# Properties

 Label 384.5.b.b Level $384$ Weight $5$ Character orbit 384.b Analytic conductor $39.694$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 384.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$39.6940658242$$ 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: $$2^{6}\cdot 3^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{3} + 12 \zeta_{12}^{3} q^{5} + ( 36 - 72 \zeta_{12}^{2} ) q^{7} + 27 q^{9} +O(q^{10})$$ $$q + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{3} + 12 \zeta_{12}^{3} q^{5} + ( 36 - 72 \zeta_{12}^{2} ) q^{7} + 27 q^{9} + ( 24 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{11} + 264 \zeta_{12}^{3} q^{13} + ( -36 + 72 \zeta_{12}^{2} ) q^{15} + 110 q^{17} + ( 120 \zeta_{12} - 60 \zeta_{12}^{3} ) q^{19} -324 \zeta_{12}^{3} q^{21} + ( -72 + 144 \zeta_{12}^{2} ) q^{23} + 481 q^{25} + ( 162 \zeta_{12} - 81 \zeta_{12}^{3} ) q^{27} -228 \zeta_{12}^{3} q^{29} + ( 828 - 1656 \zeta_{12}^{2} ) q^{31} + 108 q^{33} + ( 864 \zeta_{12} - 432 \zeta_{12}^{3} ) q^{35} -1392 \zeta_{12}^{3} q^{37} + ( -792 + 1584 \zeta_{12}^{2} ) q^{39} + 1282 q^{41} + ( 2904 \zeta_{12} - 1452 \zeta_{12}^{3} ) q^{43} + 324 \zeta_{12}^{3} q^{45} + ( -1512 + 3024 \zeta_{12}^{2} ) q^{47} -1487 q^{49} + ( 660 \zeta_{12} - 330 \zeta_{12}^{3} ) q^{51} + 4500 \zeta_{12}^{3} q^{53} + ( -144 + 288 \zeta_{12}^{2} ) q^{55} + 540 q^{57} + ( 7320 \zeta_{12} - 3660 \zeta_{12}^{3} ) q^{59} -960 \zeta_{12}^{3} q^{61} + ( 972 - 1944 \zeta_{12}^{2} ) q^{63} -3168 q^{65} + ( 3768 \zeta_{12} - 1884 \zeta_{12}^{3} ) q^{67} + 648 \zeta_{12}^{3} q^{69} + ( 3528 - 7056 \zeta_{12}^{2} ) q^{71} -3170 q^{73} + ( 2886 \zeta_{12} - 1443 \zeta_{12}^{3} ) q^{75} -1296 \zeta_{12}^{3} q^{77} + ( -900 + 1800 \zeta_{12}^{2} ) q^{79} + 729 q^{81} + ( 5112 \zeta_{12} - 2556 \zeta_{12}^{3} ) q^{83} + 1320 \zeta_{12}^{3} q^{85} + ( 684 - 1368 \zeta_{12}^{2} ) q^{87} + 1550 q^{89} + ( 19008 \zeta_{12} - 9504 \zeta_{12}^{3} ) q^{91} -7452 \zeta_{12}^{3} q^{93} + ( -720 + 1440 \zeta_{12}^{2} ) q^{95} -8018 q^{97} + ( 648 \zeta_{12} - 324 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 108q^{9} + O(q^{10})$$ $$4q + 108q^{9} + 440q^{17} + 1924q^{25} + 432q^{33} + 5128q^{41} - 5948q^{49} + 2160q^{57} - 12672q^{65} - 12680q^{73} + 2916q^{81} + 6200q^{89} - 32072q^{97} + O(q^{100})$$

## 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 −5.19615 0 12.0000i 0 62.3538i 0 27.0000 0
319.2 0 −5.19615 0 12.0000i 0 62.3538i 0 27.0000 0
319.3 0 5.19615 0 12.0000i 0 62.3538i 0 27.0000 0
319.4 0 5.19615 0 12.0000i 0 62.3538i 0 27.0000 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 384.5.b.b 4
3.b odd 2 1 1152.5.b.e 4
4.b odd 2 1 inner 384.5.b.b 4
8.b even 2 1 inner 384.5.b.b 4
8.d odd 2 1 inner 384.5.b.b 4
12.b even 2 1 1152.5.b.e 4
16.e even 4 1 768.5.g.a 2
16.e even 4 1 768.5.g.b 2
16.f odd 4 1 768.5.g.a 2
16.f odd 4 1 768.5.g.b 2
24.f even 2 1 1152.5.b.e 4
24.h odd 2 1 1152.5.b.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.5.b.b 4 1.a even 1 1 trivial
384.5.b.b 4 4.b odd 2 1 inner
384.5.b.b 4 8.b even 2 1 inner
384.5.b.b 4 8.d odd 2 1 inner
768.5.g.a 2 16.e even 4 1
768.5.g.a 2 16.f odd 4 1
768.5.g.b 2 16.e even 4 1
768.5.g.b 2 16.f odd 4 1
1152.5.b.e 4 3.b odd 2 1
1152.5.b.e 4 12.b even 2 1
1152.5.b.e 4 24.f even 2 1
1152.5.b.e 4 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 144$$ acting on $$S_{5}^{\mathrm{new}}(384, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -27 + T^{2} )^{2}$$
$5$ $$( 144 + T^{2} )^{2}$$
$7$ $$( 3888 + T^{2} )^{2}$$
$11$ $$( -432 + T^{2} )^{2}$$
$13$ $$( 69696 + T^{2} )^{2}$$
$17$ $$( -110 + T )^{4}$$
$19$ $$( -10800 + T^{2} )^{2}$$
$23$ $$( 15552 + T^{2} )^{2}$$
$29$ $$( 51984 + T^{2} )^{2}$$
$31$ $$( 2056752 + T^{2} )^{2}$$
$37$ $$( 1937664 + T^{2} )^{2}$$
$41$ $$( -1282 + T )^{4}$$
$43$ $$( -6324912 + T^{2} )^{2}$$
$47$ $$( 6858432 + T^{2} )^{2}$$
$53$ $$( 20250000 + T^{2} )^{2}$$
$59$ $$( -40186800 + T^{2} )^{2}$$
$61$ $$( 921600 + T^{2} )^{2}$$
$67$ $$( -10648368 + T^{2} )^{2}$$
$71$ $$( 37340352 + T^{2} )^{2}$$
$73$ $$( 3170 + T )^{4}$$
$79$ $$( 2430000 + T^{2} )^{2}$$
$83$ $$( -19599408 + T^{2} )^{2}$$
$89$ $$( -1550 + T )^{4}$$
$97$ $$( 8018 + T )^{4}$$