# Properties

 Label 819.2.et.c Level $819$ Weight $2$ Character orbit 819.et Analytic conductor $6.540$ Analytic rank $0$ Dimension $36$ 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: $$36$$ Relative dimension: $$9$$ over $$\Q(\zeta_{12})$$ Twist minimal: no (minimal twist has level 273) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$36q + 6q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q + 6q^{7} + 8q^{11} - 42q^{14} - 24q^{16} + 8q^{17} - 18q^{19} - 14q^{20} + 4q^{22} + 24q^{25} + 50q^{26} + 34q^{28} - 8q^{29} + 6q^{31} + 50q^{32} - 24q^{34} - 14q^{35} - 14q^{37} + 8q^{38} - 30q^{40} - 34q^{41} + 30q^{43} - 28q^{44} - 32q^{46} + 10q^{47} + 6q^{49} + 20q^{50} + 4q^{52} + 8q^{53} - 30q^{55} + 92q^{56} + 72q^{58} + 70q^{59} - 60q^{61} + 48q^{62} + 44q^{65} - 46q^{67} + 80q^{70} - 42q^{71} - 56q^{73} - 40q^{74} + 12q^{76} - 24q^{77} - 170q^{80} + 24q^{82} + 60q^{83} + 2q^{85} - 12q^{86} + 84q^{88} - 64q^{89} - 86q^{91} + 100q^{92} - 66q^{94} + 36q^{97} + 22q^{98} + O(q^{100})$$

## 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 −1.74432 + 1.74432i 0 4.08534i −0.130892 + 0.488495i 0 1.09376 + 2.40909i 3.63751 + 3.63751i 0 −0.623777 1.08041i
136.2 −1.12005 + 1.12005i 0 0.509024i 0.973479 3.63307i 0 2.28455 1.33448i −1.66997 1.66997i 0 2.97888 + 5.15957i
136.3 −1.09987 + 1.09987i 0 0.419447i −0.745735 + 2.78312i 0 −1.80794 + 1.93167i −1.73841 1.73841i 0 −2.24087 3.88130i
136.4 −0.556084 + 0.556084i 0 1.38154i −0.542987 + 2.02645i 0 −0.405927 2.61443i −1.88042 1.88042i 0 −0.824932 1.42882i
136.5 −0.374685 + 0.374685i 0 1.71922i 0.545981 2.03763i 0 −2.03549 + 1.69021i −1.39354 1.39354i 0 0.558898 + 0.968040i
136.6 0.411775 0.411775i 0 1.66088i 0.180309 0.672922i 0 2.60113 + 0.483875i 1.50746 + 1.50746i 0 −0.202846 0.351339i
136.7 1.20543 1.20543i 0 0.906108i −0.363968 + 1.35835i 0 0.864271 2.50061i 1.31861 + 1.31861i 0 1.19865 + 2.07613i
136.8 1.48267 1.48267i 0 2.39661i −0.507149 + 1.89270i 0 −0.313052 + 2.62717i −0.588043 0.588043i 0 2.05432 + 3.55819i
136.9 1.79515 1.79515i 0 4.44512i 0.590961 2.20549i 0 −2.51335 0.826467i −4.38935 4.38935i 0 −2.89833 5.02005i
145.1 −1.92842 + 1.92842i 0 5.43762i 3.41458 0.914933i 0 2.45097 0.996354i 6.62917 + 6.62917i 0 −4.82036 + 8.34912i
145.2 −1.55654 + 1.55654i 0 2.84566i −3.12520 + 0.837395i 0 1.93981 + 1.79920i 1.31631 + 1.31631i 0 3.56107 6.16796i
145.3 −0.698661 + 0.698661i 0 1.02375i 0.912136 0.244406i 0 2.61412 0.407922i −2.11257 2.11257i 0 −0.466517 + 0.808031i
145.4 −0.430820 + 0.430820i 0 1.62879i −1.97745 + 0.529856i 0 −1.23433 + 2.34018i −1.56335 1.56335i 0 0.623652 1.08020i
145.5 −0.111217 + 0.111217i 0 1.97526i 2.13060 0.570893i 0 −0.399954 2.61535i −0.442117 0.442117i 0 −0.173466 + 0.300452i
145.6 0.745928 0.745928i 0 0.887184i −3.80456 + 1.01943i 0 0.148943 2.64156i 2.15363 + 2.15363i 0 −2.07751 + 3.59835i
145.7 0.926196 0.926196i 0 0.284323i −0.409991 + 0.109857i 0 −2.25606 + 1.38209i 2.11573 + 2.11573i 0 −0.277983 + 0.481481i
145.8 1.30773 1.30773i 0 1.42031i 0.0744995 0.0199621i 0 2.55141 + 0.700217i 0.758075 + 0.758075i 0 0.0713202 0.123530i
145.9 1.74581 1.74581i 0 4.09571i 2.78539 0.746344i 0 −2.58285 + 0.573466i −3.65872 3.65872i 0 3.55979 6.16574i
271.1 −1.74432 1.74432i 0 4.08534i −0.130892 0.488495i 0 1.09376 2.40909i 3.63751 3.63751i 0 −0.623777 + 1.08041i
271.2 −1.12005 1.12005i 0 0.509024i 0.973479 + 3.63307i 0 2.28455 + 1.33448i −1.66997 + 1.66997i 0 2.97888 5.15957i
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 514.9 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.c 36
3.b odd 2 1 273.2.bt.a 36
7.d odd 6 1 819.2.gh.c 36
13.f odd 12 1 819.2.gh.c 36
21.g even 6 1 273.2.cg.a yes 36
39.k even 12 1 273.2.cg.a yes 36
91.ba even 12 1 inner 819.2.et.c 36
273.bs odd 12 1 273.2.bt.a 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bt.a 36 3.b odd 2 1
273.2.bt.a 36 273.bs odd 12 1
273.2.cg.a yes 36 21.g even 6 1
273.2.cg.a yes 36 39.k even 12 1
819.2.et.c 36 1.a even 1 1 trivial
819.2.et.c 36 91.ba even 12 1 inner
819.2.gh.c 36 7.d odd 6 1
819.2.gh.c 36 13.f odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{36} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(819, [\chi])$$.