# Properties

 Label 384.3.b.c Level $384$ Weight $3$ Character orbit 384.b Analytic conductor $10.463$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{18}$$ 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 $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{3} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{5} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{7} + 3 q^{9} +O(q^{10})$$ $$q + ( -2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{3} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{5} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{7} + 3 q^{9} + ( 8 \zeta_{24} + 8 \zeta_{24}^{2} + 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{11} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{13} + ( 4 + 6 \zeta_{24} - 6 \zeta_{24}^{3} - 8 \zeta_{24}^{4} - 6 \zeta_{24}^{5} ) q^{15} + ( -2 - 8 \zeta_{24} - 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} + 16 \zeta_{24}^{7} ) q^{17} + ( -8 \zeta_{24} - 8 \zeta_{24}^{2} - 8 \zeta_{24}^{3} + 8 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{19} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{21} + ( -16 - 4 \zeta_{24} + 4 \zeta_{24}^{3} + 32 \zeta_{24}^{4} + 4 \zeta_{24}^{5} ) q^{23} + ( -15 - 16 \zeta_{24} - 16 \zeta_{24}^{3} - 16 \zeta_{24}^{5} + 32 \zeta_{24}^{7} ) q^{25} + ( -6 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{27} + ( 10 \zeta_{24} - 10 \zeta_{24}^{3} + 10 \zeta_{24}^{5} - 20 \zeta_{24}^{6} + 20 \zeta_{24}^{7} ) q^{29} + ( 16 - 18 \zeta_{24} + 18 \zeta_{24}^{3} - 32 \zeta_{24}^{4} + 18 \zeta_{24}^{5} ) q^{31} + ( -12 - 8 \zeta_{24} - 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} + 16 \zeta_{24}^{7} ) q^{33} + ( 8 \zeta_{24} + 16 \zeta_{24}^{2} + 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} - 8 \zeta_{24}^{6} ) q^{35} + ( 16 \zeta_{24} - 16 \zeta_{24}^{3} + 16 \zeta_{24}^{5} + 4 \zeta_{24}^{6} + 32 \zeta_{24}^{7} ) q^{37} + ( -4 + 12 \zeta_{24} - 12 \zeta_{24}^{3} + 8 \zeta_{24}^{4} - 12 \zeta_{24}^{5} ) q^{39} + ( 18 - 8 \zeta_{24} - 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} + 16 \zeta_{24}^{7} ) q^{41} + ( -40 \zeta_{24} - 8 \zeta_{24}^{2} - 40 \zeta_{24}^{3} + 40 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{43} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 12 \zeta_{24}^{6} + 12 \zeta_{24}^{7} ) q^{45} + ( 32 + 12 \zeta_{24} - 12 \zeta_{24}^{3} - 64 \zeta_{24}^{4} - 12 \zeta_{24}^{5} ) q^{47} + 41 q^{49} + ( 24 \zeta_{24} + 4 \zeta_{24}^{2} + 24 \zeta_{24}^{3} - 24 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{51} + ( 14 \zeta_{24} - 14 \zeta_{24}^{3} + 14 \zeta_{24}^{5} + 36 \zeta_{24}^{6} + 28 \zeta_{24}^{7} ) q^{53} + ( -48 - 56 \zeta_{24} + 56 \zeta_{24}^{3} + 96 \zeta_{24}^{4} + 56 \zeta_{24}^{5} ) q^{55} + ( 12 + 8 \zeta_{24} + 8 \zeta_{24}^{3} + 8 \zeta_{24}^{5} - 16 \zeta_{24}^{7} ) q^{57} + ( -40 \zeta_{24}^{2} + 20 \zeta_{24}^{6} ) q^{59} + ( 8 \zeta_{24} - 8 \zeta_{24}^{3} + 8 \zeta_{24}^{5} + 44 \zeta_{24}^{6} + 16 \zeta_{24}^{7} ) q^{61} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{63} + ( -32 - 8 \zeta_{24} - 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} + 16 \zeta_{24}^{7} ) q^{65} + ( -32 \zeta_{24} + 56 \zeta_{24}^{2} - 32 \zeta_{24}^{3} + 32 \zeta_{24}^{5} - 28 \zeta_{24}^{6} ) q^{67} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 48 \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{69} + ( -16 + 68 \zeta_{24} - 68 \zeta_{24}^{3} + 32 \zeta_{24}^{4} - 68 \zeta_{24}^{5} ) q^{71} -10 q^{73} + ( 48 \zeta_{24} + 30 \zeta_{24}^{2} + 48 \zeta_{24}^{3} - 48 \zeta_{24}^{5} - 15 \zeta_{24}^{6} ) q^{75} + ( -8 \zeta_{24} + 