# Properties

 Label 384.2.f.b Level $384$ Weight $2$ Character orbit 384.f 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.f (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.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{5} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{7} + ( 1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{5} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{7} + ( 1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} -2 \zeta_{8}^{2} q^{11} + 4 \zeta_{8}^{2} q^{13} + ( -4 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{15} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{17} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{19} + ( 2 \zeta_{8} + 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{21} + 8 q^{23} + 3 q^{25} + ( \zeta_{8} + 5 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{27} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{29} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{31} + ( 2 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{33} -8 \zeta_{8}^{2} q^{35} -4 \zeta_{8}^{2} q^{37} + ( -4 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{39} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{43} + ( 2 \zeta_{8} - 8 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{45} - q^{49} + ( 4 \zeta_{8} + 8 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{51} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{53} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{55} + ( -4 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{57} -6 \zeta_{8}^{2} q^{59} + 4 \zeta_{8}^{2} q^{61} + ( -8 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{63} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{65} + ( -10 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{67} + ( -8 \zeta_{8} + 8 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{69} + 8 q^{71} -10 q^{73} + ( -3 \zeta_{8} + 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{75} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{77} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{79} + ( -7 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{81} + 6 \zeta_{8}^{2} q^{83} -16 \zeta_{8}^{2} q^{85} + ( -4 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{87} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{89} + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{91} + ( -6 \zeta_{8} - 12 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{93} + 8 q^{95} -6 q^{97} + ( -4 \zeta_{8} - 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{9} + O(q^{10})$$ $$4q + 4q^{9} - 16q^{15} + 32q^{23} + 12q^{25} + 8q^{33} - 16q^{39} - 4q^{49} - 16q^{57} - 32q^{63} + 32q^{71} - 40q^{73} - 28q^{81} - 16q^{87} + 32q^{95} - 24q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1
 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i −0.707107 − 0.707107i
0 −1.41421 1.00000i 0 2.82843 0 2.82843i 0 1.00000 + 2.82843i 0
191.2 0 −1.41421 + 1.00000i 0 2.82843 0 2.82843i 0 1.00000 2.82843i 0
191.3 0 1.41421 1.00000i 0 −2.82843 0 2.82843i 0 1.00000 2.82843i 0
191.4 0 1.41421 + 1.00000i 0 −2.82843 0 2.82843i 0 1.00000 + 2.82843i 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.b even 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.b 4
3.b odd 2 1 384.2.f.d yes 4
4.b odd 2 1 384.2.f.d yes 4
8.b even 2 1 inner 384.2.f.b 4
8.d odd 2 1 384.2.f.d yes 4
12.b even 2 1 inner 384.2.f.b 4
16.e even 4 1 768.2.c.a 2
16.e even 4 1 768.2.c.f 2
16.f odd 4 1 768.2.c.b 2
16.f odd 4 1 768.2.c.e 2
24.f even 2 1 inner 384.2.f.b 4
24.h odd 2 1 384.2.f.d yes 4
48.i odd 4 1 768.2.c.b 2
48.i odd 4 1 768.2.c.e 2
48.k even 4 1 768.2.c.a 2
48.k even 4 1 768.2.c.f 2

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 - 2 T^{2} + T^{4}$$
$5$ $$( -8 + T^{2} )^{2}$$
$7$ $$( 8 + T^{2} )^{2}$$
$11$ $$( 4 + T^{2} )^{2}$$
$13$ $$( 16 + T^{2} )^{2}$$
$17$ $$( 32 + T^{2} )^{2}$$
$19$ $$( -8 + T^{2} )^{2}$$
$23$ $$( -8 + T )^{4}$$
$29$ $$( -8 + T^{2} )^{2}$$
$31$ $$( 72 + T^{2} )^{2}$$
$37$ $$( 16 + T^{2} )^{2}$$
$41$ $$T^{4}$$
$43$ $$( -8 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$( -72 + T^{2} )^{2}$$
$59$ $$( 36 + T^{2} )^{2}$$
$61$ $$( 16 + T^{2} )^{2}$$
$67$ $$( -200 + T^{2} )^{2}$$
$71$ $$( -8 + T )^{4}$$
$73$ $$( 10 + T )^{4}$$
$79$ $$( 8 + T^{2} )^{2}$$
$83$ $$( 36 + T^{2} )^{2}$$
$89$ $$( 32 + T^{2} )^{2}$$
$97$ $$( 6 + T )^{4}$$