Embedded newform 342.3.l.a.151.17

• Label: 342.3.l.a.151.17
• Level: $342$
• Weight: $3$
• Character: 342.151
• Analytic conductor: $9.319$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• 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			151.17
Character	\(\chi\)	\(=\)	342.151
Dual form			342.3.l.a.265.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22474 - 0.707107i) q^{2} +(2.61697 - 1.46679i) q^{3} +(1.00000 + 1.73205i) q^{4} +(0.350777 + 0.607564i) q^{5} +(-4.24230 - 0.0540357i) q^{6} +(5.86746 - 10.1627i) q^{7} -2.82843i q^{8} +(4.69706 - 7.67708i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 80 q^{4} + 8 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10}) \)

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
							\(3\)	2.61697	−	1.46679i	0.872323	−	0.488929i
\(5\)	0.350777	+	0.607564i	0.0701554	+	0.121513i								0.898969	−	0.438012i	\(-0.144317\pi\)
							−0.828814	+	0.559524i	\(0.810984\pi\)
\(7\)	5.86746	−	10.1627i	0.838208	−	1.45182i	−0.0531834	−	0.998585i	\(-0.516937\pi\)
							0.891391	−	0.453234i	\(-0.149730\pi\)
\(11\)	0.375070	−	0.649641i	0.0340973	−	0.0590583i	−0.848473	−	0.529238i	\(-0.822478\pi\)
							0.882570	+	0.470180i	\(0.155811\pi\)
\(13\)	−11.4554	+	6.61377i	−0.881183	+	0.508751i	−0.871048	−	0.491197i	\(-0.836559\pi\)
							−0.0101349	+	0.999949i	\(0.503226\pi\)
\(17\)	−8.97913			−0.528184			−0.264092	−	0.964498i	\(-0.585072\pi\)
							−0.264092	+	0.964498i	\(0.585072\pi\)
\(19\)	16.3754	−	9.63559i	0.861866	−	0.507137i
							\(23\)	9.47272	+	16.4072i	0.411857	+	0.713358i	0.995093	−	0.0989453i	\(-0.0315469\pi\)
−0.583236	+	0.812303i	\(0.698214\pi\)
\(29\)	−1.14911	−	0.663439i	−0.0396245	−	0.0228772i	0.480057	−	0.877237i	\(-0.340616\pi\)
							−0.519681	+	0.854360i	\(0.673949\pi\)
\(31\)	−5.23883	+	3.02464i	−0.168994	+	0.0975690i	−0.582112	−	0.813109i	\(-0.697773\pi\)
							0.413117	+	0.910678i	\(0.364440\pi\)
\(37\)		−	42.1016i		−	1.13788i	−0.822379	−	0.568940i	\(-0.807354\pi\)
							0.822379	−	0.568940i	\(-0.192646\pi\)
\(41\)	−43.2379	+	24.9634i	−1.05458	+	0.608864i	−0.923929	−	0.382565i	\(-0.875041\pi\)
							−0.130654	+	0.991428i	\(0.541708\pi\)
\(43\)	−1.35532	+	2.34748i	−0.0315190	+	0.0545926i	−0.881355	−	0.472455i	\(-0.843368\pi\)
							0.849836	+	0.527048i	\(0.176701\pi\)
\(47\)	5.09279	−	8.82096i	0.108357	−	0.187680i	−0.806748	−	0.590896i	\(-0.798774\pi\)
							0.915105	+	0.403216i	\(0.132108\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.3.l.a.151.17	✓	80
3.2	odd	2		1026.3.l.a.721.31		80
9.4	even	3	inner	342.3.l.a.265.24	yes	80
9.5	odd	6		1026.3.l.a.37.4		80
19.18	odd	2	inner	342.3.l.a.151.24	yes	80
57.56	even	2		1026.3.l.a.721.4		80
171.94	odd	6	inner	342.3.l.a.265.17	yes	80
171.113	even	6		1026.3.l.a.37.31		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
342.3.l.a.151.17	✓	80	1.1	even	1	trivial
342.3.l.a.151.24	yes	80	19.18	odd	2	inner
342.3.l.a.265.17	yes	80	171.94	odd	6	inner
342.3.l.a.265.24	yes	80	9.4	even	3	inner
1026.3.l.a.37.4		80	9.5	odd	6
1026.3.l.a.37.31		80	171.113	even	6
1026.3.l.a.721.4		80	57.56	even	2
1026.3.l.a.721.31		80	3.2	odd	2