Embedded newform 301.2.ba.a.100.18

• Label: 301.2.ba.a.100.18
• 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			100.18
Character	\(\chi\)	\(=\)	301.100
Dual form			301.2.ba.a.298.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0546986 - 0.729902i) q^{2} +(-1.13004 + 0.770445i) q^{3} +(1.44790 + 0.218235i) q^{4} +(0.323764 - 1.41850i) q^{5} +(0.500538 + 0.866958i) q^{6} +(1.33376 - 2.28497i) q^{7} +(0.564236 - 2.47208i) q^{8} +(-0.412628 + 1.05136i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q - 4 q^{2} - 11 q^{3} + 20 q^{4} - q^{5} - 16 q^{6} - 7 q^{7} - 14 q^{8} + 40 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0546986	−	0.729902i	0.0386778	−	0.516119i	−0.943896	−	0.330244i	\(-0.892869\pi\)
							0.982573	−	0.185875i	\(-0.0595119\pi\)
\(3\)	−1.13004	+	0.770445i	−0.652426	+	0.444817i	−0.843769	−	0.536707i	\(-0.819668\pi\)
							0.191343	+	0.981523i	\(0.438716\pi\)
\(5\)	0.323764	−	1.41850i	0.144792	−	0.634374i	−0.849492	−	0.527602i	\(-0.823091\pi\)
							0.994283	−	0.106772i	\(-0.0340516\pi\)
\(7\)	1.33376	−	2.28497i	0.504114	−	0.863637i
							\(11\)	−0.615016	+	1.56704i	−0.185434	+	0.472479i	−0.993115	−	0.117141i	\(-0.962627\pi\)
0.807681	+	0.589620i	\(0.200722\pi\)
\(13\)	1.37823	+	1.27881i	0.382251	+	0.354677i	0.847773	−	0.530359i	\(-0.177943\pi\)
							−0.465522	+	0.885036i	\(0.654133\pi\)
\(17\)	−1.43402	−	6.28286i	−0.347801	−	1.52382i	−0.782160	−	0.623077i	\(-0.785882\pi\)
							0.434359	−	0.900740i	\(-0.356975\pi\)
\(19\)	4.72745	−	5.92804i	1.08455	−	1.35999i	0.156439	−	0.987688i	\(-0.449999\pi\)
							0.928113	−	0.372298i	\(-0.121430\pi\)
\(23\)	2.35764	+	2.95639i	0.491602	+	0.616450i	0.964312	−	0.264768i	\(-0.0852955\pi\)
							−0.472710	+	0.881218i	\(0.656724\pi\)
\(29\)	−0.326360	+	4.35497i	−0.0606036	+	0.808698i	0.881245	+	0.472660i	\(0.156706\pi\)
							−0.941848	+	0.336038i	\(0.890913\pi\)
\(31\)	−0.182758	+	2.43874i	−0.0328244	+	0.438011i	0.956445	+	0.291911i	\(0.0942912\pi\)
							−0.989270	+	0.146100i	\(0.953328\pi\)
\(37\)	−8.06088			−1.32520			−0.662601	−	0.748973i	\(-0.730547\pi\)
							−0.662601	+	0.748973i	\(0.730547\pi\)
\(41\)	−7.10834	+	3.42320i	−1.11014	+	0.534613i	−0.896831	−	0.442372i	\(-0.854137\pi\)
							−0.213305	+	0.976986i	\(0.568423\pi\)
\(43\)	3.55015	+	5.51330i	0.541393	+	0.840770i
							\(47\)	2.74417	+	6.99203i	0.400278	+	1.01989i	0.978908	+	0.204302i	\(0.0654924\pi\)
−0.578630	+	0.815590i	\(0.696412\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	301.2.ba.a.100.18	yes	324
7.4	even	3		301.2.z.a.186.18	✓	324
43.40	even	21		301.2.z.a.212.18	yes	324
301.298	even	21	inner	301.2.ba.a.298.18	yes	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
301.2.z.a.186.18	✓	324	7.4	even	3
301.2.z.a.212.18	yes	324	43.40	even	21
301.2.ba.a.100.18	yes	324	1.1	even	1	trivial
301.2.ba.a.298.18	yes	324	301.298	even	21	inner