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} - 32 \zeta_{24}^{6} - 16 \zeta_{24}^{7} ) q^{77} + ( 48 - 34 \zeta_{24} + 34 \zeta_{24}^{3} - 96 \zeta_{24}^{4} + 34 \zeta_{24}^{5} ) q^{79} + 9 q^{81} + ( 24 \zeta_{24} - 88 \zeta_{24}^{2} + 24 \zeta_{24}^{3} - 24 \zeta_{24}^{5} + 44 \zeta_{24}^{6} ) q^{83} + ( -36 \zeta_{24} + 36 \zeta_{24}^{3} - 36 \zeta_{24}^{5} - 104 \zeta_{24}^{6} - 72 \zeta_{24}^{7} ) q^{85} + ( -20 + 30 \zeta_{24} - 30 \zeta_{24}^{3} + 40 \zeta_{24}^{4} - 30 \zeta_{24}^{5} ) q^{87} + ( -34 + 16 \zeta_{24} + 16 \zeta_{24}^{3} + 16 \zeta_{24}^{5} - 32 \zeta_{24}^{7} ) q^{89} + ( -8 \zeta_{24} + 32 \zeta_{24}^{2} - 8 \zeta_{24}^{3} + 8 \zeta_{24}^{5} - 16 \zeta_{24}^{6} ) q^{91} + ( -18 \zeta_{24} + 18 \zeta_{24}^{3} - 18 \zeta_{24}^{5} + 48 \zeta_{24}^{6} - 36 \zeta_{24}^{7} ) q^{93} + ( 48 + 56 \zeta_{24} - 56 \zeta_{24}^{3} - 96 \zeta_{24}^{4} - 56 \zeta_{24}^{5} ) q^{95} + ( -66 + 16 \zeta_{24} + 16 \zeta_{24}^{3} + 16 \zeta_{24}^{5} - 32 \zeta_{24}^{7} ) q^{97} + ( 24 \zeta_{24} + 24 \zeta_{24}^{2} + 24 \zeta_{24}^{3} - 24 \zeta_{24}^{5} - 12 \zeta_{24}^{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 24 q^{9} + O(q^{10})$$ $$8 q + 24 q^{9} - 16 q^{17} - 120 q^{25} - 96 q^{33} + 144 q^{41} + 328 q^{49} + 96 q^{57} - 256 q^{65} - 80 q^{73} + 72 q^{81} - 272 q^{89} - 528 q^{97} + O(q^{100})$$

## 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
 0.965926 − 0.258819i −0.965926 − 0.258819i −0.965926 + 0.258819i 0.965926 + 0.258819i 0.258819 − 0.965926i −0.258819 − 0.965926i −0.258819 + 0.965926i 0.258819 + 0.965926i
0 −1.73205 0 8.89898i 0 2.82843i 0 3.00000 0
319.2 0 −1.73205 0 0.898979i 0 2.82843i 0 3.00000 0
319.3 0 −1.73205 0 0.898979i 0 2.82843i 0 3.00000 0
319.4 0 −1.73205 0 8.89898i 0 2.82843i 0 3.00000 0
319.5 0 1.73205 0 8.89898i 0 2.82843i 0 3.00000 0
319.6 0 1.73205 0 0.898979i 0 2.82843i 0 3.00000 0
319.7 0 1.73205 0 0.898979i 0 2.82843i 0 3.00000 0
319.8 0 1.73205 0 8.89898i 0 2.82843i 0 3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 319.8 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.c 8
3.b odd 2 1 1152.3.b.j 8
4.b odd 2 1 inner 384.3.b.c 8
8.b even 2 1 inner 384.3.b.c 8
8.d odd 2 1 inner 384.3.b.c 8
12.b even 2 1 1152.3.b.j 8
16.e even 4 1 768.3.g.c 4
16.e even 4 1 768.3.g.g 4
16.f odd 4 1 768.3.g.c 4
16.f odd 4 1 768.3.g.g 4
24.f even 2 1 1152.3.b.j 8
24.h odd 2 1 1152.3.b.j 8
48.i odd 4 1 2304.3.g.o 4
48.i odd 4 1 2304.3.g.x 4
48.k even 4 1 2304.3.g.o 4
48.k even 4 1 2304.3.g.x 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 80 T_{5}^{2} + 64$$ acting on $$S_{3}^{\mathrm{new}}(384, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( -3 + T^{2} )^{4}$$
$5$ $$( 64 + 80 T^{2} + T^{4} )^{2}$$
$7$ $$( 8 + T^{2} )^{4}$$
$11$ $$( 6400 - 352 T^{2} + T^{4} )^{2}$$
$13$ $$( 6400 + 224 T^{2} + T^{4} )^{2}$$
$17$ $$( -380 + 4 T + T^{2} )^{4}$$
$19$ $$( 6400 - 352 T^{2} + T^{4} )^{2}$$
$23$ $$( 541696 + 1600 T^{2} + T^{4} )^{2}$$
$29$ $$( 40000 + 2000 T^{2} + T^{4} )^{2}$$
$31$ $$( 14400 + 2832 T^{2} + T^{4} )^{2}$$
$37$ $$( 2310400 + 3104 T^{2} + T^{4} )^{2}$$
$41$ $$( -60 - 36 T + T^{2} )^{4}$$
$43$ $$( 9935104 - 6496 T^{2} + T^{4} )^{2}$$
$47$ $$( 7750656 + 6720 T^{2} + T^{4} )^{2}$$
$53$ $$( 14400 + 4944 T^{2} + T^{4} )^{2}$$
$59$ $$( -1200 + T^{2} )^{4}$$
$61$ $$( 2408704 + 4640 T^{2} + T^{4} )^{2}$$
$67$ $$( 92416 - 8800 T^{2} + T^{4} )^{2}$$
$71$ $$( 71910400 + 20032 T^{2} + T^{4} )^{2}$$
$73$ $$( 10 + T )^{8}$$
$79$ $$( 21160000 + 18448 T^{2} + T^{4} )^{2}$$
$83$ $$( 21678336 - 13920 T^{2} + T^{4} )^{2}$$
$89$ $$( -380 + 68 T + T^{2} )^{4}$$
$97$ $$( 2820 + 132 T + T^{2} )^{4}$$