Embedded newform 342.3.l.a.151.16

• Label: 342.3.l.a.151.16
• 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			151.16
Character	\(\chi\)	\(=\)	342.151
Dual form			342.3.l.a.265.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22474 - 0.707107i) q^{2} +(2.06353 + 2.17757i) q^{3} +(1.00000 + 1.73205i) q^{4} +(0.484620 + 0.839386i) q^{5} +(-0.987525 - 4.12611i) q^{6} +(1.11147 - 1.92512i) q^{7} -2.82843i q^{8} +(-0.483659 + 8.98699i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 80 q^{4} + 8 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10}) \)

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
							\(3\)	2.06353	+	2.17757i	0.687845	+	0.725858i
\(5\)	0.484620	+	0.839386i	0.0969240	+	0.167877i								0.910410	−	0.413707i	\(-0.135766\pi\)
							−0.813486	+	0.581585i	\(0.802433\pi\)
\(7\)	1.11147	−	1.92512i	0.158782	−	0.275018i	−0.775648	−	0.631166i	\(-0.782577\pi\)
							0.934430	+	0.356148i	\(0.115910\pi\)
\(11\)	10.8408	−	18.7769i	0.985531	−	1.70699i	0.345976	−	0.938243i	\(-0.387548\pi\)
							0.639555	−	0.768745i	\(-0.279119\pi\)
\(13\)	5.44332	−	3.14271i	0.418717	−	0.241747i	−0.275811	−	0.961212i	\(-0.588946\pi\)
							0.694528	+	0.719465i	\(0.255613\pi\)
\(17\)	10.1038			0.594344			0.297172	−	0.954824i	\(-0.403957\pi\)
							0.297172	+	0.954824i	\(0.403957\pi\)
\(19\)	8.54043	+	16.9724i	0.449496	+	0.893282i
							\(23\)	7.78151	+	13.4780i	0.338326	+	0.585998i	0.984118	−	0.177515i	\(-0.0568059\pi\)
−0.645792	+	0.763514i	\(0.723473\pi\)
\(29\)	23.3372	+	13.4738i	0.804732	+	0.464612i	0.845123	−	0.534571i	\(-0.179527\pi\)
							−0.0403908	+	0.999184i	\(0.512860\pi\)
\(31\)	−40.9885	+	23.6647i	−1.32221	+	0.763378i	−0.984081	−	0.177722i	\(-0.943127\pi\)
							−0.338129	+	0.941100i	\(0.609794\pi\)
\(37\)			38.0061i			1.02719i	0.858032	+	0.513595i	\(0.171687\pi\)
							−0.858032	+	0.513595i	\(0.828313\pi\)
\(41\)	−12.0945	+	6.98279i	−0.294989	+	0.170312i	−0.640190	−	0.768217i	\(-0.721144\pi\)
							0.345201	+	0.938529i	\(0.387811\pi\)
\(43\)	−20.1016	+	34.8171i	−0.467480	+	0.809699i	−0.999310	−	0.0371522i	\(-0.988171\pi\)
							0.531830	+	0.846851i	\(0.321505\pi\)
\(47\)	21.5785	−	37.3751i	0.459117	−	0.795215i	−0.539797	−	0.841795i	\(-0.681499\pi\)
							0.998915	+	0.0465806i	\(0.0148324\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.3.l.a.151.16	✓	80
3.2	odd	2		1026.3.l.a.721.37		80
9.4	even	3	inner	342.3.l.a.265.25	yes	80
9.5	odd	6		1026.3.l.a.37.19		80
19.18	odd	2	inner	342.3.l.a.151.25	yes	80
57.56	even	2		1026.3.l.a.721.19		80
171.94	odd	6	inner	342.3.l.a.265.16	yes	80
171.113	even	6		1026.3.l.a.37.37		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
342.3.l.a.151.16	✓	80	1.1	even	1	trivial
342.3.l.a.151.25	yes	80	19.18	odd	2	inner
342.3.l.a.265.16	yes	80	171.94	odd	6	inner
342.3.l.a.265.25	yes	80	9.4	even	3	inner
1026.3.l.a.37.19		80	9.5	odd	6
1026.3.l.a.37.37		80	171.113	even	6
1026.3.l.a.721.19		80	57.56	even	2
1026.3.l.a.721.37		80	3.2	odd	2