Embedded newform 342.3.l.a.151.6

• Label: 342.3.l.a.151.6
• 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			151.6
Character	\(\chi\)	\(=\)	342.151
Dual form			342.3.l.a.265.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22474 - 0.707107i) q^{2} +(-2.33101 + 1.88849i) q^{3} +(1.00000 + 1.73205i) q^{4} +(-0.682011 - 1.18128i) q^{5} +(4.19026 - 0.664650i) q^{6} +(-1.09727 + 1.90053i) q^{7} -2.82843i q^{8} +(1.86719 - 8.80418i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 80 q^{4} + 8 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.33101	+	1.88849i	−0.777003	+	0.629497i
\(5\)	−0.682011	−	1.18128i	−0.136402	−	0.236256i								0.789730	−	0.613455i	\(-0.210221\pi\)
							−0.926132	+	0.377199i	\(0.876887\pi\)
\(7\)	−1.09727	+	1.90053i	−0.156753	+	0.271505i	−0.933696	−	0.358066i	\(-0.883436\pi\)
							0.776943	+	0.629571i	\(0.216769\pi\)
\(11\)	3.73465	−	6.46860i	0.339513	−	0.588054i	−0.644828	−	0.764328i	\(-0.723071\pi\)
							0.984341	+	0.176273i	\(0.0564043\pi\)
\(13\)	−20.9012	+	12.0673i	−1.60778	+	0.928253i	−0.617917	+	0.786243i	\(0.712023\pi\)
							−0.989865	+	0.142010i	\(0.954643\pi\)
\(17\)	7.84796			0.461645			0.230822	−	0.972996i	\(-0.425858\pi\)
							0.230822	+	0.972996i	\(0.425858\pi\)
\(19\)	1.20076	−	18.9620i	0.0631981	−	0.998001i
							\(23\)	9.99324	+	17.3088i	0.434489	+	0.752557i	0.997254	−	0.0740602i	\(-0.0235957\pi\)
−0.562765	+	0.826617i	\(0.690262\pi\)
\(29\)	24.0144	+	13.8647i	0.828084	+	0.478094i	0.853196	−	0.521590i	\(-0.174661\pi\)
							−0.0251124	+	0.999685i	\(0.507994\pi\)
\(31\)	26.3792	−	15.2300i	0.850941	−	0.491291i	−0.0100270	−	0.999950i	\(-0.503192\pi\)
							0.860968	+	0.508659i	\(0.169858\pi\)
\(37\)		−	6.39162i		−	0.172746i	−0.996263	−	0.0863732i	\(-0.972472\pi\)
							0.996263	−	0.0863732i	\(-0.0275278\pi\)
\(41\)	61.0154	−	35.2272i	1.48818	−	0.859201i	0.488270	−	0.872692i	\(-0.337628\pi\)
							0.999909	+	0.0134915i	\(0.00429462\pi\)
\(43\)	−0.272147	+	0.471373i	−0.00632901	+	0.0109622i	−0.869173	−	0.494509i	\(-0.835348\pi\)
							0.862844	+	0.505471i	\(0.168681\pi\)
\(47\)	27.1482	−	47.0220i	0.577621	−	1.00047i	−0.418130	−	0.908387i	\(-0.637315\pi\)
							0.995751	−	0.0920819i	\(-0.0293521\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.3.l.a.151.6	✓	80
3.2	odd	2		1026.3.l.a.721.21		80
9.4	even	3	inner	342.3.l.a.265.35	yes	80
9.5	odd	6		1026.3.l.a.37.18		80
19.18	odd	2	inner	342.3.l.a.151.35	yes	80
57.56	even	2		1026.3.l.a.721.18		80
171.94	odd	6	inner	342.3.l.a.265.6	yes	80
171.113	even	6		1026.3.l.a.37.21		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
342.3.l.a.151.6	✓	80	1.1	even	1	trivial
342.3.l.a.151.35	yes	80	19.18	odd	2	inner
342.3.l.a.265.6	yes	80	171.94	odd	6	inner
342.3.l.a.265.35	yes	80	9.4	even	3	inner
1026.3.l.a.37.18		80	9.5	odd	6
1026.3.l.a.37.21		80	171.113	even	6
1026.3.l.a.721.18		80	57.56	even	2
1026.3.l.a.721.21		80	3.2	odd	2