# Properties

 Label 837.2.h.a Level $837$ Weight $2$ Character orbit 837.h 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.h (of order $$3$$, 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: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -2 q^{4} + ( -5 + 5 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -2 q^{4} + ( -5 + 5 \zeta_{6} ) q^{7} -5 \zeta_{6} q^{13} + 4 q^{16} + ( 7 - 7 \zeta_{6} ) q^{19} + ( 5 - 5 \zeta_{6} ) q^{25} + ( 10 - 10 \zeta_{6} ) q^{28} + ( 1 - 6 \zeta_{6} ) q^{31} + ( -11 + 11 \zeta_{6} ) q^{37} + ( 13 - 13 \zeta_{6} ) q^{43} -18 \zeta_{6} q^{49} + 10 \zeta_{6} q^{52} -13 q^{61} -8 q^{64} -11 \zeta_{6} q^{67} + 10 \zeta_{6} q^{73} + ( -14 + 14 \zeta_{6} ) q^{76} + ( 13 - 13 \zeta_{6} ) q^{79} + 25 q^{91} + 14 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{4} - 5q^{7} + O(q^{10})$$ $$2q - 4q^{4} - 5q^{7} - 5q^{13} + 8q^{16} + 7q^{19} + 5q^{25} + 10q^{28} - 4q^{31} - 11q^{37} + 13q^{43} - 18q^{49} + 10q^{52} - 26q^{61} - 16q^{64} - 11q^{67} + 10q^{73} - 14q^{76} + 13q^{79} + 50q^{91} + 28q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
676.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 −2.00000 0 0 −2.50000 4.33013i 0 0 0
811.1 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.c even 3 1 inner
93.h odd 6 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(837, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$25 + 5 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$25 + 5 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$49 - 7 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$31 + 4 T + T^{2}$$
$37$ $$121 + 11 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$169 - 13 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$( 13 + T )^{2}$$
$67$ $$121 + 11 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$100 - 10 T + T^{2}$$
$79$ $$169 - 13 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$( -14 + T )^{2}$$