# Properties

 Label 384.4.f.b Level $384$ Weight $4$ Character orbit 384.f Analytic conductor $22.657$ Analytic rank $0$ Dimension $4$ CM discriminant -8 Inner twists $8$

# 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{8} + 5 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( -23 - 10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{8} + 5 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( -23 - 10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{9} -18 \zeta_{8}^{2} q^{11} + ( -76 \zeta_{8} - 76 \zeta_{8}^{3} ) q^{17} + ( 90 \zeta_{8} - 90 \zeta_{8}^{3} ) q^{19} -125 q^{25} + ( 73 \zeta_{8} - 95 \zeta_{8}^{2} - 73 \zeta_{8}^{3} ) q^{27} + ( 90 + 18 \zeta_{8} + 18 \zeta_{8}^{3} ) q^{33} + ( 40 \zeta_{8} + 40 \zeta_{8}^{3} ) q^{41} + ( 342 \zeta_{8} - 342 \zeta_{8}^{3} ) q^{43} + 343 q^{49} + ( 380 \zeta_{8} + 152 \zeta_{8}^{2} - 380 \zeta_{8}^{3} ) q^{51} + ( -180 + 450 \zeta_{8} + 450 \zeta_{8}^{3} ) q^{57} -846 \zeta_{8}^{2} q^{59} + ( 774 \zeta_{8} - 774 \zeta_{8}^{3} ) q^{67} + 430 q^{73} + ( 125 \zeta_{8} - 625 \zeta_{8}^{2} - 125 \zeta_{8}^{3} ) q^{75} + ( 329 + 460 \zeta_{8} + 460 \zeta_{8}^{3} ) q^{81} + 1350 \zeta_{8}^{2} q^{83} + ( -940 \zeta_{8} - 940 \zeta_{8}^{3} ) q^{89} -1910 q^{97} + ( -180 \zeta_{8} + 414 \zeta_{8}^{2} + 180 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 92q^{9} + O(q^{10})$$ $$4q - 92q^{9} - 500q^{25} + 360q^{33} + 1372q^{49} - 720q^{57} + 1720q^{73} + 1316q^{81} - 7640q^{97} + O(q^{100})$$

## 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 5.00000i 0 0 0 0 0 −23.0000 + 14.1421i 0
191.2 0 −1.41421 + 5.00000i 0 0 0 0 0 −23.0000 14.1421i 0
191.3 0 1.41421 5.00000i 0 0 0 0 0 −23.0000 14.1421i 0
191.4 0 1.41421 + 5.00000i 0 0 0 0 0 −23.0000 + 14.1421i 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 $$\Q(\sqrt{-2})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 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.b 4
3.b odd 2 1 inner 384.4.f.b 4
4.b odd 2 1 inner 384.4.f.b 4
8.b even 2 1 inner 384.4.f.b 4
8.d odd 2 1 CM 384.4.f.b 4
12.b even 2 1 inner 384.4.f.b 4
16.e even 4 1 768.4.c.a 2
16.e even 4 1 768.4.c.j 2
16.f odd 4 1 768.4.c.a 2
16.f odd 4 1 768.4.c.j 2
24.f even 2 1 inner 384.4.f.b 4
24.h odd 2 1 inner 384.4.f.b 4
48.i odd 4 1 768.4.c.a 2
48.i odd 4 1 768.4.c.j 2
48.k even 4 1 768.4.c.a 2
48.k even 4 1 768.4.c.j 2

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

 $$T_{5}$$ $$T_{23}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$729 + 46 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 324 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$( 11552 + T^{2} )^{2}$$
$19$ $$( -16200 + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$( 3200 + T^{2} )^{2}$$
$43$ $$( -233928 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$( 715716 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$( -1198152 + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$( -430 + T )^{4}$$
$79$ $$T^{4}$$
$83$ $$( 1822500 + T^{2} )^{2}$$
$89$ $$( 1767200 + T^{2} )^{2}$$
$97$ $$( 1910 + T )^{4}$$