Embedded newform 343.2.k.a.100.20

• Label: 343.2.k.a.100.20
• Level: $343$
• Weight: $2$
• Character: 343.100
• Analytic conductor: $2.739$
• Analytic rank: $0$
• Dimension: $2688$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 343 = 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	343.k (of order \(147\), degree \(84\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.73886878933\)
Analytic rank:	\(0\)
Dimension:	\(2688\)
Relative dimension:	\(32\) over \(\Q(\zeta_{147})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{147}]$

Embedding invariants

Embedding label			100.20
Character	\(\chi\)	\(=\)	343.100
Dual form			343.2.k.a.319.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.523292 + 0.717022i) q^{2} +(-0.0609715 + 1.14009i) q^{3} +(0.369612 - 1.15431i) q^{4} +(1.15463 - 0.0246797i) q^{5} +(-0.849378 + 0.552884i) q^{6} +(-2.54887 + 0.709426i) q^{7} +(2.70598 - 0.898446i) q^{8} +(1.68679 + 0.180934i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2688 q - 84 q^{2} - 84 q^{3} - 84 q^{4} - 84 q^{5} - 98 q^{6} - 84 q^{7} - 84 q^{8} - 98 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/343\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{76}{147}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.523292	+	0.717022i	0.370023	+	0.507011i	0.951960	−	0.306223i	\(-0.0990654\pi\)
							−0.581937	+	0.813234i	\(0.697705\pi\)
\(3\)	−0.0609715	+	1.14009i	−0.0352019	+	0.658234i	0.925177	+	0.379536i	\(0.123916\pi\)
							−0.960379	+	0.278698i	\(0.910097\pi\)
\(5\)	1.15463	−	0.0246797i	0.516364	−	0.0110371i	0.237537	−	0.971378i	\(-0.423660\pi\)
							0.278827	+	0.960341i	\(0.410054\pi\)
\(7\)	−2.54887	+	0.709426i	−0.963381	+	0.268138i
							\(11\)	−0.695780	+	5.89210i	−0.209786	+	1.77654i	0.339211	+	0.940710i	\(0.389840\pi\)
−0.548996	+	0.835825i	\(0.684990\pi\)
\(13\)	0.561616	+	1.07651i	0.155764	+	0.298571i	0.951823	−	0.306649i	\(-0.0992077\pi\)
							−0.796059	+	0.605220i	\(0.793085\pi\)
\(17\)	7.78974	−	1.86575i	1.88929	−	0.452511i	0.889302	−	0.457321i	\(-0.151191\pi\)
							0.999988	+	0.00481036i	\(0.00153119\pi\)
\(19\)	3.38298	−	5.85948i	0.776108	−	1.34426i	−0.158062	−	0.987429i	\(-0.550525\pi\)
							0.934170	−	0.356829i	\(-0.116142\pi\)
\(23\)	−4.38703	+	3.42250i	−0.914760	+	0.713640i	−0.958406	−	0.285407i	\(-0.907871\pi\)
							0.0436464	+	0.999047i	\(0.486103\pi\)
\(29\)	−1.56536	+	0.633745i	−0.290680	+	0.117683i	−0.515965	−	0.856609i	\(-0.672567\pi\)
							0.225285	+	0.974293i	\(0.427669\pi\)
\(31\)	−0.510023	+	0.473232i	−0.0916029	+	0.0849950i	−0.724645	−	0.689122i	\(-0.757996\pi\)
							0.633042	+	0.774117i	\(0.281806\pi\)
\(37\)	−5.68677	−	5.99902i	−0.934899	−	0.986233i	0.0650333	−	0.997883i	\(-0.479285\pi\)
							−0.999932	+	0.0116505i	\(0.996291\pi\)
\(41\)	1.49959	−	4.07487i	0.234196	−	0.636387i	−0.765781	−	0.643101i	\(-0.777647\pi\)
							0.999977	+	0.00671414i	\(0.00213719\pi\)
\(43\)	−5.06674	−	1.68227i	−0.772670	−	0.256544i	−0.0985166	−	0.995135i	\(-0.531410\pi\)
							−0.674153	+	0.738591i	\(0.735491\pi\)
\(47\)	−7.71728	−	4.13160i	−1.12568	−	0.602656i	−0.199561	−	0.979885i	\(-0.563952\pi\)
							−0.926120	+	0.377230i	\(0.876877\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	343.2.k.a.100.20	✓	2688
343.319	even	147	inner	343.2.k.a.319.20	yes	2688

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
343.2.k.a.100.20	✓	2688	1.1	even	1	trivial
343.2.k.a.319.20	yes	2688	343.319	even	147	inner