# Properties

 Label 819.2.et.b Level $819$ Weight $2$ Character orbit 819.et Analytic conductor $6.540$ Analytic rank $0$ Dimension $28$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$819 = 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 819.et (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.53974792554$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$7$$ over $$\Q(\zeta_{12})$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$28q + 2q^{2} + 6q^{5} - 6q^{7} + 4q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q + 2q^{2} + 6q^{5} - 6q^{7} + 4q^{8} - 6q^{10} - 2q^{11} + 20q^{14} + 4q^{16} + 12q^{17} + 14q^{19} - 36q^{20} - 8q^{22} - 24q^{26} + 2q^{28} + 8q^{29} - 4q^{31} - 10q^{32} - 12q^{34} + 20q^{35} - 10q^{37} + 48q^{40} + 18q^{41} + 48q^{43} + 6q^{44} + 24q^{46} + 6q^{47} - 50q^{49} - 10q^{50} - 26q^{52} - 12q^{53} + 6q^{55} - 54q^{56} - 46q^{58} - 42q^{59} + 30q^{61} - 36q^{62} - 28q^{65} - 10q^{67} - 88q^{70} + 42q^{71} + 40q^{73} - 12q^{74} - 52q^{76} + 4q^{79} - 30q^{80} - 54q^{82} - 66q^{83} - 54q^{85} + 18q^{86} - 6q^{88} + 26q^{91} + 156q^{92} - 18q^{94} - 62q^{97} + 56q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
136.1 −1.74384 + 1.74384i 0 4.08193i 0.638637 2.38343i 0 −2.04541 + 1.67818i 3.63054 + 3.63054i 0 3.04263 + 5.26998i
136.2 −1.14693 + 1.14693i 0 0.630890i −0.395109 + 1.47457i 0 0.0531605 2.64522i −1.57027 1.57027i 0 −1.23806 2.14439i
136.3 −0.490988 + 0.490988i 0 1.51786i 0.00962681 0.0359277i 0 −0.176775 + 2.63984i −1.72723 1.72723i 0 0.0129135 + 0.0223668i
136.4 0.193244 0.193244i 0 1.92531i 0.383199 1.43012i 0 −2.15474 1.53528i 0.758543 + 0.758543i 0 −0.202311 0.350412i
136.5 0.270646 0.270646i 0 1.85350i −0.959617 + 3.58134i 0 1.30385 + 2.30217i 1.04293 + 1.04293i 0 0.709559 + 1.22899i
136.6 0.984398 0.984398i 0 0.0619199i 0.172312 0.643078i 0 −2.46519 0.960657i 2.02975 + 2.02975i 0 −0.463421 0.802669i
136.7 1.56744 1.56744i 0 2.91373i 0.784926 2.92938i 0 2.25305 + 1.38700i −1.43221 1.43221i 0 −3.36131 5.82195i
145.1 −1.51485 + 1.51485i 0 2.58954i 1.34505 0.360406i 0 −0.246373 + 2.63426i 0.893066 + 0.893066i 0 −1.49159 + 2.58351i
145.2 −1.28453 + 1.28453i 0 1.30006i −1.07541 + 0.288156i 0 −0.978346 2.45822i −0.899098 0.899098i 0 1.01126 1.75155i
145.3 −0.203761 + 0.203761i 0 1.91696i −0.499383 + 0.133809i 0 −2.60732 0.449339i −0.798123 0.798123i 0 0.0744896 0.129020i
145.4 0.347096 0.347096i 0 1.75905i 3.47544 0.931242i 0 0.701045 + 2.55118i 1.30475 + 1.30475i 0 0.883082 1.52954i
145.5 0.876516 0.876516i 0 0.463441i 2.51660 0.674321i 0 2.20101 1.46818i 2.15924 + 2.15924i 0 1.61479 2.79689i
145.6 1.42500 1.42500i 0 2.06123i −3.16920 + 0.849184i 0 −0.111396 + 2.64341i −0.0872533 0.0872533i 0 −3.30601 + 5.72618i
145.7 1.72056 1.72056i 0 3.92067i −0.227080 + 0.0608458i 0 1.27342 2.31914i −3.30464 3.30464i 0 −0.286016 + 0.495394i
271.1 −1.74384 1.74384i 0 4.08193i 0.638637 + 2.38343i 0 −2.04541 1.67818i 3.63054 3.63054i 0 3.04263 5.26998i
271.2 −1.14693 1.14693i 0 0.630890i −0.395109 1.47457i 0 0.0531605 + 2.64522i −1.57027 + 1.57027i 0 −1.23806 + 2.14439i
271.3 −0.490988 0.490988i 0 1.51786i 0.00962681 + 0.0359277i 0 −0.176775 2.63984i −1.72723 + 1.72723i 0 0.0129135 0.0223668i
271.4 0.193244 + 0.193244i 0 1.92531i 0.383199 + 1.43012i 0 −2.15474 + 1.53528i 0.758543 0.758543i 0 −0.202311 + 0.350412i
271.5 0.270646 + 0.270646i 0 1.85350i −0.959617 3.58134i 0 1.30385 2.30217i 1.04293 1.04293i 0 0.709559 1.22899i
271.6 0.984398 + 0.984398i 0 0.0619199i 0.172312 + 0.643078i 0 −2.46519 + 0.960657i 2.02975 2.02975i 0 −0.463421 + 0.802669i
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 514.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.ba even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.et.b 28
3.b odd 2 1 91.2.ba.a yes 28
7.d odd 6 1 819.2.gh.b 28
13.f odd 12 1 819.2.gh.b 28
21.c even 2 1 637.2.bb.a 28
21.g even 6 1 91.2.w.a 28
21.g even 6 1 637.2.bd.a 28
21.h odd 6 1 637.2.x.a 28
21.h odd 6 1 637.2.bd.b 28
39.k even 12 1 91.2.w.a 28
91.ba even 12 1 inner 819.2.et.b 28
273.bs odd 12 1 91.2.ba.a yes 28
273.bv even 12 1 637.2.bb.a 28
273.bw even 12 1 637.2.bd.a 28
273.ca odd 12 1 637.2.x.a 28
273.ch odd 12 1 637.2.bd.b 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.w.a 28 21.g even 6 1
91.2.w.a 28 39.k even 12 1
91.2.ba.a yes 28 3.b odd 2 1
91.2.ba.a yes 28 273.bs odd 12 1
637.2.x.a 28 21.h odd 6 1
637.2.x.a 28 273.ca odd 12 1
637.2.bb.a 28 21.c even 2 1
637.2.bb.a 28 273.bv even 12 1
637.2.bd.a 28 21.g even 6 1
637.2.bd.a 28 273.bw even 12 1
637.2.bd.b 28 21.h odd 6 1
637.2.bd.b 28 273.ch odd 12 1
819.2.et.b 28 1.a even 1 1 trivial
819.2.et.b 28 91.ba even 12 1 inner
819.2.gh.b 28 7.d odd 6 1
819.2.gh.b 28 13.f odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{28} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(819, [\chi])$$.