# Properties

 Label 384.3.g.b Level $384$ Weight $3$ Character orbit 384.g Analytic conductor $10.463$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 384.g (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.

 $$f(q)$$ $$=$$ $$q + ( -1 + 2 \zeta_{24}^{4} ) q^{3} + ( 2 + 2 \zeta_{24} + 4 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{5} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q + ( -1 + 2 \zeta_{24}^{4} ) q^{3} + ( 2 + 2 \zeta_{24} + 4 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{5} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{7} -3 q^{9} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 4 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{11} + ( -6 - 8 \zeta_{24}^{2} - 8 \zeta_{24}^{5} + 4 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{13} + ( -2 + 2 \zeta_{24} - 2 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 6 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{15} + ( 2 + 12 \zeta_{24} - 8 \zeta_{24}^{2} + 12 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 4 \zeta_{24}^{6} - 16 \zeta_{24}^{7} ) q^{17} + ( 4 + 12 \zeta_{24} - 12 \zeta_{24}^{3} - 8 \zeta_{24}^{4} - 4 \zeta_{24}^{5} - 12 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{19} + ( -10 \zeta_{24} + 4 \zeta_{24}^{2} - 10 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{21} + ( 12 \zeta_{24} - 12 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{23} + ( -1 + 16 \zeta_{24} + 16 \zeta_{24}^{2} + 16 \zeta_{24}^{3} - 8 \zeta_{24}^{6} - 16 \zeta_{24}^{7} ) q^{25} + ( 3 - 6 \zeta_{24}^{4} ) q^{27} + ( 10 - 18 \zeta_{24} - 20 \zeta_{24}^{2} - 18 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 10 \zeta_{24}^{6} + 16 \zeta_{24}^{7} ) q^{29} + ( 8 + 6 \zeta_{24} - 6 \zeta_{24}^{3} - 16 \zeta_{24}^{4} + 14 \zeta_{24}^{5} + 6 \zeta_{24}^{6} + 20 \zeta_{24}^{7} ) q^{31} + ( -12 \zeta_{24} - 8 \zeta_{24}^{2} - 12 \zeta_{24}^{3} + 12 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{33} + ( -4 - 4 \zeta_{24} + 4 \zeta_{24}^{3} + 8 \zeta_{24}^{4} + 28 \zeta_{24}^{5} + 12 \zeta_{24}^{6} + 24 \zeta_{24}^{7} ) q^{35} + ( 2 - 4 \zeta_{24} + 32 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 20 \zeta_{24}^{5} - 16 \zeta_{24}^{6} + 24 \zeta_{24}^{7} ) q^{37} + ( 6 + 16 \zeta_{24} - 16 \zeta_{24}^{3} - 12 \zeta_{24}^{4} - 8 \zeta_{24}^{5} - 12 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{39} + ( 10 - 12 \zeta_{24} + 40 \zeta_{24}^{2} - 12 \zeta_{24}^{3} + 28 \zeta_{24}^{5} - 20 \zeta_{24}^{6} - 16 \zeta_{24}^{7} ) q^{41} + ( -12 + 20 \zeta_{24} - 20 \zeta_{24}^{3} + 24 \zeta_{24}^{4} + 4 \zeta_{24}^{5} + 28 \zeta_{24}^{6} + 24 \zeta_{24}^{7} ) q^{43} + ( -6 - 6 \zeta_{24} - 12 \zeta_{24}^{2} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 6 \zeta_{24}^{6} ) q^{45} + ( -8 + 20 \zeta_{24} - 20 \zeta_{24}^{3} + 16 \zeta_{24}^{4} + 4 \zeta_{24}^{5} - 12 \zeta_{24}^{6} + 24 \zeta_{24}^{7} ) q^{47} + ( -11 + 24 \zeta_{24} - 64 \zeta_{24}^{2} + 24 \zeta_{24}^{3} - 8 \zeta_{24}^{5} + 32 \zeta_{24}^{6} - 16 \zeta_{24}^{7} ) q^{49} + ( -2 - 20 \zeta_{24} + 20 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 28 \zeta_{24}^{5} - 12 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{51} + ( -22 - 18 \zeta_{24} + 44 \zeta_{24}^{2} - 18 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 22 \zeta_{24}^{6} + 16 \zeta_{24}^{7} ) q^{53} + ( -24 - 24 \zeta_{24} + 24 \zeta_{24}^{3} + 48 \zeta_{24}^{4} + 40 \zeta_{24}^{5} + 8 \zeta_{24}^{6} + 16 \zeta_{24}^{7} ) q^{55} + ( 12 - 4 \zeta_{24} + 24 \zeta_{24}^{2} - 4 \zeta_{24}^{3} + 20 \zeta_{24}^{5} - 12 \zeta_{24}^{6} - 16 \zeta_{24}^{7} ) q^{57} + ( 12 - 16 \zeta_{24} + 16 \zeta_{24}^{3} - 24 \zeta_{24}^{4} + 16 \zeta_{24}^{5} - 64 \zeta_{24}^{6} ) q^{59} + ( 34 + 36 \zeta_{24} - 48 \zeta_{24}^{2} + 36 \zeta_{24}^{3} - 28 \zeta_{24}^{5} + 24 \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{61} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} - 18 \zeta_{24}^{5} + 6 \zeta_{24}^{6} - 12 \zeta_{24}^{7} ) q^{63} + ( -20 - 36 \zeta_{24} - 72 \zeta_{24}^{2} - 36 \zeta_{24}^{3} + 20 \zeta_{24}^{5} + 36 \zeta_{24}^{6} + 16 \zeta_{24}^{7} ) q^{65} + ( 4 + 48 \zeta_{24} - 48 \zeta_{24}^{3} - 8 \zeta_{24}^{4} - 48 \zeta_{24}^{5} ) q^{67} + ( -4 \zeta_{24} + 8 \zeta_{24}^{2} - 4 \zeta_{24}^{3} + 20 \zeta_{24}^{5} - 4 \zeta_{24}^{6} - 16 \zeta_{24}^{7} ) q^{69} + ( 24 + 12 \zeta_{24} - 12 \zeta_{24}^{3} - 48 \zeta_{24}^{4} - 4 \zeta_{24}^{5} + 60 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{71} + ( -2 + 40 \zeta_{24} + 64 \zeta_{24}^{2} + 40 \zeta_{24}^{3} - 56 \zeta_{24}^{5} - 32 \zeta_{24}^{6} + 16 \zeta_{24}^{7} ) q^{73} + ( 1 - 16 \zeta_{24} + 16 \zeta_{24}^{3} - 2 \zeta_{24}^{4} + 32 \zeta_{24}^{5} + 24 \zeta_{24}^{6} + 16 \zeta_{24}^{7} ) q^{75} + ( -40 - 16 \zeta_{24} - 64 \zeta_{24}^{2} - 16 \zeta_{24}^{3} + 16 \zeta_{24}^{5} + 32 \zeta_{24}^{6} ) q^{77} + ( -8 - 58 \zeta_{24} + 58 \zeta_{24}^{3} + 16 \zeta_{24}^{4} + 14 \zeta_{24}^{5} - 10 \zeta_{24}^{6} - 44 \zeta_{24}^{7} ) q^{79} + 9 q^{81} + ( -12 \zeta_{24} + 12 \zeta_{24}^{3} - 44 \zeta_{24}^{5} + 68 \zeta_{24}^{6} - 56 \zeta_{24}^{7} ) q^{83} + ( -4 + 76 \zeta_{24} + 56 \zeta_{24}^{2} + 76 \zeta_{24}^{3} - 44 \zeta_{24}^{5} - 28 \zeta_{24}^{6} - 32 \zeta_{24}^{7} ) q^{85} + ( -10 + 14 \zeta_{24} - 14 \zeta_{24}^{3} + 20 \zeta_{24}^{4} - 34 \zeta_{24}^{5} - 30 \zeta_{24}^{6} - 20 \zeta_{24}^{7} ) q^{87} + ( -30 - 8 \zeta_{24} + 48 \zeta_{24}^{2} - 8 \zeta_{24}^{3} - 24 \zeta_{24}^{5} - 24 \zeta_{24}^{6} + 32 \zeta_{24}^{7} ) q^{89} + ( 8 + 36 \zeta_{24} - 36 \zeta_{24}^{3} - 16 \zeta_{24}^{4} - 76 \zeta_{24}^{5} - 4 \zeta_{24}^{6} - 40 \zeta_{24}^{7} ) q^{91} + ( 24 - 34 \zeta_{24} - 12 \zeta_{24}^{2} - 34 \zeta_{24}^{3} + 26 \zeta_{24}^{5} + 6 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{93} + ( 16 - 32 \zeta_{24}^{4} - 32 \zeta_{24}^{5} - 80 \zeta_{24}^{6} - 32 \zeta_{24}^{7} ) q^{95} + ( 50 + 88 \zeta_{24} + 16 \zeta_{24}^{2} + 88 \zeta_{24}^{3} - 88 \zeta_{24}^{5} - 8 \zeta_{24}^{6} ) q^{97} + ( -12 \zeta_{24} + 12 \zeta_{24}^{3} - 