236 1.30193i 0 −0.569745 + 3.96266i 0 0.363865 + 3.81057i 0 −0.390019 + 2.97454i 0
41.8 0 −1.05930 + 1.37036i 0 −0.533599 + 3.71127i 0 0.465124 + 4.87100i 0 −0.755762 2.90324i 0
41.9 0 −0.709719 1.57997i 0 0.569745 3.96266i 0 0.363865 + 3.81057i 0 −1.99260 + 2.24267i 0
41.10 0 −0.468055 1.66761i 0 −0.102406 + 0.712247i 0 0.0729329 + 0.763788i 0 −2.56185 + 1.56107i 0
41.11 0 −0.313881 + 1.70337i 0 0.391534 2.72318i 0 0.116340 + 1.21837i 0 −2.80296 1.06931i 0
41.12 0 −0.0450273 + 1.73147i 0 −0.306661 + 2.13288i 0 −0.233782 2.44828i 0 −2.99595 0.155926i 0
41.13 0 0.354927 1.69530i 0 −0.606307 + 4.21696i 0 −0.295017 3.08956i 0 −2.74805 1.20341i 0
41.14 0 0.399410 1.68537i 0 0.202812 1.41059i 0 −0.147283 1.54242i 0 −2.68094 1.34631i 0
41.15 0 0.774432 + 1.54928i 0 −0.177548 + 1.23488i 0 0.271050 + 2.83857i 0 −1.80051 + 2.39962i 0
41.16 0 0.995897 1.41711i 0 −0.152781 + 1.06262i 0 0.324444 + 3.39773i 0 −1.01638 2.82258i 0
41.17 0 1.08756 + 1.34804i 0 0.177548 1.23488i 0 0.271050 + 2.83857i 0 −0.634425 + 2.93215i 0
41.18 0 1.44123 0.960654i 0 0.412679 2.87025i 0 −0.273979 2.86924i 0 1.15429 2.76905i 0
41.19 0 1.59370 + 0.678318i 0 0.306661 2.13288i 0 −0.233782 2.44828i 0 2.07977 + 2.16207i 0
41.20 0 1.63546 0.570332i 0 −0.253886 + 1.76582i 0 −0.160706 1.68299i 0 2.34944 1.86551i 0
See next 80 embeddings (of 440 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 785.22 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.h odd 66 1 inner
201.p even 66 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 804.2.ba.b 440
3.b odd 2 1 inner 804.2.ba.b 440
67.h odd 66 1 inner 804.2.ba.b 440
201.p even 66 1 inner 804.2.ba.b 440

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
804.2.ba.b 440 1.a even 1 1 trivial
804.2.ba.b 440 3.b odd 2 1 inner
804.2.ba.b 440 67.h odd 66 1 inner
804.2.ba.b 440 201.p even 66 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{440} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(804, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database