Embedded newform 192.3.q.a.101.26

• Label: 192.3.q.a.101.26
• Level: $192$
• Weight: $3$
• Character: 192.101
• 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			101.26
Character	\(\chi\)	\(=\)	192.101
Dual form			192.3.q.a.173.26

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.515953 + 1.93230i) q^{2} +(0.149422 + 2.99628i) q^{3} +(-3.46759 - 1.99395i) q^{4} +(-6.20358 - 1.23397i) q^{5} +(-5.86681 - 1.25721i) q^{6} +(1.70415 - 4.11418i) q^{7} +(5.64203 - 5.67164i) q^{8} +(-8.95535 + 0.895421i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(65\)	\(127\)	\(133\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{9}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.515953	+	1.93230i	−0.257976	+	0.966151i
							\(3\)	0.149422	+	2.99628i	0.0498074	+	0.998759i
\(5\)	−6.20358	−	1.23397i	−1.24072	−	0.246794i								−0.469251	−	0.883065i	\(-0.655476\pi\)
							−0.771466	+	0.636271i	\(0.780476\pi\)
\(7\)	1.70415	−	4.11418i	0.243450	−	0.587740i	−0.754171	−	0.656678i	\(-0.771961\pi\)
							0.997621	+	0.0689379i	\(0.0219610\pi\)
\(11\)	4.91805	+	7.36039i	0.447096	+	0.669126i	0.984737	−	0.174048i	\(-0.0556846\pi\)
							−0.537642	+	0.843174i	\(0.680685\pi\)
\(13\)	−4.36230	−	21.9308i	−0.335562	−	1.68698i	−0.668243	−	0.743944i	\(-0.732953\pi\)
							0.332681	−	0.943040i	\(-0.392047\pi\)
\(17\)	0.122994	+	0.122994i	0.00723491	+	0.00723491i	0.710715	−	0.703480i	\(-0.248372\pi\)
							−0.703480	+	0.710715i	\(0.748372\pi\)
\(19\)	−14.6726	+	2.91857i	−0.772245	+	0.153609i	−0.565458	−	0.824777i	\(-0.691301\pi\)
							−0.206787	+	0.978386i	\(0.566301\pi\)
\(23\)	−1.69566	−	4.09369i	−0.0737244	−	0.177986i	0.882721	−	0.469897i	\(-0.155709\pi\)
							−0.956446	+	0.291911i	\(0.905709\pi\)
\(29\)	−25.7279	+	38.5045i	−0.887170	+	1.32774i	0.0570311	+	0.998372i	\(0.481837\pi\)
							−0.944201	+	0.329371i	\(0.893163\pi\)
\(31\)		−	53.5712i		−	1.72810i	−0.503403	−	0.864052i	\(-0.667919\pi\)
							0.503403	−	0.864052i	\(-0.332081\pi\)
\(37\)	10.3499	+	2.05872i	0.279727	+	0.0556411i	0.332960	−	0.942941i	\(-0.391953\pi\)
							−0.0532331	+	0.998582i	\(0.516953\pi\)
\(41\)	−23.4804	−	56.6867i	−0.572693	−	1.38260i	−0.899254	−	0.437427i	\(-0.855890\pi\)
							0.326561	−	0.945176i	\(-0.394110\pi\)
\(43\)	−19.0388	−	28.4936i	−0.442764	−	0.662643i	0.541224	−	0.840878i	\(-0.317961\pi\)
							−0.983988	+	0.178236i	\(0.942961\pi\)
\(47\)	−28.0757	−	28.0757i	−0.597356	−	0.597356i	0.342252	−	0.939608i	\(-0.388810\pi\)
							−0.939608	+	0.342252i	\(0.888810\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.3.q.a.101.26	✓	496
3.2	odd	2	inner	192.3.q.a.101.37	yes	496
64.45	even	16	inner	192.3.q.a.173.37	yes	496
192.173	odd	16	inner	192.3.q.a.173.26	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.3.q.a.101.26	✓	496	1.1	even	1	trivial
192.3.q.a.101.37	yes	496	3.2	odd	2	inner
192.3.q.a.173.26	yes	496	192.173	odd	16	inner
192.3.q.a.173.37	yes	496	64.45	even	16	inner