# Properties

 Label 999.2.c.e Level $999$ Weight $2$ Character orbit 999.c Analytic conductor $7.977$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$999 = 3^{3} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 999.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.97705516193$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{37}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/999\mathbb{Z}\right)^\times$$.

 $$n$$ $$298$$ $$704$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
406.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 2.00000 0 0 1.00000 0 0 0
406.2 0 0 2.00000 0 0 1.00000 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 $$\Q(\sqrt{-3})$$
37.b even 2 1 inner
111.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 999.2.c.e 2
3.b odd 2 1 CM 999.2.c.e 2
37.b even 2 1 inner 999.2.c.e 2
111.d odd 2 1 inner 999.2.c.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
999.2.c.e 2 1.a even 1 1 trivial
999.2.c.e 2 3.b odd 2 1 CM
999.2.c.e 2 37.b even 2 1 inner
999.2.c.e 2 111.d odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(999, [\chi])$$:

 $$T_{2}$$ $$T_{5}$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$27 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$27 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$108 + T^{2}$$
$37$ $$37 - 11 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$108 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$243 + T^{2}$$
$67$ $$( 5 + T )^{2}$$
$71$ $$T^{2}$$
$73$ $$( -7 + T )^{2}$$
$79$ $$27 + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$27 + T^{2}$$