# Properties

 Label 819.2.bm.h Level $819$ Weight $2$ Character orbit 819.bm Analytic conductor $6.540$ Analytic rank $0$ Dimension $36$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$819 = 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 819.bm (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.53974792554$$ Analytic rank: $$0$$ Dimension: $$36$$ Relative dimension: $$18$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$36 q - 44 q^{4} + 8 q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36 q - 44 q^{4} + 8 q^{7} + 8 q^{10} + 52 q^{16} - 36 q^{19} + 2 q^{22} + 22 q^{25} + 16 q^{28} - 18 q^{31} - 34 q^{40} + 4 q^{43} + 12 q^{49} + 74 q^{52} - 22 q^{55} + 84 q^{58} - 54 q^{61} - 100 q^{64} + 36 q^{67} - 72 q^{70} - 30 q^{73} + 42 q^{76} + 40 q^{79} + 18 q^{82} + 12 q^{88} + 32 q^{91} - 56 q^{94} - 36 q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
478.1 2.69813i 0 −5.27990 −0.933651 0.539043i 0 2.30689 + 1.29548i 8.84961i 0 −1.45441 + 2.51911i
478.2 2.62688i 0 −4.90051 2.99000 + 1.72628i 0 −2.63476 + 0.240965i 7.61930i 0 4.53473 7.85438i
478.3 2.24196i 0 −3.02638 −0.550805 0.318007i 0 0.958080 2.46619i 2.30110i 0 −0.712960 + 1.23488i
478.4 1.78625i 0 −1.19071 0.972470 + 0.561456i 0 −2.33277 + 1.24828i 1.44561i 0 1.00290 1.73708i
478.5 1.70960i 0 −0.922748 −1.84265 1.06385i 0 1.43697 + 2.22151i 1.84167i 0 −1.81877 + 3.15020i
478.6 1.46062i 0 −0.133410 2.51646 + 1.45288i 0 1.03970 2.43290i 2.72638i 0 2.12210 3.67559i
478.7 1.16108i 0 0.651901 −3.43209 1.98152i 0 −2.10943 1.59697i 3.07906i 0 −2.30069 + 3.98492i
478.8 0.411691i 0 1.83051 1.54520 + 0.892120i 0 2.44833 1.00283i 1.57699i 0 0.367278 0.636145i
478.9 0.169557i 0 1.97125 2.65412 + 1.53236i 0 0.886985 + 2.49264i 0.673355i 0 0.259822 0.450026i
478.10 0.169557i 0 1.97125 −2.65412 1.53236i 0 0.886985 + 2.49264i 0.673355i 0 0.259822 0.450026i
478.11 0.411691i 0 1.83051 −1.54520 0.892120i 0 2.44833 1.00283i 1.57699i 0 0.367278 0.636145i
478.12 1.16108i 0 0.651901 3.43209 + 1.98152i 0 −2.10943 1.59697i 3.07906i 0 −2.30069 + 3.98492i
478.13 1.46062i 0 −0.133410 −2.51646 1.45288i 0 1.03970 2.43290i 2.72638i 0 2.12210 3.67559i
478.14 1.70960i 0 −0.922748 1.84265 + 1.06385i 0 1.43697 + 2.22151i 1.84167i 0 −1.81877 + 3.15020i
478.15 1.78625i 0 −1.19071 −0.972470 0.561456i 0 −2.33277 + 1.24828i 1.44561i 0 1.00290 1.73708i
478.16 2.24196i 0 −3.02638 0.550805 + 0.318007i 0 0.958080 2.46619i 2.30110i 0 −0.712960 + 1.23488i
478.17 2.62688i 0 −4.90051 −2.99000 1.72628i 0 −2.63476 + 0.240965i 7.61930i 0 4.53473 7.85438i
478.18 2.69813i 0 −5.27990 0.933651 + 0.539043i 0 2.30689 + 1.29548i 8.84961i 0 −1.45441 + 2.51911i
550.1 2.69813i 0 −5.27990 0.933651 0.539043i 0 2.30689 1.29548i 8.84961i 0 −1.45441 2.51911i
550.2 2.62688i 0 −4.90051 −2.99000 + 1.72628i 0 −2.63476 0.240965i 7.61930i 0 4.53473 + 7.85438i
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 550.18 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
91.k even 6 1 inner
273.bp odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.bm.h 36
3.b odd 2 1 inner 819.2.bm.h 36
7.c even 3 1 819.2.do.h yes 36
13.e even 6 1 819.2.do.h yes 36
21.h odd 6 1 819.2.do.h yes 36
39.h odd 6 1 819.2.do.h yes 36
91.k even 6 1 inner 819.2.bm.h 36
273.bp odd 6 1 inner 819.2.bm.h 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.bm.h 36 1.a even 1 1 trivial
819.2.bm.h 36 3.b odd 2 1 inner
819.2.bm.h 36 91.k even 6 1 inner
819.2.bm.h 36 273.bp odd 6 1 inner
819.2.do.h yes 36 7.c even 3 1
819.2.do.h yes 36 13.e even 6 1
819.2.do.h yes 36 21.h odd 6 1
819.2.do.h yes 36 39.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(819, [\chi])$$:

 $$T_{2}^{18} + \cdots$$ $$T_{5}^{36} - \cdots$$