Embedded newform 342.3.l.a.151.10

• Label: 342.3.l.a.151.10
• 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			151.10
Character	\(\chi\)	\(=\)	342.151
Dual form			342.3.l.a.265.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22474 - 0.707107i) q^{2} +(-0.212313 + 2.99248i) q^{3} +(1.00000 + 1.73205i) q^{4} +(-4.45237 - 7.71173i) q^{5} +(2.37603 - 3.51489i) q^{6} +(-6.19455 + 10.7293i) q^{7} -2.82843i q^{8} +(-8.90985 - 1.27068i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 80 q^{4} + 8 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.212313	+	2.99248i	−0.0707709	+	0.997493i
\(5\)	−4.45237	−	7.71173i	−0.890474	−	1.54235i								−0.839308	−	0.543656i	\(-0.817040\pi\)
							−0.0511656	−	0.998690i	\(-0.516294\pi\)
\(7\)	−6.19455	+	10.7293i	−0.884936	+	1.53275i	−0.0391480	+	0.999233i	\(0.512464\pi\)
							−0.845788	+	0.533520i	\(0.820869\pi\)
\(11\)	2.11807	−	3.66861i	0.192552	−	0.333510i	−0.753543	−	0.657398i	\(-0.771657\pi\)
							0.946095	+	0.323889i	\(0.104990\pi\)
\(13\)	14.4996	−	8.37133i	1.11535	−	0.643949i	0.175142	−	0.984543i	\(-0.443962\pi\)
							0.940210	+	0.340594i	\(0.110628\pi\)
\(17\)	−0.129909			−0.00764171			−0.00382086	−	0.999993i	\(-0.501216\pi\)
							−0.00382086	+	0.999993i	\(0.501216\pi\)
\(19\)	18.8321	+	2.52054i	0.991162	+	0.132660i
							\(23\)	−8.83731	−	15.3067i	−0.384231	−	0.665507i	0.607431	−	0.794372i	\(-0.292200\pi\)
−0.991662	+	0.128865i	\(0.958867\pi\)
\(29\)	31.2335	+	18.0327i	1.07702	+	0.621816i	0.930090	−	0.367331i	\(-0.119728\pi\)
							0.146927	+	0.989147i	\(0.453062\pi\)
\(31\)	32.5389	−	18.7863i	1.04964	−	0.606011i	0.127093	−	0.991891i	\(-0.459435\pi\)
							0.922549	+	0.385880i	\(0.126102\pi\)
\(37\)			12.2869i			0.332079i	0.986119	+	0.166040i	\(0.0530980\pi\)
							−0.986119	+	0.166040i	\(0.946902\pi\)
\(41\)	22.7159	−	13.1150i	0.554046	−	0.319878i	−0.196707	−	0.980462i	\(-0.563025\pi\)
							0.750752	+	0.660584i	\(0.229691\pi\)
\(43\)	32.1357	−	55.6607i	0.747342	−	1.29444i	−0.201750	−	0.979437i	\(-0.564663\pi\)
							0.949092	−	0.314998i	\(-0.102004\pi\)
\(47\)	5.67521	−	9.82976i	0.120749	−	0.209144i	−0.799314	−	0.600913i	\(-0.794804\pi\)
							0.920063	+	0.391770i	\(0.128137\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.3.l.a.151.10	✓	80
3.2	odd	2		1026.3.l.a.721.38		80
9.4	even	3	inner	342.3.l.a.265.31	yes	80
9.5	odd	6		1026.3.l.a.37.1		80
19.18	odd	2	inner	342.3.l.a.151.31	yes	80
57.56	even	2		1026.3.l.a.721.1		80
171.94	odd	6	inner	342.3.l.a.265.10	yes	80
171.113	even	6		1026.3.l.a.37.38		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
342.3.l.a.151.10	✓	80	1.1	even	1	trivial
342.3.l.a.151.31	yes	80	19.18	odd	2	inner
342.3.l.a.265.10	yes	80	171.94	odd	6	inner
342.3.l.a.265.31	yes	80	9.4	even	3	inner
1026.3.l.a.37.1		80	9.5	odd	6
1026.3.l.a.37.38		80	171.113	even	6
1026.3.l.a.721.1		80	57.56	even	2
1026.3.l.a.721.38		80	3.2	odd	2