Embedded newform 342.3.l.a.151.9

• Label: 342.3.l.a.151.9
• 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			151.9
Character	\(\chi\)	\(=\)	342.151
Dual form			342.3.l.a.265.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22474 - 0.707107i) q^{2} +(-0.749940 - 2.90475i) q^{3} +(1.00000 + 1.73205i) q^{4} +(-2.70510 - 4.68538i) q^{5} +(-1.13549 + 4.08787i) q^{6} +(-0.415272 + 0.719272i) q^{7} -2.82843i q^{8} +(-7.87518 + 4.35678i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 80 q^{4} + 8 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.749940	−	2.90475i	−0.249980	−	0.968251i
\(5\)	−2.70510	−	4.68538i	−0.541021	−	0.937076i								−0.998846	−	0.0480334i	\(-0.984705\pi\)
							0.457825	−	0.889042i	\(-0.348629\pi\)
\(7\)	−0.415272	+	0.719272i	−0.0593245	+	0.102753i	−0.894162	−	0.447743i	\(-0.852228\pi\)
							0.834838	+	0.550496i	\(0.185561\pi\)
\(11\)	−9.84646	+	17.0546i	−0.895133	+	1.55042i	−0.0614927	+	0.998108i	\(0.519586\pi\)
							−0.833640	+	0.552308i	\(0.813747\pi\)
\(13\)	7.55354	−	4.36104i	0.581042	−	0.335465i	−0.180506	−	0.983574i	\(-0.557773\pi\)
							0.761547	+	0.648109i	\(0.224440\pi\)
\(17\)	−21.4426			−1.26133			−0.630666	−	0.776054i	\(-0.717218\pi\)
							−0.630666	+	0.776054i	\(0.717218\pi\)
\(19\)	14.8709	−	11.8261i	0.782677	−	0.622428i
							\(23\)	16.1443	+	27.9627i	0.701925	+	1.21577i	0.967790	+	0.251760i	\(0.0810092\pi\)
−0.265865	+	0.964010i	\(0.585657\pi\)
\(29\)	−35.2849	−	20.3718i	−1.21672	−	0.702475i	−0.252507	−	0.967595i	\(-0.581255\pi\)
							−0.964215	+	0.265120i	\(0.914588\pi\)
\(31\)	−6.87107	+	3.96701i	−0.221647	+	0.127968i	−0.606713	−	0.794921i	\(-0.707512\pi\)
							0.385065	+	0.922889i	\(0.374179\pi\)
\(37\)			68.0923i			1.84033i	0.391527	+	0.920167i	\(0.371947\pi\)
							−0.391527	+	0.920167i	\(0.628053\pi\)
\(41\)	26.5094	−	15.3052i	0.646571	−	0.373298i	−0.140570	−	0.990071i	\(-0.544894\pi\)
							0.787141	+	0.616773i	\(0.211560\pi\)
\(43\)	7.09853	−	12.2950i	0.165082	−	0.285931i	−0.771602	−	0.636105i	\(-0.780544\pi\)
							0.936684	+	0.350175i	\(0.113878\pi\)
\(47\)	20.9521	−	36.2902i	0.445790	−	0.772131i	−0.552317	−	0.833634i	\(-0.686256\pi\)
							0.998107	+	0.0615033i	\(0.0195895\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.3.l.a.151.9	✓	80
3.2	odd	2		1026.3.l.a.721.24		80
9.4	even	3	inner	342.3.l.a.265.32	yes	80
9.5	odd	6		1026.3.l.a.37.10		80
19.18	odd	2	inner	342.3.l.a.151.32	yes	80
57.56	even	2		1026.3.l.a.721.10		80
171.94	odd	6	inner	342.3.l.a.265.9	yes	80
171.113	even	6		1026.3.l.a.37.24		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
342.3.l.a.151.9	✓	80	1.1	even	1	trivial
342.3.l.a.151.32	yes	80	19.18	odd	2	inner
342.3.l.a.265.9	yes	80	171.94	odd	6	inner
342.3.l.a.265.32	yes	80	9.4	even	3	inner
1026.3.l.a.37.10		80	9.5	odd	6
1026.3.l.a.37.24		80	171.113	even	6
1026.3.l.a.721.10		80	57.56	even	2
1026.3.l.a.721.24		80	3.2	odd	2