# Properties

 Label 684.5.h.b Level $684$ Weight $5$ Character orbit 684.h Self dual yes Analytic conductor $70.705$ 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: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 684.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$70.7050547493$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ Defining polynomial: $$x^{2} - x - 14$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ 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 $$\beta = \frac{1}{2}(-1 + 3\sqrt{57})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 17 + 3 \beta ) q^{5} + ( 39 + 5 \beta ) q^{7} +O(q^{10})$$ $$q + ( 17 + 3 \beta ) q^{5} + ( 39 + 5 \beta ) q^{7} + ( -119 - 5 \beta ) q^{11} + ( -159 + 35 \beta ) q^{17} + 361 q^{19} + 158 q^{23} + ( 816 + 93 \beta ) q^{25} + ( 2583 + 187 \beta ) q^{35} + ( -1721 + 85 \beta ) q^{43} + ( 441 - 325 \beta ) q^{47} + ( 2320 + 365 \beta ) q^{49} + ( -3943 - 427 \beta ) q^{55} + ( -1841 - 515 \beta ) q^{61} + ( 4879 - 275 \beta ) q^{73} + ( -7841 - 765 \beta ) q^{77} + 5678 q^{83} + ( 10737 + 13 \beta ) q^{85} + ( 6137 + 1083 \beta ) q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 31q^{5} + 73q^{7} + O(q^{10})$$ $$2q + 31q^{5} + 73q^{7} - 233q^{11} - 353q^{17} + 722q^{19} + 316q^{23} + 1539q^{25} + 4979q^{35} - 3527q^{43} + 1207q^{47} + 4275q^{49} - 7459q^{55} - 3167q^{61} + 10033q^{73} - 14917q^{77} + 11356q^{83} + 21461q^{85} + 11191q^{95} + O(q^{100})$$

## 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
 −3.27492 4.27492
0 0 0 −18.4743 0 −20.1238 0 0 0
37.2 0 0 0 49.4743 0 93.1238 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 684.5.h.b 2
3.b odd 2 1 76.5.c.a 2
12.b even 2 1 304.5.e.b 2
19.b odd 2 1 CM 684.5.h.b 2
57.d even 2 1 76.5.c.a 2
228.b odd 2 1 304.5.e.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.5.c.a 2 3.b odd 2 1
76.5.c.a 2 57.d even 2 1
304.5.e.b 2 12.b even 2 1
304.5.e.b 2 228.b odd 2 1
684.5.h.b 2 1.a even 1 1 trivial
684.5.h.b 2 19.b odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 31 T_{5} - 914$$ acting on $$S_{5}^{\mathrm{new}}(684, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-914 - 31 T + T^{2}$$
$7$ $$-1874 - 73 T + T^{2}$$
$11$ $$10366 + 233 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$-125954 + 353 T + T^{2}$$
$19$ $$( -361 + T )^{2}$$
$23$ $$( -158 + T )^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$2183326 + 3527 T + T^{2}$$
$47$ $$-13182194 - 1207 T + T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$-31507634 + 3167 T + T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$15466366 - 10033 T + T^{2}$$
$79$ $$T^{2}$$
$83$ $$( -5678 + T )^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$