12 \zeta_{24}^{5} - 12 \zeta_{24}^{6} - 24 \zeta_{24}^{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 16q^{5} - 24q^{9} + O(q^{10})$$ $$8q + 16q^{5} - 24q^{9} - 48q^{13} + 16q^{17} - 8q^{25} + 80q^{29} + 16q^{37} + 80q^{41} - 48q^{45} - 88q^{49} - 176q^{53} + 96q^{57} + 272q^{61} - 160q^{65} - 16q^{73} - 320q^{77} + 72q^{81} - 32q^{85} - 240q^{89} + 192q^{93} + 400q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 0.258819 + 0.965926i −0.258819 − 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i 0.258819 − 0.965926i −0.258819 + 0.965926i −0.965926 − 0.258819i 0.965926 + 0.258819i
0 1.73205i 0 −4.29253 0 2.75787i 0 −3.00000 0
127.2 0 1.73205i 0 1.36433 0 1.24213i 0 −3.00000 0
127.3 0 1.73205i 0 2.63567 0 12.5558i 0 −3.00000 0
127.4 0 1.73205i 0 8.29253 0 8.55583i 0 −3.00000 0
127.5 0 1.73205i 0 −4.29253 0 2.75787i 0 −3.00000 0
127.6 0 1.73205i 0 1.36433 0 1.24213i 0 −3.00000 0
127.7 0 1.73205i 0 2.63567 0 12.5558i 0 −3.00000 0
127.8 0 1.73205i 0 8.29253 0 8.55583i 0 −3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 127.8 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

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 8 T_{5}^{3} - 16 T_{5}^{2} + 128 T_{5} - 128$$ acting on $$S_{3}^{\mathrm{new}}(384, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 3 + T^{2} )^{4}$$
$5$ $$( -128 + 128 T - 16 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$7$ $$135424 + 108288 T^{2} + 13664 T^{4} + 240 T^{6} + T^{8}$$
$11$ $$( 6400 + 224 T^{2} + T^{4} )^{2}$$
$13$ $$( -35696 - 6432 T - 136 T^{2} + 24 T^{3} + T^{4} )^{2}$$
$17$ $$( 70288 + 9824 T - 904 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$19$ $$22351446016 + 246677504 T^{2} + 983040 T^{4} + 1664 T^{6} + T^{8}$$
$23$ $$34668544 + 6930432 T^{2} + 218624 T^{4} + 960 T^{6} + T^{8}$$
$29$ $$( -777344 + 69760 T - 1168 T^{2} - 40 T^{3} + T^{4} )^{2}$$
$31$ $$1115025664 + 1078935296 T^{2} + 3628896 T^{4} + 3440 T^{6} + T^{8}$$
$37$ $$( -553712 + 87776 T - 3496 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$41$ $$( -3231344 + 244960 T - 4168 T^{2} - 40 T^{3} + T^{4} )^{2}$$
$43$ $$10858553933824 + 28354805760 T^{2} + 25296896 T^{4} + 8832 T^{6} + T^{8}$$
$47$ $$102236225536 + 1132249088 T^{2} + 4154880 T^{4} + 5312 T^{6} + T^{8}$$
$53$ $$( -3148928 - 221056 T - 1168 T^{2} + 88 T^{3} + T^{4} )^{2}$$
$59$ $$81909160935424 + 240437477376 T^{2} + 119707136 T^{4} + 20160 T^{6} + T^{8}$$
$61$ $$( -13242608 + 516832 T - 808 T^{2} - 136 T^{3} + T^{4} )^{2}$$
$67$ $$( 20793600 + 9312 T^{2} + T^{4} )^{2}$$
$71$ $$11090924732416 + 69089804288 T^{2} + 112264704 T^{4} + 22208 T^{6} + T^{8}$$
$73$ $$( -2276336 + 525344 T - 16104 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$79$ $$3915755776 + 5149011712 T^{2} + 109548384 T^{4} + 23152 T^{6} + T^{8}$$
$83$ $$86180178755584 + 347354677248 T^{2} + 368735744 T^{4} + 39360 T^{6} + T^{8}$$
$89$ $$( -1654256 - 151584 T - 1384 T^{2} + 120 T^{3} + T^{4} )^{2}$$
$97$ $$( 161817616 + 2636000 T - 16360 T^{2} - 200 T^{3} + T^{4} )^{2}$$