# Properties

 Label 864.3.h.b Level $864$ Weight $3$ Character orbit 864.h Self dual yes Analytic conductor $23.542$ Analytic rank $0$ Dimension $2$ CM discriminant -24 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 864.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$23.5422948407$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 216) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 6\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{5} + ( -5 + \beta ) q^{7} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{5} + ( -5 + \beta ) q^{7} + ( 5 + 2 \beta ) q^{11} + ( 48 + 2 \beta ) q^{25} -50 q^{29} + ( 19 - 5 \beta ) q^{31} + ( 67 - 4 \beta ) q^{35} + ( 48 - 10 \beta ) q^{49} + ( -47 - 5 \beta ) q^{53} + ( 149 + 7 \beta ) q^{55} -10 q^{59} + ( -25 + 14 \beta ) q^{73} + ( 119 - 5 \beta ) q^{77} + 58 q^{79} + ( -67 - 10 \beta ) q^{83} + ( 95 - 4 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} - 10q^{7} + O(q^{10})$$ $$2q + 2q^{5} - 10q^{7} + 10q^{11} + 96q^{25} - 100q^{29} + 38q^{31} + 134q^{35} + 96q^{49} - 94q^{53} + 298q^{55} - 20q^{59} - 50q^{73} + 238q^{77} + 116q^{79} - 134q^{83} + 190q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$353$$ $$703$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
593.1
 −1.41421 1.41421
0 0 0 −7.48528 0 −13.4853 0 0 0
593.2 0 0 0 9.48528 0 3.48528 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
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.3.h.b 2
3.b odd 2 1 864.3.h.a 2
4.b odd 2 1 216.3.h.b yes 2
8.b even 2 1 864.3.h.a 2
8.d odd 2 1 216.3.h.a 2
12.b even 2 1 216.3.h.a 2
24.f even 2 1 216.3.h.b yes 2
24.h odd 2 1 CM 864.3.h.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.3.h.a 2 8.d odd 2 1
216.3.h.a 2 12.b even 2 1
216.3.h.b yes 2 4.b odd 2 1
216.3.h.b yes 2 24.f even 2 1
864.3.h.a 2 3.b odd 2 1
864.3.h.a 2 8.b even 2 1
864.3.h.b 2 1.a even 1 1 trivial
864.3.h.b 2 24.h odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 2 T_{5} - 71$$ acting on $$S_{3}^{\mathrm{new}}(864, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-71 - 2 T + T^{2}$$
$7$ $$-47 + 10 T + T^{2}$$
$11$ $$-263 - 10 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$( 50 + T )^{2}$$
$31$ $$-1439 - 38 T + T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$409 + 94 T + T^{2}$$
$59$ $$( 10 + T )^{2}$$
$61$ $$T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$-13487 + 50 T + T^{2}$$
$79$ $$( -58 + T )^{2}$$
$83$ $$-2711 + 134 T + T^{2}$$
$89$ $$T^{2}$$
$97$ $$7873 - 190 T + T^{2}$$