# Properties

 Label 192.3.b.b Level $192$ Weight $3$ Character orbit 192.b Analytic conductor $5.232$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 192.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.23162107572$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( 2 - 4 \zeta_{12}^{2} ) q^{5} + 10 \zeta_{12}^{3} q^{7} + 3 q^{9} +O(q^{10})$$ $$q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( 2 - 4 \zeta_{12}^{2} ) q^{5} + 10 \zeta_{12}^{3} q^{7} + 3 q^{9} + ( 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{11} + ( 4 - 8 \zeta_{12}^{2} ) q^{13} + 6 \zeta_{12}^{3} q^{15} + 30 q^{17} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{19} + ( 10 - 20 \zeta_{12}^{2} ) q^{21} + 12 \zeta_{12}^{3} q^{23} + 13 q^{25} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + ( -30 + 60 \zeta_{12}^{2} ) q^{29} + 14 \zeta_{12}^{3} q^{31} -24 q^{33} + ( 40 \zeta_{12} - 20 \zeta_{12}^{3} ) q^{35} + ( 32 - 64 \zeta_{12}^{2} ) q^{37} + 12 \zeta_{12}^{3} q^{39} -6 q^{41} + ( -72 \zeta_{12} + 36 \zeta_{12}^{3} ) q^{43} + ( 6 - 12 \zeta_{12}^{2} ) q^{45} -84 \zeta_{12}^{3} q^{47} -51 q^{49} + ( -60 \zeta_{12} + 30 \zeta_{12}^{3} ) q^{51} + ( -10 + 20 \zeta_{12}^{2} ) q^{53} -48 \zeta_{12}^{3} q^{55} -12 q^{57} + ( 72 \zeta_{12} - 36 \zeta_{12}^{3} ) q^{59} + ( -56 + 112 \zeta_{12}^{2} ) q^{61} + 30 \zeta_{12}^{3} q^{63} -24 q^{65} + ( -56 \zeta_{12} + 28 \zeta_{12}^{3} ) q^{67} + ( 12 - 24 \zeta_{12}^{2} ) q^{69} -60 \zeta_{12}^{3} q^{71} -86 q^{73} + ( -26 \zeta_{12} + 13 \zeta_{12}^{3} ) q^{75} + ( -80 + 160 \zeta_{12}^{2} ) q^{77} + 38 \zeta_{12}^{3} q^{79} + 9 q^{81} + ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{83} + ( 60 - 120 \zeta_{12}^{2} ) q^{85} -90 \zeta_{12}^{3} q^{87} + 78 q^{89} + ( 80 \zeta_{12} - 40 \zeta_{12}^{3} ) q^{91} + ( 14 - 28 \zeta_{12}^{2} ) q^{93} -24 \zeta_{12}^{3} q^{95} + 62 q^{97} + ( 48 \zeta_{12} - 24 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{9} + O(q^{10})$$ $$4q + 12q^{9} + 120q^{17} + 52q^{25} - 96q^{33} - 24q^{41} - 204q^{49} - 48q^{57} - 96q^{65} - 344q^{73} + 36q^{81} + 312q^{89} + 248q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$133$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i
0 −1.73205 0 3.46410i 0 10.0000i 0 3.00000 0
31.2 0 −1.73205 0 3.46410i 0 10.0000i 0 3.00000 0
31.3 0 1.73205 0 3.46410i 0 10.0000i 0 3.00000 0
31.4 0 1.73205 0 3.46410i 0 10.0000i 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 192.3.b.b 4
3.b odd 2 1 576.3.b.a 4
4.b odd 2 1 inner 192.3.b.b 4
8.b even 2 1 inner 192.3.b.b 4
8.d odd 2 1 inner 192.3.b.b 4
12.b even 2 1 576.3.b.a 4
16.e even 4 2 768.3.g.f 4
16.f odd 4 2 768.3.g.f 4
24.f even 2 1 576.3.b.a 4
24.h odd 2 1 576.3.b.a 4
48.i odd 4 2 2304.3.g.q 4
48.k even 4 2 2304.3.g.q 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 100$$ acting on $$S_{3}^{\mathrm{new}}(192, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -3 + T^{2} )^{2}$$
$5$ $$( 12 + T^{2} )^{2}$$
$7$ $$( 100 + T^{2} )^{2}$$
$11$ $$( -192 + T^{2} )^{2}$$
$13$ $$( 48 + T^{2} )^{2}$$
$17$ $$( -30 + T )^{4}$$
$19$ $$( -48 + T^{2} )^{2}$$
$23$ $$( 144 + T^{2} )^{2}$$
$29$ $$( 2700 + T^{2} )^{2}$$
$31$ $$( 196 + T^{2} )^{2}$$
$37$ $$( 3072 + T^{2} )^{2}$$
$41$ $$( 6 + T )^{4}$$
$43$ $$( -3888 + T^{2} )^{2}$$
$47$ $$( 7056 + T^{2} )^{2}$$
$53$ $$( 300 + T^{2} )^{2}$$
$59$ $$( -3888 + T^{2} )^{2}$$
$61$ $$( 9408 + T^{2} )^{2}$$
$67$ $$( -2352 + T^{2} )^{2}$$
$71$ $$( 3600 + T^{2} )^{2}$$
$73$ $$( 86 + T )^{4}$$
$79$ $$( 1444 + T^{2} )^{2}$$
$83$ $$( -192 + T^{2} )^{2}$$
$89$ $$( -78 + T )^{4}$$
$97$ $$( -62 + T )^{4}$$