# Properties

 Label 684.3.y.e 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.

 $$f(q)$$ $$=$$ $$q + ( 6 - 6 \zeta_{6} ) q^{5} -5 q^{7} +O(q^{10})$$ $$q + ( 6 - 6 \zeta_{6} ) q^{5} -5 q^{7} + ( -22 + 11 \zeta_{6} ) q^{13} + ( 6 - 6 \zeta_{6} ) q^{17} + ( -5 - 16 \zeta_{6} ) q^{19} -24 \zeta_{6} q^{23} -11 \zeta_{6} q^{25} + ( -36 + 18 \zeta_{6} ) q^{29} + ( -17 + 34 \zeta_{6} ) q^{31} + ( -30 + 30 \zeta_{6} ) q^{35} + ( -35 + 70 \zeta_{6} ) q^{37} + ( 24 + 24 \zeta_{6} ) q^{41} + ( 25 - 25 \zeta_{6} ) q^{43} -42 \zeta_{6} q^{47} -24 q^{49} + ( -72 + 36 \zeta_{6} ) q^{53} + ( -42 - 42 \zeta_{6} ) q^{59} -43 \zeta_{6} q^{61} + ( -66 + 132 \zeta_{6} ) q^{65} + ( 66 - 33 \zeta_{6} ) q^{67} + ( -36 - 36 \zeta_{6} ) q^{71} + ( -11 + 11 \zeta_{6} ) q^{73} + ( 1 + \zeta_{6} ) q^{79} -126 q^{83} -36 \zeta_{6} q^{85} + ( 12 - 6 \zeta_{6} ) q^{89} + ( 110 - 55 \zeta_{6} ) q^{91} + ( -126 + 30 \zeta_{6} ) q^{95} + ( -76 - 76 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{5} - 10 q^{7} + O(q^{10})$$ $$2 q + 6 q^{5} - 10 q^{7} - 33 q^{13} + 6 q^{17} - 26 q^{19} - 24 q^{23} - 11 q^{25} - 54 q^{29} - 30 q^{35} + 72 q^{41} + 25 q^{43} - 42 q^{47} - 48 q^{49} - 108 q^{53} - 126 q^{59} - 43 q^{61} + 99 q^{67} - 108 q^{71} - 11 q^{73} + 3 q^{79} - 252 q^{83} - 36 q^{85} + 18 q^{89} + 165 q^{91} - 222 q^{95} - 228 q^{97} + O(q^{100})$$

## 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 3.00000 + 5.19615i 0 −5.00000 0 0 0
217.1 0 0 0 3.00000 5.19615i 0 −5.00000 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.e 2
3.b odd 2 1 228.3.l.a 2
12.b even 2 1 912.3.be.a 2
19.d odd 6 1 inner 684.3.y.e 2
57.f even 6 1 228.3.l.a 2
228.n odd 6 1 912.3.be.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.3.l.a 2 3.b odd 2 1
228.3.l.a 2 57.f even 6 1
684.3.y.e 2 1.a even 1 1 trivial
684.3.y.e 2 19.d odd 6 1 inner
912.3.be.a 2 12.b even 2 1
912.3.be.a 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 $$S_{3}^{\mathrm{new}}(684, [\chi])$$:

 $$T_{5}^{2} - 6 T_{5} + 36$$ $$T_{7} + 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$36 - 6 T + T^{2}$$
$7$ $$( 5 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$363 + 33 T + T^{2}$$
$17$ $$36 - 6 T + T^{2}$$
$19$ $$361 + 26 T + T^{2}$$
$23$ $$576 + 24 T + T^{2}$$
$29$ $$972 + 54 T + T^{2}$$
$31$ $$867 + T^{2}$$
$37$ $$3675 + T^{2}$$
$41$ $$1728 - 72 T + T^{2}$$
$43$ $$625 - 25 T + T^{2}$$
$47$ $$1764 + 42 T + T^{2}$$
$53$ $$3888 + 108 T + T^{2}$$
$59$ $$5292 + 126 T + T^{2}$$
$61$ $$1849 + 43 T + T^{2}$$
$67$ $$3267 - 99 T + T^{2}$$
$71$ $$3888 + 108 T + T^{2}$$
$73$ $$121 + 11 T + T^{2}$$
$79$ $$3 - 3 T + T^{2}$$
$83$ $$( 126 + T )^{2}$$
$89$ $$108 - 18 T + T^{2}$$
$97$ $$17328 + 228 T + T^{2}$$