# Properties

 Label 384.3.h.a Level $384$ Weight $3$ Character orbit 384.h Self dual yes Analytic conductor $10.463$ Analytic rank $0$ Dimension $2$ CM discriminant -24 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 384.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$10.4632421514$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ Defining polynomial: $$x^{2} - 6$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 4\sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -3 q^{3} + \beta q^{5} + \beta q^{7} + 9 q^{9} +O(q^{10})$$ $$q -3 q^{3} + \beta q^{5} + \beta q^{7} + 9 q^{9} -10 q^{11} -3 \beta q^{15} -3 \beta q^{21} + 71 q^{25} -27 q^{27} -3 \beta q^{29} + 5 \beta q^{31} + 30 q^{33} + 96 q^{35} + 9 \beta q^{45} + 47 q^{49} + 5 \beta q^{53} -10 \beta q^{55} + 10 q^{59} + 9 \beta q^{63} -50 q^{73} -213 q^{75} -10 \beta q^{77} -15 \beta q^{79} + 81 q^{81} + 134 q^{83} + 9 \beta q^{87} -15 \beta q^{93} -190 q^{97} -90 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{3} + 18q^{9} + O(q^{10})$$ $$2q - 6q^{3} + 18q^{9} - 20q^{11} + 142q^{25} - 54q^{27} + 60q^{33} + 192q^{35} + 94q^{49} + 20q^{59} - 100q^{73} - 426q^{75} + 162q^{81} + 268q^{83} - 380q^{97} - 180q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$133$$ $$257$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 −2.44949 2.44949
0 −3.00000 0 −9.79796 0 −9.79796 0 9.00000 0
65.2 0 −3.00000 0 9.79796 0 9.79796 0 9.00000 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 $$\Q(\sqrt{-6})$$
8.d odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.3.h.a 2
3.b odd 2 1 384.3.h.d yes 2
4.b odd 2 1 384.3.h.d yes 2
8.b even 2 1 384.3.h.d yes 2
8.d odd 2 1 inner 384.3.h.a 2
12.b even 2 1 inner 384.3.h.a 2
16.e even 4 2 768.3.e.j 4
16.f odd 4 2 768.3.e.j 4
24.f even 2 1 384.3.h.d yes 2
24.h odd 2 1 CM 384.3.h.a 2
48.i odd 4 2 768.3.e.j 4
48.k even 4 2 768.3.e.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.3.h.a 2 1.a even 1 1 trivial
384.3.h.a 2 8.d odd 2 1 inner
384.3.h.a 2 12.b even 2 1 inner
384.3.h.a 2 24.h odd 2 1 CM
384.3.h.d yes 2 3.b odd 2 1
384.3.h.d yes 2 4.b odd 2 1
384.3.h.d yes 2 8.b even 2 1
384.3.h.d yes 2 24.f even 2 1
768.3.e.j 4 16.e even 4 2
768.3.e.j 4 16.f odd 4 2
768.3.e.j 4 48.i odd 4 2
768.3.e.j 4 48.k even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(384, [\chi])$$:

 $$T_{5}^{2} - 96$$ $$T_{11} + 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 3 + T )^{2}$$
$5$ $$-96 + T^{2}$$
$7$ $$-96 + T^{2}$$
$11$ $$( 10 + T )^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$-864 + T^{2}$$
$31$ $$-2400 + T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$-2400 + T^{2}$$
$59$ $$( -10 + T )^{2}$$
$61$ $$T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 50 + T )^{2}$$
$79$ $$-21600 + T^{2}$$
$83$ $$( -134 + T )^{2}$$
$89$ $$T^{2}$$
$97$ $$( 190 + T )^{2}$$