# Properties

 Label 384.4.f.c Level $384$ Weight $4$ Character orbit 384.f Analytic conductor $22.657$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 384.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.6567334422$$ 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: $$2^{3}\cdot 3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 3 \zeta_{8} + 3 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{3} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{5} + ( 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{7} + ( 9 + 18 \zeta_{8} + 18 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( 3 \zeta_{8} + 3 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{3} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{5} + ( 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{7} + ( 9 + 18 \zeta_{8} + 18 \zeta_{8}^{3} ) q^{9} -30 \zeta_{8}^{2} q^{11} + 72 \zeta_{8}^{2} q^{13} + ( -24 - 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{15} + ( -36 \zeta_{8} - 36 \zeta_{8}^{3} ) q^{17} + ( 18 \zeta_{8} - 18 \zeta_{8}^{3} ) q^{19} + ( -36 \zeta_{8} + 72 \zeta_{8}^{2} + 36 \zeta_{8}^{3} ) q^{21} -144 q^{23} -93 q^{25} + ( -27 \zeta_{8} + 135 \zeta_{8}^{2} + 27 \zeta_{8}^{3} ) q^{27} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{29} + ( 156 \zeta_{8} + 156 \zeta_{8}^{3} ) q^{31} + ( 90 - 90 \zeta_{8} - 90 \zeta_{8}^{3} ) q^{33} -96 \zeta_{8}^{2} q^{35} -72 \zeta_{8}^{2} q^{37} + ( -216 + 216 \zeta_{8} + 216 \zeta_{8}^{3} ) q^{39} + ( 216 \zeta_{8} + 216 \zeta_{8}^{3} ) q^{41} + ( -162 \zeta_{8} + 162 \zeta_{8}^{3} ) q^{43} + ( -36 \zeta_{8} - 144 \zeta_{8}^{2} + 36 \zeta_{8}^{3} ) q^{45} -576 q^{47} + 55 q^{49} + ( 108 \zeta_{8} - 216 \zeta_{8}^{2} - 108 \zeta_{8}^{3} ) q^{51} + ( 364 \zeta_{8} - 364 \zeta_{8}^{3} ) q^{53} + ( 120 \zeta_{8} + 120 \zeta_{8}^{3} ) q^{55} + ( 108 + 54 \zeta_{8} + 54 \zeta_{8}^{3} ) q^{57} + 414 \zeta_{8}^{2} q^{59} -504 \zeta_{8}^{2} q^{61} + ( -432 + 108 \zeta_{8} + 108 \zeta_{8}^{3} ) q^{63} + ( -288 \zeta_{8} - 288 \zeta_{8}^{3} ) q^{65} + ( 558 \zeta_{8} - 558 \zeta_{8}^{3} ) q^{67} + ( -432 \zeta_{8} - 432 \zeta_{8}^{2} + 432 \zeta_{8}^{3} ) q^{69} -720 q^{71} -178 q^{73} + ( -279 \zeta_{8} - 279 \zeta_{8}^{2} + 279 \zeta_{8}^{3} ) q^{75} + ( 360 \zeta_{8} - 360 \zeta_{8}^{3} ) q^{77} + ( 684 \zeta_{8} + 684 \zeta_{8}^{3} ) q^{79} + ( -567 + 324 \zeta_{8} + 324 \zeta_{8}^{3} ) q^{81} -438 \zeta_{8}^{2} q^{83} + 288 \zeta_{8}^{2} q^{85} + ( -24 - 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{87} + ( -612 \zeta_{8} - 612 \zeta_{8}^{3} ) q^{89} + ( -864 \zeta_{8} + 864 \zeta_{8}^{3} ) q^{91} + ( -468 \zeta_{8} + 936 \zeta_{8}^{2} + 468 \zeta_{8}^{3} ) q^{93} -144 q^{95} + 650 q^{97} + ( 540 \zeta_{8} - 270 \zeta_{8}^{2} - 540 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 36q^{9} + O(q^{10})$$ $$4q + 36q^{9} - 96q^{15} - 576q^{23} - 372q^{25} + 360q^{33} - 864q^{39} - 2304q^{47} + 220q^{49} + 432q^{57} - 1728q^{63} - 2880q^{71} - 712q^{73} - 2268q^{81} - 96q^{87} - 576q^{95} + 2600q^{97} + O(q^{100})$$

## 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 −4.24264 3.00000i 0 5.65685 0 16.9706i 0 9.00000 + 25.4558i 0
191.2 0 −4.24264 + 3.00000i 0 5.65685 0 16.9706i 0 9.00000 25.4558i 0
191.3 0 4.24264 3.00000i 0 −5.65685 0 16.9706i 0 9.00000 25.4558i 0
191.4 0 4.24264 + 3.00000i 0 −5.65685 0 16.9706i 0 9.00000 + 25.4558i 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 384.4.f.c 4
3.b odd 2 1 384.4.f.d yes 4
4.b odd 2 1 384.4.f.d yes 4
8.b even 2 1 inner 384.4.f.c 4
8.d odd 2 1 384.4.f.d yes 4
12.b even 2 1 inner 384.4.f.c 4
16.e even 4 1 768.4.c.b 2
16.e even 4 1 768.4.c.i 2
16.f odd 4 1 768.4.c.c 2
16.f odd 4 1 768.4.c.h 2
24.f even 2 1 inner 384.4.f.c 4
24.h odd 2 1 384.4.f.d yes 4
48.i odd 4 1 768.4.c.c 2
48.i odd 4 1 768.4.c.h 2
48.k even 4 1 768.4.c.b 2
48.k even 4 1 768.4.c.i 2

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

 $$T_{5}^{2} - 32$$ $$T_{23} + 144$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$729 - 18 T^{2} + T^{4}$$
$5$ $$( -32 + T^{2} )^{2}$$
$7$ $$( 288 + T^{2} )^{2}$$
$11$ $$( 900 + T^{2} )^{2}$$
$13$ $$( 5184 + T^{2} )^{2}$$
$17$ $$( 2592 + T^{2} )^{2}$$
$19$ $$( -648 + T^{2} )^{2}$$
$23$ $$( 144 + T )^{4}$$
$29$ $$( -32 + T^{2} )^{2}$$
$31$ $$( 48672 + T^{2} )^{2}$$
$37$ $$( 5184 + T^{2} )^{2}$$
$41$ $$( 93312 + T^{2} )^{2}$$
$43$ $$( -52488 + T^{2} )^{2}$$
$47$ $$( 576 + T )^{4}$$
$53$ $$( -264992 + T^{2} )^{2}$$
$59$ $$( 171396 + T^{2} )^{2}$$
$61$ $$( 254016 + T^{2} )^{2}$$
$67$ $$( -622728 + T^{2} )^{2}$$
$71$ $$( 720 + T )^{4}$$
$73$ $$( 178 + T )^{4}$$
$79$ $$( 935712 + T^{2} )^{2}$$
$83$ $$( 191844 + T^{2} )^{2}$$
$89$ $$( 749088 + T^{2} )^{2}$$
$97$ $$( -650 + T )^{4}$$