Embedded newform 301.2.ba.a.100.26

• Label: 301.2.ba.a.100.26
• 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			100.26
Character	\(\chi\)	\(=\)	301.100
Dual form			301.2.ba.a.298.26

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.194170 - 2.59102i) q^{2} +(-1.61413 + 1.10049i) q^{3} +(-4.69804 - 0.708115i) q^{4} +(-0.618669 + 2.71057i) q^{5} +(2.53799 + 4.39592i) q^{6} +(2.38881 - 1.13736i) q^{7} +(-1.59061 + 6.96894i) q^{8} +(0.298296 - 0.760047i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q - 4 q^{2} - 11 q^{3} + 20 q^{4} - q^{5} - 16 q^{6} - 7 q^{7} - 14 q^{8} + 40 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.194170	−	2.59102i	0.137299	−	1.83213i	−0.325356	−	0.945592i	\(-0.605484\pi\)
							0.462655	−	0.886538i	\(-0.346897\pi\)
\(3\)	−1.61413	+	1.10049i	−0.931916	+	0.635370i	−0.931153	−	0.364630i	\(-0.881195\pi\)
							−0.000763742		1.00000i	\(0.500243\pi\)
\(5\)	−0.618669	+	2.71057i	−0.276677	+	1.21220i	0.625288	+	0.780394i	\(0.284981\pi\)
							−0.901965	+	0.431808i	\(0.857876\pi\)
\(7\)	2.38881	−	1.13736i	0.902885	−	0.429881i
							\(11\)	1.50780	−	3.84180i	0.454618	−	1.15835i	−0.501446	−	0.865189i	\(-0.667198\pi\)
0.956064	−	0.293159i	\(-0.0947064\pi\)
\(13\)	3.69055	+	3.42433i	1.02358	+	0.949739i	0.998752	−	0.0499526i	\(-0.0159070\pi\)
							0.0248239	+	0.999692i	\(0.492097\pi\)
\(17\)	0.868265	+	3.80412i	0.210585	+	0.922634i	0.964170	+	0.265284i	\(0.0854657\pi\)
							−0.753585	+	0.657350i	\(0.771677\pi\)
\(19\)	−0.570944	+	0.715941i	−0.130983	+	0.164248i	−0.842998	−	0.537917i	\(-0.819212\pi\)
							0.712015	+	0.702165i	\(0.247783\pi\)
\(23\)	4.36777	+	5.47701i	0.910743	+	1.14204i	0.989412	+	0.145134i	\(0.0463615\pi\)
							−0.0786693	+	0.996901i	\(0.525067\pi\)
\(29\)	−0.0729655	+	0.973657i	−0.0135494	+	0.180804i	0.986323	+	0.164824i	\(0.0527057\pi\)
							−0.999872	+	0.0159794i	\(0.994913\pi\)
\(31\)	−0.401391	+	5.35619i	−0.0720920	+	0.962000i	0.837484	+	0.546462i	\(0.184026\pi\)
							−0.909576	+	0.415538i	\(0.863593\pi\)
\(37\)	1.87010			0.307443			0.153721	−	0.988114i	\(-0.450874\pi\)
							0.153721	+	0.988114i	\(0.450874\pi\)
\(41\)	−5.83318	+	2.80911i	−0.910990	+	0.438710i	−0.829846	−	0.557993i	\(-0.811572\pi\)
							−0.0811441	+	0.996702i	\(0.525857\pi\)
\(43\)	6.51689	+	0.728071i	0.993817	+	0.111030i
							\(47\)	−0.492493	−	1.25485i	−0.0718374	−	0.183039i	0.890445	−	0.455090i	\(-0.150393\pi\)
−0.962283	+	0.272052i	\(0.912298\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	301.2.ba.a.100.26	yes	324
7.4	even	3		301.2.z.a.186.26	✓	324
43.40	even	21		301.2.z.a.212.26	yes	324
301.298	even	21	inner	301.2.ba.a.298.26	yes	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
301.2.z.a.186.26	✓	324	7.4	even	3
301.2.z.a.212.26	yes	324	43.40	even	21
301.2.ba.a.100.26	yes	324	1.1	even	1	trivial
301.2.ba.a.298.26	yes	324	301.298	even	21	inner