# Properties

 Label 384.2.d.c Level $384$ Weight $2$ Character orbit 384.d Analytic conductor $3.066$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.06625543762$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{8}^{2} q^{3} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{5} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{7} - q^{9} +O(q^{10})$$ $$q + \zeta_{8}^{2} q^{3} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{5} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{7} - q^{9} -4 \zeta_{8}^{2} q^{11} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{13} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{15} -2 q^{17} + 4 \zeta_{8}^{2} q^{19} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{21} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{23} -3 q^{25} -\zeta_{8}^{2} q^{27} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{29} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{31} + 4 q^{33} + 8 \zeta_{8}^{2} q^{35} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{39} + 10 q^{41} -12 \zeta_{8}^{2} q^{43} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{45} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{47} + q^{49} -2 \zeta_{8}^{2} q^{51} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{53} + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{55} -4 q^{57} + 4 \zeta_{8}^{2} q^{59} + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{61} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{63} -16 q^{65} + 4 \zeta_{8}^{2} q^{67} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{69} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{71} -2 q^{73} -3 \zeta_{8}^{2} q^{75} + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{77} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{79} + q^{81} -4 \zeta_{8}^{2} q^{83} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{85} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{87} + 6 q^{89} + 16 \zeta_{8}^{2} q^{91} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{93} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{95} + 14 q^{97} + 4 \zeta_{8}^{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{9} + O(q^{10})$$ $$4q - 4q^{9} - 8q^{17} - 12q^{25} + 16q^{33} + 40q^{41} + 4q^{49} - 16q^{57} - 64q^{65} - 8q^{73} + 4q^{81} + 24q^{89} + 56q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 0.707107 − 0.707107i −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
0 1.00000i 0 2.82843i 0 2.82843 0 −1.00000 0
193.2 0 1.00000i 0 2.82843i 0 −2.82843 0 −1.00000 0
193.3 0 1.00000i 0 2.82843i 0 −2.82843 0 −1.00000 0
193.4 0 1.00000i 0 2.82843i 0 2.82843 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.2.d.c 4
3.b odd 2 1 1152.2.d.h 4
4.b odd 2 1 inner 384.2.d.c 4
8.b even 2 1 inner 384.2.d.c 4
8.d odd 2 1 inner 384.2.d.c 4
12.b even 2 1 1152.2.d.h 4
16.e even 4 1 768.2.a.i 2
16.e even 4 1 768.2.a.l 2
16.f odd 4 1 768.2.a.i 2
16.f odd 4 1 768.2.a.l 2
24.f even 2 1 1152.2.d.h 4
24.h odd 2 1 1152.2.d.h 4
48.i odd 4 1 2304.2.a.r 2
48.i odd 4 1 2304.2.a.x 2
48.k even 4 1 2304.2.a.r 2
48.k even 4 1 2304.2.a.x 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.d.c 4 1.a even 1 1 trivial
384.2.d.c 4 4.b odd 2 1 inner
384.2.d.c 4 8.b even 2 1 inner
384.2.d.c 4 8.d odd 2 1 inner
768.2.a.i 2 16.e even 4 1
768.2.a.i 2 16.f odd 4 1
768.2.a.l 2 16.e even 4 1
768.2.a.l 2 16.f odd 4 1
1152.2.d.h 4 3.b odd 2 1
1152.2.d.h 4 12.b even 2 1
1152.2.d.h 4 24.f even 2 1
1152.2.d.h 4 24.h odd 2 1
2304.2.a.r 2 48.i odd 4 1
2304.2.a.r 2 48.k even 4 1
2304.2.a.x 2 48.i odd 4 1
2304.2.a.x 2 48.k even 4 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$( 8 + T^{2} )^{2}$$
$7$ $$( -8 + T^{2} )^{2}$$
$11$ $$( 16 + T^{2} )^{2}$$
$13$ $$( 32 + T^{2} )^{2}$$
$17$ $$( 2 + T )^{4}$$
$19$ $$( 16 + T^{2} )^{2}$$
$23$ $$( -32 + T^{2} )^{2}$$
$29$ $$( 8 + T^{2} )^{2}$$
$31$ $$( -72 + T^{2} )^{2}$$
$37$ $$T^{4}$$
$41$ $$( -10 + T )^{4}$$
$43$ $$( 144 + T^{2} )^{2}$$
$47$ $$( -32 + T^{2} )^{2}$$
$53$ $$( 8 + T^{2} )^{2}$$
$59$ $$( 16 + T^{2} )^{2}$$
$61$ $$( 128 + T^{2} )^{2}$$
$67$ $$( 16 + T^{2} )^{2}$$
$71$ $$( -32 + T^{2} )^{2}$$
$73$ $$( 2 + T )^{4}$$
$79$ $$( -72 + T^{2} )^{2}$$
$83$ $$( 16 + T^{2} )^{2}$$
$89$ $$( -6 + T )^{4}$$
$97$ $$( -14 + T )^{4}$$