Embedded newform 342.3.l.a.265.7

• Label: 342.3.l.a.265.7
• Level: $342$
• Weight: $3$
• Character: 342.265
• 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			265.7
Character	\(\chi\)	\(=\)	342.265
Dual form			342.3.l.a.151.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22474 + 0.707107i) q^{2} +(-1.69573 - 2.47477i) q^{3} +(1.00000 - 1.73205i) q^{4} +(4.69486 - 8.13173i) q^{5} +(3.82677 + 1.83190i) q^{6} +(5.67137 + 9.82311i) q^{7} +2.82843i q^{8} +(-3.24900 + 8.39309i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 80 q^{4} + 8 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).

\(n\)	\(191\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(-1\)

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
							\(3\)	−1.69573	−	2.47477i	−0.565243	−	0.824924i
\(5\)	4.69486	−	8.13173i	0.938971	−	1.62635i								0.171577	−	0.985171i	\(-0.445114\pi\)
							0.767395	−	0.641175i	\(-0.221553\pi\)
\(7\)	5.67137	+	9.82311i	0.810196	+	1.40330i	0.912726	+	0.408571i	\(0.133973\pi\)
							−0.102530	+	0.994730i	\(0.532694\pi\)
\(11\)	3.82304	+	6.62171i	0.347549	+	0.601973i	0.985814	−	0.167844i	\(-0.0536805\pi\)
							−0.638264	+	0.769817i	\(0.720347\pi\)
\(13\)	15.6637	+	9.04345i	1.20490	+	0.695650i	0.961641	−	0.274311i	\(-0.0884498\pi\)
							0.243260	+	0.969961i	\(0.421783\pi\)
\(17\)	21.1249			1.24264			0.621320	−	0.783557i	\(-0.286597\pi\)
							0.621320	+	0.783557i	\(0.286597\pi\)
\(19\)	−0.118426	+	18.9996i	−0.00623296	+	0.999981i
							\(23\)	−8.37354	+	14.5034i	−0.364067	+	0.630582i	−0.988626	−	0.150395i	\(-0.951945\pi\)
0.624559	+	0.780978i	\(0.285279\pi\)
\(29\)	12.3614	−	7.13685i	0.426255	−	0.246098i	−0.271495	−	0.962440i	\(-0.587518\pi\)
							0.697750	+	0.716341i	\(0.254185\pi\)
\(31\)	−15.1408	−	8.74155i	−0.488413	−	0.281986i	0.235503	−	0.971874i	\(-0.424326\pi\)
							−0.723916	+	0.689888i	\(0.757660\pi\)
\(37\)			10.6584i			0.288066i	0.989573	+	0.144033i	\(0.0460071\pi\)
							−0.989573	+	0.144033i	\(0.953993\pi\)
\(41\)	1.55194	+	0.896012i	0.0378521	+	0.0218539i	0.518807	−	0.854892i	\(-0.326376\pi\)
							−0.480955	+	0.876746i	\(0.659710\pi\)
\(43\)	16.7356	+	28.9869i	0.389200	+	0.674113i	0.992342	−	0.123520i	\(-0.0394183\pi\)
							−0.603143	+	0.797633i	\(0.706085\pi\)
\(47\)	−10.3261	−	17.8853i	−0.219704	−	0.380538i	0.735014	−	0.678052i	\(-0.237176\pi\)
							−0.954717	+	0.297514i	\(0.903842\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.3.l.a.265.7	yes	80
3.2	odd	2		1026.3.l.a.37.25		80
9.2	odd	6		1026.3.l.a.721.6		80
9.7	even	3	inner	342.3.l.a.151.34	yes	80
19.18	odd	2	inner	342.3.l.a.265.34	yes	80
57.56	even	2		1026.3.l.a.37.6		80
171.56	even	6		1026.3.l.a.721.25		80
171.151	odd	6	inner	342.3.l.a.151.7	✓	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
342.3.l.a.151.7	✓	80	171.151	odd	6	inner
342.3.l.a.151.34	yes	80	9.7	even	3	inner
342.3.l.a.265.7	yes	80	1.1	even	1	trivial
342.3.l.a.265.34	yes	80	19.18	odd	2	inner
1026.3.l.a.37.6		80	57.56	even	2
1026.3.l.a.37.25		80	3.2	odd	2
1026.3.l.a.721.6		80	9.2	odd	6
1026.3.l.a.721.25		80	171.56	even	6