Embedded newform 301.2.ba.a.100.21

• Label: 301.2.ba.a.100.21
• 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			100.21
Character	\(\chi\)	\(=\)	301.100
Dual form			301.2.ba.a.298.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.116490 - 1.55445i) q^{2} +(-2.22207 + 1.51498i) q^{3} +(-0.425069 - 0.0640688i) q^{4} +(-0.322249 + 1.41187i) q^{5} +(2.09611 + 3.63056i) q^{6} +(-2.55639 - 0.681809i) q^{7} +(0.544626 - 2.38616i) q^{8} +(1.54640 - 3.94016i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q - 4 q^{2} - 11 q^{3} + 20 q^{4} - q^{5} - 16 q^{6} - 7 q^{7} - 14 q^{8} + 40 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.116490	−	1.55445i	0.0823706	−	1.09916i	−0.790907	−	0.611936i	\(-0.790391\pi\)
							0.873278	−	0.487223i	\(-0.161990\pi\)
\(3\)	−2.22207	+	1.51498i	−1.28291	+	0.874675i	−0.996513	−	0.0834357i	\(-0.973411\pi\)
							−0.286399	+	0.958110i	\(0.592458\pi\)
\(5\)	−0.322249	+	1.41187i	−0.144114	+	0.631406i	0.850340	+	0.526234i	\(0.176396\pi\)
							−0.994454	+	0.105172i	\(0.966461\pi\)
\(7\)	−2.55639	−	0.681809i	−0.966225	−	0.257700i
							\(11\)	1.81445	−	4.62313i	0.547076	−	1.39393i	−0.343526	−	0.939143i	\(-0.611621\pi\)
0.890602	−	0.454783i	\(-0.150283\pi\)
\(13\)	−3.13406	−	2.90798i	−0.869231	−	0.806529i	0.113119	−	0.993581i	\(-0.463916\pi\)
							−0.982350	+	0.187053i	\(0.940106\pi\)
\(17\)	−0.263006	−	1.15230i	−0.0637883	−	0.279475i	0.932967	−	0.359961i	\(-0.117210\pi\)
							−0.996756	+	0.0804859i	\(0.974353\pi\)
\(19\)	1.00804	−	1.26404i	0.231260	−	0.289990i	−0.652639	−	0.757669i	\(-0.726338\pi\)
							0.883899	+	0.467679i	\(0.154910\pi\)
\(23\)	−2.26336	−	2.83817i	−0.471944	−	0.591799i	0.487702	−	0.873010i	\(-0.337835\pi\)
							−0.959646	+	0.281211i	\(0.909264\pi\)
\(29\)	−0.710491	+	9.48084i	−0.131935	+	1.76055i	0.402576	+	0.915387i	\(0.368115\pi\)
							−0.534511	+	0.845162i	\(0.679504\pi\)
\(31\)	0.381871	−	5.09572i	0.0685861	−	0.915218i	−0.851715	−	0.524006i	\(-0.824437\pi\)
							0.920301	−	0.391212i	\(-0.127944\pi\)
\(37\)	−5.88146			−0.966906			−0.483453	−	0.875370i	\(-0.660618\pi\)
							−0.483453	+	0.875370i	\(0.660618\pi\)
\(41\)	−4.74097	+	2.28313i	−0.740415	+	0.356565i	−0.765771	−	0.643113i	\(-0.777642\pi\)
							0.0253559	+	0.999678i	\(0.491928\pi\)
\(43\)	−1.46474	−	6.39176i	−0.223371	−	0.974734i
							\(47\)	−1.28728	−	3.27994i	−0.187769	−	0.478429i	0.805728	−	0.592286i	\(-0.201774\pi\)
−0.993497	+	0.113858i	\(0.963679\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	301.2.ba.a.100.21	yes	324
7.4	even	3		301.2.z.a.186.21	✓	324
43.40	even	21		301.2.z.a.212.21	yes	324
301.298	even	21	inner	301.2.ba.a.298.21	yes	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
301.2.z.a.186.21	✓	324	7.4	even	3
301.2.z.a.212.21	yes	324	43.40	even	21
301.2.ba.a.100.21	yes	324	1.1	even	1	trivial
301.2.ba.a.298.21	yes	324	301.298	even	21	inner