# Properties

 Label 384.2.f.a Level $384$ Weight $2$ Character orbit 384.f Analytic conductor $3.066$ Analytic rank $0$ Dimension $4$ CM discriminant -24 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.06625543762$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + \beta_{1} q^{5} -\beta_{3} q^{7} -3 q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + \beta_{1} q^{5} -\beta_{3} q^{7} -3 q^{9} -2 \beta_{2} q^{11} -\beta_{3} q^{15} -3 \beta_{1} q^{21} + 3 q^{25} -3 \beta_{2} q^{27} + \beta_{1} q^{29} -\beta_{3} q^{31} + 6 q^{33} + 8 \beta_{2} q^{35} -3 \beta_{1} q^{45} -17 q^{49} + 5 \beta_{1} q^{53} + 2 \beta_{3} q^{55} -6 \beta_{2} q^{59} + 3 \beta_{3} q^{63} + 14 q^{73} + 3 \beta_{2} q^{75} + 6 \beta_{1} q^{77} + 3 \beta_{3} q^{79} + 9 q^{81} -10 \beta_{2} q^{83} -\beta_{3} q^{87} -3 \beta_{1} q^{93} + 2 q^{97} + 6 \beta_{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 12q^{9} + O(q^{10})$$ $$4q - 12q^{9} + 12q^{25} + 24q^{33} - 68q^{49} + 56q^{73} + 36q^{81} + 8q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3}$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 4 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1
 0.707107 − 1.22474i −0.707107 + 1.22474i 0.707107 + 1.22474i −0.707107 − 1.22474i
0 1.73205i 0 −2.82843 0 4.89898i 0 −3.00000 0
191.2 0 1.73205i 0 2.82843 0 4.89898i 0 −3.00000 0
191.3 0 1.73205i 0 −2.82843 0 4.89898i 0 −3.00000 0
191.4 0 1.73205i 0 2.82843 0 4.89898i 0 −3.00000 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 $$\Q(\sqrt{-6})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.2.f.a 4
3.b odd 2 1 inner 384.2.f.a 4
4.b odd 2 1 inner 384.2.f.a 4
8.b even 2 1 inner 384.2.f.a 4
8.d odd 2 1 inner 384.2.f.a 4
12.b even 2 1 inner 384.2.f.a 4
16.e even 4 2 768.2.c.j 4
16.f odd 4 2 768.2.c.j 4
24.f even 2 1 inner 384.2.f.a 4
24.h odd 2 1 CM 384.2.f.a 4
48.i odd 4 2 768.2.c.j 4
48.k even 4 2 768.2.c.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.f.a 4 1.a even 1 1 trivial
384.2.f.a 4 3.b odd 2 1 inner
384.2.f.a 4 4.b odd 2 1 inner
384.2.f.a 4 8.b even 2 1 inner
384.2.f.a 4 8.d odd 2 1 inner
384.2.f.a 4 12.b even 2 1 inner
384.2.f.a 4 24.f even 2 1 inner
384.2.f.a 4 24.h odd 2 1 CM
768.2.c.j 4 16.e even 4 2
768.2.c.j 4 16.f odd 4 2
768.2.c.j 4 48.i odd 4 2
768.2.c.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 $$S_{2}^{\mathrm{new}}(384, [\chi])$$:

 $$T_{5}^{2} - 8$$ $$T_{7}^{2} + 24$$ $$T_{23}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 + T^{2} )^{2}$$
$5$ $$( -8 + T^{2} )^{2}$$
$7$ $$( 24 + T^{2} )^{2}$$
$11$ $$( 12 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$( -8 + T^{2} )^{2}$$
$31$ $$( 24 + T^{2} )^{2}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$( -200 + T^{2} )^{2}$$
$59$ $$( 108 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$( -14 + T )^{4}$$
$79$ $$( 216 + T^{2} )^{2}$$
$83$ $$( 300 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$( -2 + T )^{4}$$