Embedded newform 342.3.l.a.151.20

• Label: 342.3.l.a.151.20
• 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			151.20
Character	\(\chi\)	\(=\)	342.151
Dual form			342.3.l.a.265.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22474 - 0.707107i) q^{2} +(2.96233 - 0.473898i) q^{3} +(1.00000 + 1.73205i) q^{4} +(4.58731 + 7.94545i) q^{5} +(-3.96320 - 1.51428i) q^{6} +(-1.84045 + 3.18775i) q^{7} -2.82843i q^{8} +(8.55084 - 2.80769i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 80 q^{4} + 8 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10}) \)

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
							\(3\)	2.96233	−	0.473898i	0.987445	−	0.157966i
\(5\)	4.58731	+	7.94545i	0.917462	+	1.58909i								0.803257	+	0.595633i	\(0.203099\pi\)
							0.114205	+	0.993457i	\(0.463568\pi\)
\(7\)	−1.84045	+	3.18775i	−0.262921	+	0.455393i	−0.967017	−	0.254712i	\(-0.918019\pi\)
							0.704096	+	0.710105i	\(0.251353\pi\)
\(11\)	−0.0446586	+	0.0773509i	−0.00405987	+	0.00703190i	−0.868048	−	0.496480i	\(-0.834626\pi\)
							0.863988	+	0.503512i	\(0.167959\pi\)
\(13\)	0.729014	−	0.420896i	0.0560780	−	0.0323766i	−0.471699	−	0.881760i	\(-0.656359\pi\)
							0.527777	+	0.849383i	\(0.323026\pi\)
\(17\)	−0.505636			−0.0297433			−0.0148717	−	0.999889i	\(-0.504734\pi\)
							−0.0148717	+	0.999889i	\(0.504734\pi\)
\(19\)	−14.3885	+	12.4085i	−0.757290	+	0.653079i
							\(23\)	4.07909	+	7.06519i	0.177352	+	0.307182i	0.940973	−	0.338483i	\(-0.109914\pi\)
−0.763621	+	0.645665i	\(0.776580\pi\)
\(29\)	−21.2563	−	12.2723i	−0.732975	−	0.423183i	0.0865345	−	0.996249i	\(-0.472421\pi\)
							−0.819510	+	0.573065i	\(0.805754\pi\)
\(31\)	17.1632	−	9.90920i	0.553653	−	0.319652i	−0.196941	−	0.980415i	\(-0.563101\pi\)
							0.750594	+	0.660764i	\(0.229767\pi\)
\(37\)			16.3163i			0.440982i	0.975389	+	0.220491i	\(0.0707660\pi\)
							−0.975389	+	0.220491i	\(0.929234\pi\)
\(41\)	24.2204	−	13.9836i	0.590741	−	0.341064i	−0.174650	−	0.984631i	\(-0.555879\pi\)
							0.765390	+	0.643566i	\(0.222546\pi\)
\(43\)	−8.41135	+	14.5689i	−0.195613	+	0.338811i	−0.947101	−	0.320935i	\(-0.896003\pi\)
							0.751488	+	0.659746i	\(0.229336\pi\)
\(47\)	46.0021	−	79.6779i	0.978767	−	1.69527i	0.311870	−	0.950125i	\(-0.399045\pi\)
							0.666897	−	0.745150i	\(-0.267622\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.3.l.a.151.20	✓	80
3.2	odd	2		1026.3.l.a.721.32		80
9.4	even	3	inner	342.3.l.a.265.21	yes	80
9.5	odd	6		1026.3.l.a.37.9		80
19.18	odd	2	inner	342.3.l.a.151.21	yes	80
57.56	even	2		1026.3.l.a.721.9		80
171.94	odd	6	inner	342.3.l.a.265.20	yes	80
171.113	even	6		1026.3.l.a.37.32		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
342.3.l.a.151.20	✓	80	1.1	even	1	trivial
342.3.l.a.151.21	yes	80	19.18	odd	2	inner
342.3.l.a.265.20	yes	80	171.94	odd	6	inner
342.3.l.a.265.21	yes	80	9.4	even	3	inner
1026.3.l.a.37.9		80	9.5	odd	6
1026.3.l.a.37.32		80	171.113	even	6
1026.3.l.a.721.9		80	57.56	even	2
1026.3.l.a.721.32		80	3.2	odd	2