# Properties

 Label 384.3.b.a Level $384$ Weight $3$ Character orbit 384.b Analytic conductor $10.463$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$10.4632421514$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{6}$$ 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + 4 \zeta_{12}^{3} q^{5} + ( -4 + 8 \zeta_{12}^{2} ) q^{7} + 3 q^{9} +O(q^{10})$$ $$q + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + 4 \zeta_{12}^{3} q^{5} + ( -4 + 8 \zeta_{12}^{2} ) q^{7} + 3 q^{9} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{11} + ( -4 + 8 \zeta_{12}^{2} ) q^{15} -18 q^{17} + ( -24 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{19} + 12 \zeta_{12}^{3} q^{21} + ( -24 + 48 \zeta_{12}^{2} ) q^{23} + 9 q^{25} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + 4 \zeta_{12}^{3} q^{29} + ( -28 + 56 \zeta_{12}^{2} ) q^{31} + 12 q^{33} + ( -32 \zeta_{12} + 16 \zeta_{12}^{3} ) q^{35} -72 \zeta_{12}^{3} q^{37} + 18 q^{41} + ( 72 \zeta_{12} - 36 \zeta_{12}^{3} ) q^{43} + 12 \zeta_{12}^{3} q^{45} + ( -24 + 48 \zeta_{12}^{2} ) q^{47} + q^{49} + ( -36 \zeta_{12} + 18 \zeta_{12}^{3} ) q^{51} + 44 \zeta_{12}^{3} q^{53} + ( -16 + 32 \zeta_{12}^{2} ) q^{55} -36 q^{57} + ( 72 \zeta_{12} - 36 \zeta_{12}^{3} ) q^{59} -72 \zeta_{12}^{3} q^{61} + ( -12 + 24 \zeta_{12}^{2} ) q^{63} + ( -24 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{67} + 72 \zeta_{12}^{3} q^{69} + ( 24 - 48 \zeta_{12}^{2} ) q^{71} -82 q^{73} + ( 18 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{75} + 48 \zeta_{12}^{3} q^{77} + ( 36 - 72 \zeta_{12}^{2} ) q^{79} + 9 q^{81} + ( -152 \zeta_{12} + 76 \zeta_{12}^{3} ) q^{83} -72 \zeta_{12}^{3} q^{85} + ( -4 + 8 \zeta_{12}^{2} ) q^{87} + 126 q^{89} + 84 \zeta_{12}^{3} q^{93} + ( 48 - 96 \zeta_{12}^{2} ) q^{95} + 110 q^{97} + ( 24 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{9} + O(q^{10})$$ $$4q + 12q^{9} - 72q^{17} + 36q^{25} + 48q^{33} + 72q^{41} + 4q^{49} - 144q^{57} - 328q^{73} + 36q^{81} + 504q^{89} + 440q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
319.1
 −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i
0 −1.73205 0 4.00000i 0 6.92820i 0 3.00000 0
319.2 0 −1.73205 0 4.00000i 0 6.92820i 0 3.00000 0
319.3 0 1.73205 0 4.00000i 0 6.92820i 0 3.00000 0
319.4 0 1.73205 0 4.00000i 0 6.92820i 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
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 384.3.b.a 4
3.b odd 2 1 1152.3.b.h 4
4.b odd 2 1 inner 384.3.b.a 4
8.b even 2 1 inner 384.3.b.a 4
8.d odd 2 1 inner 384.3.b.a 4
12.b even 2 1 1152.3.b.h 4
16.e even 4 1 768.3.g.a 2
16.e even 4 1 768.3.g.b 2
16.f odd 4 1 768.3.g.a 2
16.f odd 4 1 768.3.g.b 2
24.f even 2 1 1152.3.b.h 4
24.h odd 2 1 1152.3.b.h 4
48.i odd 4 1 2304.3.g.g 2
48.i odd 4 1 2304.3.g.n 2
48.k even 4 1 2304.3.g.g 2
48.k even 4 1 2304.3.g.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.3.b.a 4 1.a even 1 1 trivial
384.3.b.a 4 4.b odd 2 1 inner
384.3.b.a 4 8.b even 2 1 inner
384.3.b.a 4 8.d odd 2 1 inner
768.3.g.a 2 16.e even 4 1
768.3.g.a 2 16.f odd 4 1
768.3.g.b 2 16.e even 4 1
768.3.g.b 2 16.f odd 4 1
1152.3.b.h 4 3.b odd 2 1
1152.3.b.h 4 12.b even 2 1
1152.3.b.h 4 24.f even 2 1
1152.3.b.h 4 24.h odd 2 1
2304.3.g.g 2 48.i odd 4 1
2304.3.g.g 2 48.k even 4 1
2304.3.g.n 2 48.i odd 4 1
2304.3.g.n 2 48.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 16$$ acting on $$S_{3}^{\mathrm{new}}(384, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -3 + T^{2} )^{2}$$
$5$ $$( 16 + T^{2} )^{2}$$
$7$ $$( 48 + T^{2} )^{2}$$
$11$ $$( -48 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$( 18 + T )^{4}$$
$19$ $$( -432 + T^{2} )^{2}$$
$23$ $$( 1728 + T^{2} )^{2}$$
$29$ $$( 16 + T^{2} )^{2}$$
$31$ $$( 2352 + T^{2} )^{2}$$
$37$ $$( 5184 + T^{2} )^{2}$$
$41$ $$( -18 + T )^{4}$$
$43$ $$( -3888 + T^{2} )^{2}$$
$47$ $$( 1728 + T^{2} )^{2}$$
$53$ $$( 1936 + T^{2} )^{2}$$
$59$ $$( -3888 + T^{2} )^{2}$$
$61$ $$( 5184 + T^{2} )^{2}$$
$67$ $$( -432 + T^{2} )^{2}$$
$71$ $$( 1728 + T^{2} )^{2}$$
$73$ $$( 82 + T )^{4}$$
$79$ $$( 3888 + T^{2} )^{2}$$
$83$ $$( -17328 + T^{2} )^{2}$$
$89$ $$( -126 + T )^{4}$$
$97$ $$( -110 + T )^{4}$$