Embedded newform 342.3.l.a.151.15

• Label: 342.3.l.a.151.15
• 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			151.15
Character	\(\chi\)	\(=\)	342.151
Dual form			342.3.l.a.265.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22474 - 0.707107i) q^{2} +(1.96099 - 2.27035i) q^{3} +(1.00000 + 1.73205i) q^{4} +(-1.90135 - 3.29324i) q^{5} +(-4.00710 + 1.39397i) q^{6} +(-4.83213 + 8.36949i) q^{7} -2.82843i q^{8} +(-1.30902 - 8.90429i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 80 q^{4} + 8 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10}) \)

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
							\(3\)	1.96099	−	2.27035i	0.653664	−	0.756785i
\(5\)	−1.90135	−	3.29324i	−0.380271	−	0.658649i								0.610830	−	0.791762i	\(-0.290836\pi\)
							−0.991101	+	0.133113i	\(0.957503\pi\)
\(7\)	−4.83213	+	8.36949i	−0.690304	+	1.19564i	0.281435	+	0.959580i	\(0.409190\pi\)
							−0.971738	+	0.236061i	\(0.924144\pi\)
\(11\)	0.423662	−	0.733805i	0.0385147	−	0.0667095i	−0.846125	−	0.532984i	\(-0.821071\pi\)
							0.884640	+	0.466274i	\(0.154404\pi\)
\(13\)	−5.15772	+	2.97781i	−0.396748	+	0.229063i	−0.685080	−	0.728468i	\(-0.740233\pi\)
							0.288332	+	0.957531i	\(0.406899\pi\)
\(17\)	−32.6621			−1.92130			−0.960651	−	0.277758i	\(-0.910409\pi\)
							−0.960651	+	0.277758i	\(0.910409\pi\)
\(19\)	−14.6983	+	12.0400i	−0.773594	+	0.633682i
							\(23\)	−3.69647	−	6.40247i	−0.160716	−	0.278368i	0.774410	−	0.632684i	\(-0.218047\pi\)
−0.935126	+	0.354316i	\(0.884714\pi\)
\(29\)	37.5642	+	21.6877i	1.29532	+	0.747851i	0.979591	−	0.201000i	\(-0.0644190\pi\)
							0.315725	+	0.948851i	\(0.397752\pi\)
\(31\)	−31.5817	+	18.2337i	−1.01876	+	0.588183i	−0.913745	−	0.406287i	\(-0.866823\pi\)
							−0.105018	+	0.994470i	\(0.533490\pi\)
\(37\)		−	51.1367i		−	1.38207i	−0.722819	−	0.691037i	\(-0.757154\pi\)
							0.722819	−	0.691037i	\(-0.242846\pi\)
\(41\)	15.1912	−	8.77065i	0.370517	−	0.213918i	−0.303167	−	0.952937i	\(-0.598044\pi\)
							0.673684	+	0.739019i	\(0.264711\pi\)
\(43\)	−41.9444	+	72.6498i	−0.975451	+	1.68953i	−0.297013	+	0.954873i	\(0.595990\pi\)
							−0.678438	+	0.734658i	\(0.737343\pi\)
\(47\)	8.80651	−	15.2533i	0.187373	−	0.324539i	−0.757001	−	0.653414i	\(-0.773336\pi\)
							0.944373	+	0.328875i	\(0.106669\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.3.l.a.151.15	✓	80
3.2	odd	2		1026.3.l.a.721.39		80
9.4	even	3	inner	342.3.l.a.265.26	yes	80
9.5	odd	6		1026.3.l.a.37.13		80
19.18	odd	2	inner	342.3.l.a.151.26	yes	80
57.56	even	2		1026.3.l.a.721.13		80
171.94	odd	6	inner	342.3.l.a.265.15	yes	80
171.113	even	6		1026.3.l.a.37.39		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
342.3.l.a.151.15	✓	80	1.1	even	1	trivial
342.3.l.a.151.26	yes	80	19.18	odd	2	inner
342.3.l.a.265.15	yes	80	171.94	odd	6	inner
342.3.l.a.265.26	yes	80	9.4	even	3	inner
1026.3.l.a.37.13		80	9.5	odd	6
1026.3.l.a.37.39		80	171.113	even	6
1026.3.l.a.721.13		80	57.56	even	2
1026.3.l.a.721.39		80	3.2	odd	2