# Properties

 Label 384.3.b.b 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: $$2^{6}\cdot 3$$ 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -4 + 8 \zeta_{12}^{2} ) q^{5} -12 \zeta_{12}^{3} q^{7} + 3 q^{9} +O(q^{10})$$ $$q + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -4 + 8 \zeta_{12}^{2} ) q^{5} -12 \zeta_{12}^{3} q^{7} + 3 q^{9} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{11} + ( 8 - 16 \zeta_{12}^{2} ) q^{13} + 12 \zeta_{12}^{3} q^{15} + 14 q^{17} + ( 40 \zeta_{12} - 20 \zeta_{12}^{3} ) q^{19} + ( 12 - 24 \zeta_{12}^{2} ) q^{21} + 24 \zeta_{12}^{3} q^{23} -23 q^{25} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + ( -20 + 40 \zeta_{12}^{2} ) q^{29} + 12 \zeta_{12}^{3} q^{31} + 12 q^{33} + ( 96 \zeta_{12} - 48 \zeta_{12}^{3} ) q^{35} + ( -16 + 32 \zeta_{12}^{2} ) q^{37} -24 \zeta_{12}^{3} q^{39} -14 q^{41} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{43} + ( -12 + 24 \zeta_{12}^{2} ) q^{45} -72 \zeta_{12}^{3} q^{47} -95 q^{49} + ( 28 \zeta_{12} - 14 \zeta_{12}^{3} ) q^{51} + ( 36 - 72 \zeta_{12}^{2} ) q^{53} + 48 \zeta_{12}^{3} q^{55} + 60 q^{57} + ( -56 \zeta_{12} + 28 \zeta_{12}^{3} ) q^{59} + ( 32 - 64 \zeta_{12}^{2} ) q^{61} -36 \zeta_{12}^{3} q^{63} + 96 q^{65} + ( 104 \zeta_{12} - 52 \zeta_{12}^{3} ) q^{67} + ( -24 + 48 \zeta_{12}^{2} ) q^{69} -24 \zeta_{12}^{3} q^{71} -50 q^{73} + ( -46 \zeta_{12} + 23 \zeta_{12}^{3} ) q^{75} + ( 48 - 96 \zeta_{12}^{2} ) q^{77} + 12 \zeta_{12}^{3} q^{79} + 9 q^{81} + ( -24 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{83} + ( -56 + 112 \zeta_{12}^{2} ) q^{85} + 60 \zeta_{12}^{3} q^{87} + 62 q^{89} + ( -192 \zeta_{12} + 96 \zeta_{12}^{3} ) q^{91} + ( -12 + 24 \zeta_{12}^{2} ) q^{93} + 240 \zeta_{12}^{3} q^{95} -146 q^{97} + ( 24 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{9} + O(q^{10})$$ $$4q + 12q^{9} + 56q^{17} - 92q^{25} + 48q^{33} - 56q^{41} - 380q^{49} + 240q^{57} + 384q^{65} - 200q^{73} + 36q^{81} + 248q^{89} - 584q^{97} + O(q^{100})$$

## 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 6.92820i 0 12.0000i 0 3.00000 0
319.2 0 −1.73205 0 6.92820i 0 12.0000i 0 3.00000 0
319.3 0 1.73205 0 6.92820i 0 12.0000i 0 3.00000 0
319.4 0 1.73205 0 6.92820i 0 12.0000i 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.b 4
3.b odd 2 1 1152.3.b.e 4
4.b odd 2 1 inner 384.3.b.b 4
8.b even 2 1 inner 384.3.b.b 4
8.d odd 2 1 inner 384.3.b.b 4
12.b even 2 1 1152.3.b.e 4
16.e even 4 2 768.3.g.e 4
16.f odd 4 2 768.3.g.e 4
24.f even 2 1 1152.3.b.e 4
24.h odd 2 1 1152.3.b.e 4
48.i odd 4 2 2304.3.g.r 4
48.k even 4 2 2304.3.g.r 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 48$$ acting on $$S_{3}^{\mathrm{new}}(384, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -3 + T^{2} )^{2}$$
$5$ $$( 48 + T^{2} )^{2}$$
$7$ $$( 144 + T^{2} )^{2}$$
$11$ $$( -48 + T^{2} )^{2}$$
$13$ $$( 192 + T^{2} )^{2}$$
$17$ $$( -14 + T )^{4}$$
$19$ $$( -1200 + T^{2} )^{2}$$
$23$ $$( 576 + T^{2} )^{2}$$
$29$ $$( 1200 + T^{2} )^{2}$$
$31$ $$( 144 + T^{2} )^{2}$$
$37$ $$( 768 + T^{2} )^{2}$$
$41$ $$( 14 + T )^{4}$$
$43$ $$( -48 + T^{2} )^{2}$$
$47$ $$( 5184 + T^{2} )^{2}$$
$53$ $$( 3888 + T^{2} )^{2}$$
$59$ $$( -2352 + T^{2} )^{2}$$
$61$ $$( 3072 + T^{2} )^{2}$$
$67$ $$( -8112 + T^{2} )^{2}$$
$71$ $$( 576 + T^{2} )^{2}$$
$73$ $$( 50 + T )^{4}$$
$79$ $$( 144 + T^{2} )^{2}$$
$83$ $$( -432 + T^{2} )^{2}$$
$89$ $$( -62 + T )^{4}$$
$97$ $$( 146 + T )^{4}$$