Embedded newform 192.3.q.a.5.17

• Label: 192.3.q.a.5.17
• Level: $192$
• Weight: $3$
• Character: 192.5
• Analytic conductor: $5.232$
• Analytic rank: $0$
• Dimension: $496$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	192.q (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.23162107572\)
Analytic rank:	\(0\)
Dimension:	\(496\)
Relative dimension:	\(62\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			5.17
Character	\(\chi\)	\(=\)	192.5
Dual form			192.3.q.a.77.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.32849 + 1.49503i) q^{2} +(1.59078 - 2.54350i) q^{3} +(-0.470240 - 3.97226i) q^{4} +(0.758965 - 3.81558i) q^{5} +(1.68928 + 5.75729i) q^{6} +(-2.88939 + 6.97561i) q^{7} +(6.56337 + 4.57408i) q^{8} +(-3.93881 - 8.09233i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(65\)	\(127\)	\(133\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{1}{16}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.32849	+	1.49503i	−0.664244	+	0.747516i
							\(3\)	1.59078	−	2.54350i	0.530261	−	0.847834i
\(5\)	0.758965	−	3.81558i	0.151793	−	0.763115i								−0.827626	−	0.561280i	\(-0.810309\pi\)
							0.979419	−	0.201836i	\(-0.0646907\pi\)
\(7\)	−2.88939	+	6.97561i	−0.412771	+	0.996516i	0.571620	+	0.820518i	\(0.306315\pi\)
							−0.984391	+	0.175998i	\(0.943685\pi\)
\(11\)	14.8347	−	9.91225i	1.34861	−	0.901114i	0.349257	−	0.937027i	\(-0.386434\pi\)
							0.999355	+	0.0359130i	\(0.0114339\pi\)
\(13\)	−4.73710	+	0.942268i	−0.364392	+	0.0724822i	−0.373891	−	0.927473i	\(-0.621977\pi\)
							0.00949835	+	0.999955i	\(0.496977\pi\)
\(17\)	−18.8620	−	18.8620i	−1.10953	−	1.10953i	−0.993212	−	0.116320i	\(-0.962890\pi\)
							−0.116320	−	0.993212i	\(-0.537110\pi\)
\(19\)	−3.71208	−	18.6619i	−0.195373	−	0.982205i	−0.946661	−	0.322231i	\(-0.895567\pi\)
							0.751288	−	0.659974i	\(-0.229433\pi\)
\(23\)	−13.7955	−	33.3052i	−0.599803	−	1.44805i	−0.873782	−	0.486318i	\(-0.838340\pi\)
							0.273979	−	0.961736i	\(-0.411660\pi\)
\(29\)	30.4849	+	20.3694i	1.05120	+	0.702392i	0.956090	−	0.293074i	\(-0.0946782\pi\)
							0.0951148	+	0.995466i	\(0.469678\pi\)
\(31\)		−	0.578241i		−	0.0186529i	−0.999957	−	0.00932647i	\(-0.997031\pi\)
							0.999957	−	0.00932647i	\(-0.00296875\pi\)
\(37\)	1.69002	−	8.49630i	0.0456762	−	0.229630i	−0.951204	−	0.308564i	\(-0.900152\pi\)
							0.996880	+	0.0789341i	\(0.0251517\pi\)
\(41\)	26.9832	+	65.1432i	0.658126	+	1.58886i	0.800695	+	0.599072i	\(0.204464\pi\)
							−0.142569	+	0.989785i	\(0.545536\pi\)
\(43\)	−9.04907	+	6.04640i	−0.210444	+	0.140614i	−0.656325	−	0.754478i	\(-0.727890\pi\)
							0.445882	+	0.895092i	\(0.352890\pi\)
\(47\)	−6.16047	−	6.16047i	−0.131074	−	0.131074i	0.638526	−	0.769600i	\(-0.279544\pi\)
							−0.769600	+	0.638526i	\(0.779544\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.3.q.a.5.17	✓	496
3.2	odd	2	inner	192.3.q.a.5.46	yes	496
64.13	even	16	inner	192.3.q.a.77.46	yes	496
192.77	odd	16	inner	192.3.q.a.77.17	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.3.q.a.5.17	✓	496	1.1	even	1	trivial
192.3.q.a.5.46	yes	496	3.2	odd	2	inner
192.3.q.a.77.17	yes	496	192.77	odd	16	inner
192.3.q.a.77.46	yes	496	64.13	even	16	inner