# Properties

 Label 684.3.y.d Level $684$ Weight $3$ Character orbit 684.y Analytic conductor $18.638$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# 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: $$\Z[a_1, \ldots, a_{19}]$$ 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.

 $$f(q)$$ $$=$$ $$q + 13 q^{7} +O(q^{10})$$ $$q + 13 q^{7} + ( 14 - 7 \zeta_{6} ) q^{13} + ( -5 - 16 \zeta_{6} ) q^{19} + 25 \zeta_{6} q^{25} + ( -11 + 22 \zeta_{6} ) q^{31} + ( 7 - 14 \zeta_{6} ) q^{37} + ( 61 - 61 \zeta_{6} ) q^{43} + 120 q^{49} + 47 \zeta_{6} q^{61} + ( 90 - 45 \zeta_{6} ) q^{67} + ( 97 - 97 \zeta_{6} ) q^{73} + ( 91 + 91 \zeta_{6} ) q^{79} + ( 182 - 91 \zeta_{6} ) q^{91} + ( -112 - 112 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 26q^{7} + O(q^{10})$$ $$2q + 26q^{7} + 21q^{13} - 26q^{19} + 25q^{25} + 61q^{43} + 240q^{49} + 47q^{61} + 135q^{67} + 97q^{73} + 273q^{79} + 273q^{91} - 336q^{97} + O(q^{100})$$

## 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 0 0 13.0000 0 0 0
217.1 0 0 0 0 0 13.0000 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 $$\Q(\sqrt{-3})$$
19.d odd 6 1 inner
57.f even 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.d 2
3.b odd 2 1 CM 684.3.y.d 2
19.d odd 6 1 inner 684.3.y.d 2
57.f even 6 1 inner 684.3.y.d 2

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

 $$T_{5}$$ $$T_{7} - 13$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$( -13 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$147 - 21 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$361 + 26 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$363 + T^{2}$$
$37$ $$147 + T^{2}$$
$41$ $$T^{2}$$
$43$ $$3721 - 61 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$2209 - 47 T + T^{2}$$
$67$ $$6075 - 135 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$9409 - 97 T + T^{2}$$
$79$ $$24843 - 273 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$37632 + 336 T + T^{2}$$