# Properties

 Label 684.3.h.b Level $684$ Weight $3$ Character orbit 684.h 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.h (of order $$2$$, degree $$1$$, 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: $$2^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q -2 q^{7} +O(q^{10})$$ $$q -2 q^{7} + ( -8 + 16 \zeta_{6} ) q^{13} + ( -5 - 16 \zeta_{6} ) q^{19} -25 q^{25} + ( -24 + 48 \zeta_{6} ) q^{31} + ( -40 + 80 \zeta_{6} ) q^{37} -22 q^{43} -45 q^{49} -74 q^{61} + ( -32 + 64 \zeta_{6} ) q^{67} -46 q^{73} + ( -40 + 80 \zeta_{6} ) q^{79} + ( 16 - 32 \zeta_{6} ) q^{91} + ( -112 + 224 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{7} + O(q^{10})$$ $$2 q - 4 q^{7} - 26 q^{19} - 50 q^{25} - 44 q^{43} - 90 q^{49} - 148 q^{61} - 92 q^{73} + O(q^{100})$$

## Character values

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

 $$n$$ $$325$$ $$343$$ $$533$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 0 0 −2.00000 0 0 0
37.2 0 0 0 0 0 −2.00000 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 $$\Q(\sqrt{-3})$$
19.b odd 2 1 inner
57.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.3.h.b 2
3.b odd 2 1 CM 684.3.h.b 2
4.b odd 2 1 2736.3.o.g 2
12.b even 2 1 2736.3.o.g 2
19.b odd 2 1 inner 684.3.h.b 2
57.d even 2 1 inner 684.3.h.b 2
76.d even 2 1 2736.3.o.g 2
228.b odd 2 1 2736.3.o.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.3.h.b 2 1.a even 1 1 trivial
684.3.h.b 2 3.b odd 2 1 CM
684.3.h.b 2 19.b odd 2 1 inner
684.3.h.b 2 57.d even 2 1 inner
2736.3.o.g 2 4.b odd 2 1
2736.3.o.g 2 12.b even 2 1
2736.3.o.g 2 76.d even 2 1
2736.3.o.g 2 228.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{3}^{\mathrm{new}}(684, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$( 2 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$192 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$361 + 26 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$1728 + T^{2}$$
$37$ $$4800 + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( 22 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$( 74 + T )^{2}$$
$67$ $$3072 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 46 + T )^{2}$$
$79$ $$4800 + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$37632 + T^{2}$$