# Properties

 Label 384.1.h.b Level $384$ Weight $1$ Character orbit 384.h Self dual yes Analytic conductor $0.192$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -8, -24, 12 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$0.191640964851$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\sqrt{-2}, \sqrt{3})$$ Artin image $D_4$ Artin field Galois closure of 4.0.3072.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{9} + O(q^{10})$$ $$q + q^{3} + q^{9} - 2q^{11} - q^{25} + q^{27} - 2q^{33} - q^{49} + 2q^{59} - 2q^{73} - q^{75} + q^{81} - 2q^{83} + 2q^{97} - 2q^{99} + O(q^{100})$$

## 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
 0
0 1.00000 0 0 0 0 0 1.00000 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
12.b even 2 1 RM by $$\Q(\sqrt{3})$$
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 384.1.h.b yes 1
3.b odd 2 1 384.1.h.a 1
4.b odd 2 1 384.1.h.a 1
8.b even 2 1 384.1.h.a 1
8.d odd 2 1 CM 384.1.h.b yes 1
12.b even 2 1 RM 384.1.h.b yes 1
16.e even 4 2 768.1.e.c 2
16.f odd 4 2 768.1.e.c 2
24.f even 2 1 384.1.h.a 1
24.h odd 2 1 CM 384.1.h.b yes 1
32.g even 8 4 3072.1.i.f 4
32.h odd 8 4 3072.1.i.f 4
48.i odd 4 2 768.1.e.c 2
48.k even 4 2 768.1.e.c 2
96.o even 8 4 3072.1.i.f 4
96.p odd 8 4 3072.1.i.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.1.h.a 1 3.b odd 2 1
384.1.h.a 1 4.b odd 2 1
384.1.h.a 1 8.b even 2 1
384.1.h.a 1 24.f even 2 1
384.1.h.b yes 1 1.a even 1 1 trivial
384.1.h.b yes 1 8.d odd 2 1 CM
384.1.h.b yes 1 12.b even 2 1 RM
384.1.h.b yes 1 24.h odd 2 1 CM
768.1.e.c 2 16.e even 4 2
768.1.e.c 2 16.f odd 4 2
768.1.e.c 2 48.i odd 4 2
768.1.e.c 2 48.k even 4 2
3072.1.i.f 4 32.g even 8 4
3072.1.i.f 4 32.h odd 8 4
3072.1.i.f 4 96.o even 8 4
3072.1.i.f 4 96.p odd 8 4

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11} + 2$$ acting on $$S_{1}^{\mathrm{new}}(384, [\chi])$$.

## Hecke characteristic polynomials

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