Embedded newform 301.2.ba.a.100.17

• Label: 301.2.ba.a.100.17
• 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			100.17
Character	\(\chi\)	\(=\)	301.100
Dual form			301.2.ba.a.298.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0471151 - 0.628707i) q^{2} +(-2.19521 + 1.49667i) q^{3} +(1.58461 + 0.238841i) q^{4} +(0.324651 - 1.42239i) q^{5} +(0.837537 + 1.45066i) q^{6} +(-0.105083 + 2.64366i) q^{7} +(0.505406 - 2.21433i) q^{8} +(1.48290 - 3.77836i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q - 4 q^{2} - 11 q^{3} + 20 q^{4} - q^{5} - 16 q^{6} - 7 q^{7} - 14 q^{8} + 40 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0471151	−	0.628707i	0.0333154	−	0.444563i	−0.955457	−	0.295131i	\(-0.904637\pi\)
							0.988772	−	0.149432i	\(-0.0477444\pi\)
\(3\)	−2.19521	+	1.49667i	−1.26740	+	0.864100i	−0.995173	−	0.0981321i	\(-0.968713\pi\)
							−0.272229	+	0.962232i	\(0.587761\pi\)
\(5\)	0.324651	−	1.42239i	0.145188	−	0.636111i	−0.848994	−	0.528402i	\(-0.822791\pi\)
							0.994182	−	0.107709i	\(-0.0343515\pi\)
\(7\)	−0.105083	+	2.64366i	−0.0397176	+	0.999211i
							\(11\)	−0.146751	+	0.373916i	−0.0442472	+	0.112740i	−0.951289	−	0.308301i	\(-0.900240\pi\)
0.907042	+	0.421041i	\(0.138335\pi\)
\(13\)	3.21023	+	2.97866i	0.890359	+	0.826132i	0.985560	−	0.169326i	\(-0.0541591\pi\)
							−0.0952015	+	0.995458i	\(0.530350\pi\)
\(17\)	0.631052	+	2.76482i	0.153053	+	0.670568i	0.991988	+	0.126336i	\(0.0403217\pi\)
							−0.838935	+	0.544232i	\(0.816821\pi\)
\(19\)	−2.98492	+	3.74297i	−0.684788	+	0.858696i	−0.995785	−	0.0917142i	\(-0.970765\pi\)
							0.310998	+	0.950411i	\(0.399337\pi\)
\(23\)	3.90300	+	4.89421i	0.813832	+	1.02051i	0.999283	+	0.0378595i	\(0.0120539\pi\)
							−0.185451	+	0.982653i	\(0.559375\pi\)
\(29\)	0.459424	−	6.13058i	0.0853128	−	1.13842i	−0.776132	−	0.630570i	\(-0.782821\pi\)
							0.861445	−	0.507851i	\(-0.169560\pi\)
\(31\)	0.0113313	−	0.151205i	0.00203516	−	0.0271573i	−0.996094	−	0.0883018i	\(-0.971856\pi\)
							0.998129	+	0.0611445i	\(0.0194751\pi\)
\(37\)	5.92869			0.974671			0.487335	−	0.873215i	\(-0.337969\pi\)
							0.487335	+	0.873215i	\(0.337969\pi\)
\(41\)	−2.81300	+	1.35467i	−0.439317	+	0.211564i	−0.640450	−	0.768000i	\(-0.721252\pi\)
							0.201132	+	0.979564i	\(0.435538\pi\)
\(43\)	−6.48617	+	0.964128i	−0.989132	+	0.147028i
							\(47\)	−0.360743	−	0.919158i	−0.0526198	−	0.134073i	0.902147	−	0.431429i	\(-0.141990\pi\)
−0.954767	+	0.297356i	\(0.903895\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	301.2.ba.a.100.17	yes	324
7.4	even	3		301.2.z.a.186.17	✓	324
43.40	even	21		301.2.z.a.212.17	yes	324
301.298	even	21	inner	301.2.ba.a.298.17	yes	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
301.2.z.a.186.17	✓	324	7.4	even	3
301.2.z.a.212.17	yes	324	43.40	even	21
301.2.ba.a.100.17	yes	324	1.1	even	1	trivial
301.2.ba.a.298.17	yes	324	301.298	even	21	inner