# Properties

 Label 837.2.j.c Level $837$ Weight $2$ Character orbit 837.j Analytic conductor $6.683$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$837 = 3^{3} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 837.j (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.68347864918$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{4} + ( -5 + 5 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + 2 q^{4} + ( -5 + 5 \zeta_{6} ) q^{7} + ( 6 - 3 \zeta_{6} ) q^{13} + 4 q^{16} + ( -7 + 7 \zeta_{6} ) q^{19} + ( -5 + 5 \zeta_{6} ) q^{25} + ( -10 + 10 \zeta_{6} ) q^{28} + ( -1 + 6 \zeta_{6} ) q^{31} + ( 3 + 3 \zeta_{6} ) q^{37} + ( 1 + \zeta_{6} ) q^{43} -18 \zeta_{6} q^{49} + ( 12 - 6 \zeta_{6} ) q^{52} + ( 5 - 10 \zeta_{6} ) q^{61} + 8 q^{64} + 11 \zeta_{6} q^{67} + ( 16 - 8 \zeta_{6} ) q^{73} + ( -14 + 14 \zeta_{6} ) q^{76} + ( -7 - 7 \zeta_{6} ) q^{79} + ( -15 + 30 \zeta_{6} ) q^{91} + 14 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{4} - 5q^{7} + O(q^{10})$$ $$2q + 4q^{4} - 5q^{7} + 9q^{13} + 8q^{16} - 7q^{19} - 5q^{25} - 10q^{28} + 4q^{31} + 9q^{37} + 3q^{43} - 18q^{49} + 18q^{52} + 16q^{64} + 11q^{67} + 24q^{73} - 14q^{76} - 21q^{79} + 28q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/837\mathbb{Z}\right)^\times$$.

 $$n$$ $$218$$ $$406$$ $$\chi(n)$$ $$-1$$ $$\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 2.00000 0 0 −2.50000 + 4.33013i 0 0 0
161.1 0 0 2.00000 0 0 −2.50000 4.33013i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 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 CM by $$\Q(\sqrt{-3})$$
31.e odd 6 1 inner
93.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 837.2.j.c 2
3.b odd 2 1 CM 837.2.j.c 2
31.e odd 6 1 inner 837.2.j.c 2
93.g even 6 1 inner 837.2.j.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
837.2.j.c 2 1.a even 1 1 trivial
837.2.j.c 2 3.b odd 2 1 CM
837.2.j.c 2 31.e odd 6 1 inner
837.2.j.c 2 93.g even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(837, [\chi])$$:

 $$T_{2}$$ $$T_{5}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$25 + 5 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$27 - 9 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$49 + 7 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$31 - 4 T + T^{2}$$
$37$ $$27 - 9 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$3 - 3 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$75 + T^{2}$$
$67$ $$121 - 11 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$192 - 24 T + T^{2}$$
$79$ $$147 + 21 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$( -14 + T )^{2}$$