# Properties

 Label 684.3.y.a Level $684$ Weight $3$ Character orbit 684.y Analytic conductor $18.638$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$684 = 2^{2} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 684.y (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.6376500822$$ 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: no (minimal twist has level 228) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -2 + 2 \zeta_{6} ) q^{5} - q^{7} +O(q^{10})$$ $$q + ( -2 + 2 \zeta_{6} ) q^{5} - q^{7} -16 q^{11} + ( 18 - 9 \zeta_{6} ) q^{13} + ( 22 - 22 \zeta_{6} ) q^{17} + 19 q^{19} + 40 \zeta_{6} q^{23} + 21 \zeta_{6} q^{25} + ( -20 + 10 \zeta_{6} ) q^{29} + ( -29 + 58 \zeta_{6} ) q^{31} + ( 2 - 2 \zeta_{6} ) q^{35} + ( 9 - 18 \zeta_{6} ) q^{37} + ( 8 + 8 \zeta_{6} ) q^{41} + ( 49 - 49 \zeta_{6} ) q^{43} + 46 \zeta_{6} q^{47} -48 q^{49} + ( 56 - 28 \zeta_{6} ) q^{53} + ( 32 - 32 \zeta_{6} ) q^{55} + ( 38 + 38 \zeta_{6} ) q^{59} + 97 \zeta_{6} q^{61} + ( -18 + 36 \zeta_{6} ) q^{65} + ( -30 + 15 \zeta_{6} ) q^{67} + ( 28 + 28 \zeta_{6} ) q^{71} + ( -35 + 35 \zeta_{6} ) q^{73} + 16 q^{77} + ( -51 - 51 \zeta_{6} ) q^{79} + 146 q^{83} + 44 \zeta_{6} q^{85} + ( 44 - 22 \zeta_{6} ) q^{89} + ( -18 + 9 \zeta_{6} ) q^{91} + ( -38 + 38 \zeta_{6} ) q^{95} + ( 36 + 36 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} - 2 q^{7} + O(q^{10})$$ $$2 q - 2 q^{5} - 2 q^{7} - 32 q^{11} + 27 q^{13} + 22 q^{17} + 38 q^{19} + 40 q^{23} + 21 q^{25} - 30 q^{29} + 2 q^{35} + 24 q^{41} + 49 q^{43} + 46 q^{47} - 96 q^{49} + 84 q^{53} + 32 q^{55} + 114 q^{59} + 97 q^{61} - 45 q^{67} + 84 q^{71} - 35 q^{73} + 32 q^{77} - 153 q^{79} + 292 q^{83} + 44 q^{85} + 66 q^{89} - 27 q^{91} - 38 q^{95} + 108 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$343$$ $$533$$ $$\chi(n)$$ $$\zeta_{6}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
145.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 −1.00000 1.73205i 0 −1.00000 0 0 0
217.1 0 0 0 −1.00000 + 1.73205i 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
19.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.3.y.a 2
3.b odd 2 1 228.3.l.b 2
12.b even 2 1 912.3.be.c 2
19.d odd 6 1 inner 684.3.y.a 2
57.f even 6 1 228.3.l.b 2
228.n odd 6 1 912.3.be.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.3.l.b 2 3.b odd 2 1
228.3.l.b 2 57.f even 6 1
684.3.y.a 2 1.a even 1 1 trivial
684.3.y.a 2 19.d odd 6 1 inner
912.3.be.c 2 12.b even 2 1
912.3.be.c 2 228.n odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(684, [\chi])$$:

 $$T_{5}^{2} + 2 T_{5} + 4$$ $$T_{7} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$4 + 2 T + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$( 16 + T )^{2}$$
$13$ $$243 - 27 T + T^{2}$$
$17$ $$484 - 22 T + T^{2}$$
$19$ $$( -19 + T )^{2}$$
$23$ $$1600 - 40 T + T^{2}$$
$29$ $$300 + 30 T + T^{2}$$
$31$ $$2523 + T^{2}$$
$37$ $$243 + T^{2}$$
$41$ $$192 - 24 T + T^{2}$$
$43$ $$2401 - 49 T + T^{2}$$
$47$ $$2116 - 46 T + T^{2}$$
$53$ $$2352 - 84 T + T^{2}$$
$59$ $$4332 - 114 T + T^{2}$$
$61$ $$9409 - 97 T + T^{2}$$
$67$ $$675 + 45 T + T^{2}$$
$71$ $$2352 - 84 T + T^{2}$$
$73$ $$1225 + 35 T + T^{2}$$
$79$ $$7803 + 153 T + T^{2}$$
$83$ $$( -146 + T )^{2}$$
$89$ $$1452 - 66 T + T^{2}$$
$97$ $$3888 - 108 T + T^{2}$$