Embedded newform 301.2.ba.a.100.4

• Label: 301.2.ba.a.100.4
• Level: $301$
• Weight: $2$
• Character: 301.100
• Analytic conductor: $2.403$
• Analytic rank: $0$
• Dimension: $324$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 301 = 7 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	301.ba (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.40349710084\)
Analytic rank:	\(0\)
Dimension:	\(324\)
Relative dimension:	\(27\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			100.4
Character	\(\chi\)	\(=\)	301.100
Dual form			301.2.ba.a.298.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.178479 + 2.38164i) q^{2} +(1.39326 - 0.949907i) q^{3} +(-3.66271 - 0.552064i) q^{4} +(-0.253321 + 1.10987i) q^{5} +(2.01367 + 3.48778i) q^{6} +(-2.32959 + 1.25420i) q^{7} +(0.905635 - 3.96785i) q^{8} +(-0.0571793 + 0.145691i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q - 4 q^{2} - 11 q^{3} + 20 q^{4} - q^{5} - 16 q^{6} - 7 q^{7} - 14 q^{8} + 40 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/301\mathbb{Z}\right)^\times\).

\(n\)	\(87\)	\(218\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{10}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.178479	+	2.38164i	−0.126204	+	1.68408i	0.471115	+	0.882072i	\(0.343852\pi\)
							−0.597319	+	0.802004i	\(0.703767\pi\)
\(3\)	1.39326	−	0.949907i	0.804398	−	0.548429i	−0.0898742	−	0.995953i	\(-0.528646\pi\)
							0.894272	+	0.447524i	\(0.147694\pi\)
\(5\)	−0.253321	+	1.10987i	−0.113289	+	0.496351i	0.886167	+	0.463366i	\(0.153358\pi\)
							−0.999456	+	0.0329847i	\(0.989499\pi\)
\(7\)	−2.32959	+	1.25420i	−0.880502	+	0.474043i
							\(11\)	−0.792086	+	2.01820i	−0.238823	+	0.608511i	−0.999158	−	0.0410199i	\(-0.986939\pi\)
0.760335	+	0.649531i	\(0.225035\pi\)
\(13\)	3.67769	+	3.41239i	1.02001	+	0.946428i	0.998576	−	0.0533428i	\(-0.0169876\pi\)
							0.0214301	+	0.999770i	\(0.493178\pi\)
\(17\)	−0.228404	−	1.00070i	−0.0553960	−	0.242706i	0.939648	−	0.342143i	\(-0.111153\pi\)
							−0.995044	+	0.0994375i	\(0.968296\pi\)
\(19\)	−0.818180	+	1.02597i	−0.187703	+	0.235373i	−0.866775	−	0.498699i	\(-0.833811\pi\)
							0.679072	+	0.734072i	\(0.262383\pi\)
\(23\)	−0.389561	−	0.488495i	−0.0812292	−	0.101858i	0.739556	−	0.673095i	\(-0.235035\pi\)
							−0.820786	+	0.571236i	\(0.806464\pi\)
\(29\)	0.705693	−	9.41682i	0.131044	−	1.74866i	−0.414023	−	0.910266i	\(-0.635877\pi\)
							0.545067	−	0.838393i	\(-0.316504\pi\)
\(31\)	−0.106225	+	1.41747i	−0.0190786	+	0.254586i	0.979507	+	0.201412i	\(0.0645529\pi\)
							−0.998585	+	0.0531742i	\(0.983066\pi\)
\(37\)	3.25333			0.534844			0.267422	−	0.963579i	\(-0.413828\pi\)
							0.267422	+	0.963579i	\(0.413828\pi\)
\(41\)	6.81352	−	3.28122i	1.06409	−	0.512440i	0.181894	−	0.983318i	\(-0.441777\pi\)
							0.882198	+	0.470878i	\(0.156063\pi\)
\(43\)	6.36829	−	1.56362i	0.971155	−	0.238450i
							\(47\)	−2.47499	−	6.30617i	−0.361014	−	0.919850i	−0.989825	−	0.142291i	\(-0.954553\pi\)
0.628810	−	0.777559i	\(-0.283542\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	301.2.ba.a.100.4	yes	324
7.4	even	3		301.2.z.a.186.4	✓	324
43.40	even	21		301.2.z.a.212.4	yes	324
301.298	even	21	inner	301.2.ba.a.298.4	yes	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
301.2.z.a.186.4	✓	324	7.4	even	3
301.2.z.a.212.4	yes	324	43.40	even	21
301.2.ba.a.100.4	yes	324	1.1	even	1	trivial
301.2.ba.a.298.4	yes	324	301.298	even	21	inner