# Properties

 Label 684.3.h.a Level $684$ Weight $3$ Character orbit 684.h Self dual yes Analytic conductor $18.638$ Analytic rank $0$ Dimension $2$ CM discriminant -19 Inner twists $2$

# 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: yes Analytic conductor: $$18.6376500822$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ Defining polynomial: $$x^{2} - x - 14$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 76) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{57})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -4 - \beta ) q^{5} + ( 4 - 3 \beta ) q^{7} +O(q^{10})$$ $$q + ( -4 - \beta ) q^{5} + ( 4 - 3 \beta ) q^{7} + ( 4 - 5 \beta ) q^{11} + ( 4 + 7 \beta ) q^{17} -19 q^{19} + 30 q^{23} + ( 5 + 9 \beta ) q^{25} + ( 26 + 11 \beta ) q^{35} + ( 44 - 3 \beta ) q^{43} + ( 44 - 13 \beta ) q^{47} + ( 93 - 15 \beta ) q^{49} + ( 54 + 21 \beta ) q^{55} + ( -44 - 15 \beta ) q^{61} + ( -4 + 33 \beta ) q^{73} + ( 226 - 17 \beta ) q^{77} -90 q^{83} + ( -114 - 39 \beta ) q^{85} + ( 76 + 19 \beta ) q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 9 q^{5} + 5 q^{7} + O(q^{10})$$ $$2 q - 9 q^{5} + 5 q^{7} + 3 q^{11} + 15 q^{17} - 38 q^{19} + 60 q^{23} + 19 q^{25} + 63 q^{35} + 85 q^{43} + 75 q^{47} + 171 q^{49} + 129 q^{55} - 103 q^{61} + 25 q^{73} + 435 q^{77} - 180 q^{83} - 267 q^{85} + 171 q^{95} + O(q^{100})$$

## 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
 4.27492 −3.27492
0 0 0 −8.27492 0 −8.82475 0 0 0
37.2 0 0 0 −0.725083 0 13.8248 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.b odd 2 1 CM by $$\Q(\sqrt{-19})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.3.h.a 2
3.b odd 2 1 76.3.c.b 2
4.b odd 2 1 2736.3.o.c 2
12.b even 2 1 304.3.e.e 2
15.d odd 2 1 1900.3.e.a 2
15.e even 4 2 1900.3.g.a 4
19.b odd 2 1 CM 684.3.h.a 2
24.f even 2 1 1216.3.e.e 2
24.h odd 2 1 1216.3.e.f 2
57.d even 2 1 76.3.c.b 2
76.d even 2 1 2736.3.o.c 2
228.b odd 2 1 304.3.e.e 2
285.b even 2 1 1900.3.e.a 2
285.j odd 4 2 1900.3.g.a 4
456.l odd 2 1 1216.3.e.e 2
456.p even 2 1 1216.3.e.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.c.b 2 3.b odd 2 1
76.3.c.b 2 57.d even 2 1
304.3.e.e 2 12.b even 2 1
304.3.e.e 2 228.b odd 2 1
684.3.h.a 2 1.a even 1 1 trivial
684.3.h.a 2 19.b odd 2 1 CM
1216.3.e.e 2 24.f even 2 1
1216.3.e.e 2 456.l odd 2 1
1216.3.e.f 2 24.h odd 2 1
1216.3.e.f 2 456.p even 2 1
1900.3.e.a 2 15.d odd 2 1
1900.3.e.a 2 285.b even 2 1
1900.3.g.a 4 15.e even 4 2
1900.3.g.a 4 285.j odd 4 2
2736.3.o.c 2 4.b odd 2 1
2736.3.o.c 2 76.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 9 T_{5} + 6$$ acting on $$S_{3}^{\mathrm{new}}(684, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$6 + 9 T + T^{2}$$
$7$ $$-122 - 5 T + T^{2}$$
$11$ $$-354 - 3 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$-642 - 15 T + T^{2}$$
$19$ $$( 19 + T )^{2}$$
$23$ $$( -30 + T )^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$1678 - 85 T + T^{2}$$
$47$ $$-1002 - 75 T + T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$-554 + 103 T + T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$-15362 - 25 T + T^{2}$$
$79$ $$T^{2}$$
$83$ $$( 90 + T )^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$