Embedded newform 342.3.l.a.151.1

• Label: 342.3.l.a.151.1
• Level: $342$
• Weight: $3$
• Character: 342.151
• Analytic conductor: $9.319$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• 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			151.1
Character	\(\chi\)	\(=\)	342.151
Dual form			342.3.l.a.265.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22474 - 0.707107i) q^{2} +(-2.99547 - 0.164850i) q^{3} +(1.00000 + 1.73205i) q^{4} +(-3.44954 - 5.97479i) q^{5} +(3.55212 + 2.32001i) q^{6} +(3.48951 - 6.04402i) q^{7} -2.82843i q^{8} +(8.94565 + 0.987607i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 80 q^{4} + 8 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.99547	−	0.164850i	−0.998489	−	0.0549501i
\(5\)	−3.44954	−	5.97479i	−0.689909	−	1.19496i								−0.971867	−	0.235531i	\(-0.924317\pi\)
							0.281958	−	0.959427i	\(-0.409016\pi\)
\(7\)	3.48951	−	6.04402i	0.498502	−	0.863431i	−0.501496	−	0.865160i	\(-0.667217\pi\)
							0.999999	+	0.00172875i	\(0.000550279\pi\)
\(11\)	4.33172	−	7.50277i	0.393793	−	0.682070i	−0.599153	−	0.800634i	\(-0.704496\pi\)
							0.992946	+	0.118565i	\(0.0378293\pi\)
\(13\)	6.57022	−	3.79332i	0.505401	−	0.291794i	−0.225540	−	0.974234i	\(-0.572415\pi\)
							0.730941	+	0.682440i	\(0.239081\pi\)
\(17\)	−5.75011			−0.338242			−0.169121	−	0.985595i	\(-0.554093\pi\)
							−0.169121	+	0.985595i	\(0.554093\pi\)
\(19\)	18.7366	−	3.15304i	0.986134	−	0.165949i
							\(23\)	−11.5575	−	20.0182i	−0.502500	−	0.870356i	−0.999996	−	0.00288964i	\(-0.999080\pi\)
0.497495	−	0.867467i	\(-0.334253\pi\)
\(29\)	−22.1512	−	12.7890i	−0.763835	−	0.441000i	0.0668361	−	0.997764i	\(-0.478710\pi\)
							−0.830671	+	0.556764i	\(0.812043\pi\)
\(31\)	−23.5126	+	13.5750i	−0.758470	+	0.437903i	−0.828746	−	0.559625i	\(-0.810945\pi\)
							0.0702760	+	0.997528i	\(0.477612\pi\)
\(37\)		−	40.7417i		−	1.10113i	−0.834793	−	0.550563i	\(-0.814413\pi\)
							0.834793	−	0.550563i	\(-0.185587\pi\)
\(41\)	21.1993	−	12.2394i	0.517055	−	0.298522i	−0.218674	−	0.975798i	\(-0.570173\pi\)
							0.735729	+	0.677276i	\(0.236840\pi\)
\(43\)	−33.5714	+	58.1474i	−0.780731	+	1.35227i	0.150785	+	0.988567i	\(0.451820\pi\)
							−0.931516	+	0.363700i	\(0.881513\pi\)
\(47\)	−17.8471	+	30.9121i	−0.379726	+	0.657705i	−0.991022	−	0.133697i	\(-0.957315\pi\)
							0.611296	+	0.791402i	\(0.290648\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.3.l.a.151.1	✓	80
3.2	odd	2		1026.3.l.a.721.35		80
9.4	even	3	inner	342.3.l.a.265.40	yes	80
9.5	odd	6		1026.3.l.a.37.15		80
19.18	odd	2	inner	342.3.l.a.151.40	yes	80
57.56	even	2		1026.3.l.a.721.15		80
171.94	odd	6	inner	342.3.l.a.265.1	yes	80
171.113	even	6		1026.3.l.a.37.35		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
342.3.l.a.151.1	✓	80	1.1	even	1	trivial
342.3.l.a.151.40	yes	80	19.18	odd	2	inner
342.3.l.a.265.1	yes	80	171.94	odd	6	inner
342.3.l.a.265.40	yes	80	9.4	even	3	inner
1026.3.l.a.37.15		80	9.5	odd	6
1026.3.l.a.37.35		80	171.113	even	6
1026.3.l.a.721.15		80	57.56	even	2
1026.3.l.a.721.35		80	3.2	odd	2