Embedded newform 342.3.l.a.151.4

• Label: 342.3.l.a.151.4
• 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			151.4
Character	\(\chi\)	\(=\)	342.151
Dual form			342.3.l.a.265.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22474 - 0.707107i) q^{2} +(-2.56998 - 1.54765i) q^{3} +(1.00000 + 1.73205i) q^{4} +(0.947945 + 1.64189i) q^{5} +(2.05321 + 3.71273i) q^{6} +(4.99227 - 8.64687i) q^{7} -2.82843i q^{8} +(4.20955 + 7.95486i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 80 q^{4} + 8 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.56998	−	1.54765i	−0.856658	−	0.515884i
\(5\)	0.947945	+	1.64189i	0.189589	+	0.328378i								0.945113	−	0.326743i	\(-0.105951\pi\)
							−0.755524	+	0.655121i	\(0.772618\pi\)
\(7\)	4.99227	−	8.64687i	0.713182	−	1.23527i	−0.250475	−	0.968123i	\(-0.580587\pi\)
							0.963657	−	0.267144i	\(-0.0860798\pi\)
\(11\)	−9.00971	+	15.6053i	−0.819065	+	1.41866i	0.0873073	+	0.996181i	\(0.472174\pi\)
							−0.906372	+	0.422480i	\(0.861160\pi\)
\(13\)	−2.54923	+	1.47180i	−0.196095	+	0.113215i	−0.594833	−	0.803850i	\(-0.702782\pi\)
							0.398738	+	0.917065i	\(0.369448\pi\)
\(17\)	27.8440			1.63788			0.818941	−	0.573878i	\(-0.194562\pi\)
							0.818941	+	0.573878i	\(0.194562\pi\)
\(19\)	−14.6889	+	12.0514i	−0.773099	+	0.634286i
							\(23\)	3.49003	+	6.04491i	0.151740	+	0.262822i	0.931867	−	0.362799i	\(-0.118179\pi\)
−0.780127	+	0.625621i	\(0.784846\pi\)
\(29\)	30.5247	+	17.6234i	1.05257	+	0.607704i	0.923369	−	0.383914i	\(-0.125424\pi\)
							0.129206	+	0.991618i	\(0.458757\pi\)
\(31\)	9.80005	−	5.65806i	0.316131	−	0.182518i	−0.333536	−	0.942737i	\(-0.608242\pi\)
							0.649667	+	0.760219i	\(0.274909\pi\)
\(37\)		−	16.3168i		−	0.440995i	−0.975388	−	0.220497i	\(-0.929232\pi\)
							0.975388	−	0.220497i	\(-0.0707681\pi\)
\(41\)	64.1077	−	37.0126i	1.56360	−	0.902746i	0.566714	−	0.823914i	\(-0.308214\pi\)
							0.996888	−	0.0788316i	\(-0.0251189\pi\)
\(43\)	−2.41104	+	4.17604i	−0.0560706	+	0.0971171i	−0.892698	−	0.450655i	\(-0.851191\pi\)
							0.836628	+	0.547772i	\(0.184524\pi\)
\(47\)	33.7294	−	58.4210i	0.717647	−	1.24300i	−0.244283	−	0.969704i	\(-0.578553\pi\)
							0.961930	−	0.273297i	\(-0.0881142\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.3.l.a.151.4	✓	80
3.2	odd	2		1026.3.l.a.721.40		80
9.4	even	3	inner	342.3.l.a.265.37	yes	80
9.5	odd	6		1026.3.l.a.37.20		80
19.18	odd	2	inner	342.3.l.a.151.37	yes	80
57.56	even	2		1026.3.l.a.721.20		80
171.94	odd	6	inner	342.3.l.a.265.4	yes	80
171.113	even	6		1026.3.l.a.37.40		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
342.3.l.a.151.4	✓	80	1.1	even	1	trivial
342.3.l.a.151.37	yes	80	19.18	odd	2	inner
342.3.l.a.265.4	yes	80	171.94	odd	6	inner
342.3.l.a.265.37	yes	80	9.4	even	3	inner
1026.3.l.a.37.20		80	9.5	odd	6
1026.3.l.a.37.40		80	171.113	even	6
1026.3.l.a.721.20		80	57.56	even	2
1026.3.l.a.721.40		80	3.2	odd	2