# Properties

 Label 384.2.a.a Level $384$ Weight $2$ Character orbit 384.a Self dual yes Analytic conductor $3.066$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 384.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.06625543762$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} - 4q^{5} + 2q^{7} + q^{9} + O(q^{10})$$ $$q - q^{3} - 4q^{5} + 2q^{7} + q^{9} + 4q^{11} + 2q^{13} + 4q^{15} - 2q^{17} + 8q^{19} - 2q^{21} + 4q^{23} + 11q^{25} - q^{27} - 6q^{31} - 4q^{33} - 8q^{35} - 2q^{37} - 2q^{39} + 6q^{41} - 4q^{45} + 4q^{47} - 3q^{49} + 2q^{51} - 16q^{55} - 8q^{57} - 4q^{59} + 14q^{61} + 2q^{63} - 8q^{65} + 4q^{67} - 4q^{69} + 12q^{71} - 10q^{73} - 11q^{75} + 8q^{77} + 10q^{79} + q^{81} - 12q^{83} + 8q^{85} - 14q^{89} + 4q^{91} + 6q^{93} - 32q^{95} + 10q^{97} + 4q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 −1.00000 0 −4.00000 0 2.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.2.a.a 1
3.b odd 2 1 1152.2.a.t 1
4.b odd 2 1 384.2.a.e yes 1
5.b even 2 1 9600.2.a.bk 1
8.b even 2 1 384.2.a.h yes 1
8.d odd 2 1 384.2.a.d yes 1
12.b even 2 1 1152.2.a.s 1
16.e even 4 2 768.2.d.c 2
16.f odd 4 2 768.2.d.f 2
20.d odd 2 1 9600.2.a.t 1
24.f even 2 1 1152.2.a.a 1
24.h odd 2 1 1152.2.a.b 1
40.e odd 2 1 9600.2.a.bz 1
40.f even 2 1 9600.2.a.e 1
48.i odd 4 2 2304.2.d.f 2
48.k even 4 2 2304.2.d.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.a.a 1 1.a even 1 1 trivial
384.2.a.d yes 1 8.d odd 2 1
384.2.a.e yes 1 4.b odd 2 1
384.2.a.h yes 1 8.b even 2 1
768.2.d.c 2 16.e even 4 2
768.2.d.f 2 16.f odd 4 2
1152.2.a.a 1 24.f even 2 1
1152.2.a.b 1 24.h odd 2 1
1152.2.a.s 1 12.b even 2 1
1152.2.a.t 1 3.b odd 2 1
2304.2.d.f 2 48.i odd 4 2
2304.2.d.o 2 48.k even 4 2
9600.2.a.e 1 40.f even 2 1
9600.2.a.t 1 20.d odd 2 1
9600.2.a.bk 1 5.b even 2 1
9600.2.a.bz 1 40.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(384))$$:

 $$T_{5} + 4$$ $$T_{7} - 2$$ $$T_{11} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$1 + T$$
$5$ $$4 + T$$
$7$ $$-2 + T$$
$11$ $$-4 + T$$
$13$ $$-2 + T$$
$17$ $$2 + T$$
$19$ $$-8 + T$$
$23$ $$-4 + T$$
$29$ $$T$$
$31$ $$6 + T$$
$37$ $$2 + T$$
$41$ $$-6 + T$$
$43$ $$T$$
$47$ $$-4 + T$$
$53$ $$T$$
$59$ $$4 + T$$
$61$ $$-14 + T$$
$67$ $$-4 + T$$
$71$ $$-12 + T$$
$73$ $$10 + T$$
$79$ $$-10 + T$$
$83$ $$12 + T$$
$89$ $$14 + T$$
$97$ $$-10 + T$$