Embedded newform 342.3.l.a.151.3

• Label: 342.3.l.a.151.3
• Level: $342$
• Weight: $3$
• Character: 342.151
• Analytic conductor: $9.319$
• Analytic rank: $0$
• Dimension: $80$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	342.l (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.31882504112\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(40\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			151.3
Character	\(\chi\)	\(=\)	342.151
Dual form			342.3.l.a.265.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22474 - 0.707107i) q^{2} +(-2.70026 + 1.30712i) q^{3} +(1.00000 + 1.73205i) q^{4} +(2.01557 + 3.49107i) q^{5} +(4.23141 + 0.308482i) q^{6} +(-5.00320 + 8.66581i) q^{7} -2.82843i q^{8} +(5.58285 - 7.05916i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 80 q^{4} + 8 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).

\(n\)	\(191\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.22474	−	0.707107i	−0.612372	−	0.353553i
							\(3\)	−2.70026	+	1.30712i	−0.900088	+	0.435708i
\(5\)	2.01557	+	3.49107i	0.403114	+	0.698213i								0.994100	−	0.108468i	\(-0.0345946\pi\)
							−0.590986	+	0.806682i	\(0.701261\pi\)
\(7\)	−5.00320	+	8.66581i	−0.714744	+	1.23797i	0.248315	+	0.968679i	\(0.420123\pi\)
							−0.963058	+	0.269293i	\(0.913210\pi\)
\(11\)	−9.23211	+	15.9905i	−0.839283	+	1.45368i	0.0512118	+	0.998688i	\(0.483692\pi\)
							−0.890495	+	0.454993i	\(0.849642\pi\)
\(13\)	4.30462	−	2.48528i	0.331125	−	0.191175i	−0.325216	−	0.945640i	\(-0.605437\pi\)
							0.656340	+	0.754465i	\(0.272104\pi\)
\(17\)	11.3774			0.669257			0.334629	−	0.942350i	\(-0.391389\pi\)
							0.334629	+	0.942350i	\(0.391389\pi\)
\(19\)	15.6097	+	10.8322i	0.821565	+	0.570115i
							\(23\)	−19.7470	−	34.2028i	−0.858565	−	1.48708i	−0.873298	−	0.487187i	\(-0.838023\pi\)
0.0147330	−	0.999891i	\(-0.495310\pi\)
\(29\)	−26.2776	−	15.1714i	−0.906126	−	0.523152i	−0.0269431	−	0.999637i	\(-0.508577\pi\)
							−0.879183	+	0.476485i	\(0.841911\pi\)
\(31\)	−25.5235	+	14.7360i	−0.823338	+	0.475354i	−0.851566	−	0.524247i	\(-0.824347\pi\)
							0.0282281	+	0.999602i	\(0.491014\pi\)
\(37\)			23.9993i			0.648631i	0.945949	+	0.324315i	\(0.105134\pi\)
							−0.945949	+	0.324315i	\(0.894866\pi\)
\(41\)	−58.5776	+	33.8198i	−1.42872	+	0.824873i	−0.997020	−	0.0771425i	\(-0.975420\pi\)
							−0.431703	+	0.902016i	\(0.642087\pi\)
\(43\)	−24.9656	+	43.2417i	−0.580595	+	1.00562i	0.414814	+	0.909906i	\(0.363847\pi\)
							−0.995409	+	0.0957138i	\(0.969487\pi\)
\(47\)	30.5957	−	52.9933i	0.650972	−	1.12752i	−0.331915	−	0.943309i	\(-0.607695\pi\)
							0.982887	−	0.184208i	\(-0.0589720\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.3.l.a.151.3	✓	80
3.2	odd	2		1026.3.l.a.721.28		80
9.4	even	3	inner	342.3.l.a.265.38	yes	80
9.5	odd	6		1026.3.l.a.37.12		80
19.18	odd	2	inner	342.3.l.a.151.38	yes	80
57.56	even	2		1026.3.l.a.721.12		80
171.94	odd	6	inner	342.3.l.a.265.3	yes	80
171.113	even	6		1026.3.l.a.37.28		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
342.3.l.a.151.3	✓	80	1.1	even	1	trivial
342.3.l.a.151.38	yes	80	19.18	odd	2	inner
342.3.l.a.265.3	yes	80	171.94	odd	6	inner
342.3.l.a.265.38	yes	80	9.4	even	3	inner
1026.3.l.a.37.12		80	9.5	odd	6
1026.3.l.a.37.28		80	171.113	even	6
1026.3.l.a.721.12		80	57.56	even	2
1026.3.l.a.721.28		80	3.2	odd	2