Embedded newform 301.2.ba.a.100.2

• Label: 301.2.ba.a.100.2
• 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			100.2
Character	\(\chi\)	\(=\)	301.100
Dual form			301.2.ba.a.298.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.190317 + 2.53960i) q^{2} +(1.73832 - 1.18517i) q^{3} +(-4.43570 - 0.668574i) q^{4} +(0.544306 - 2.38476i) q^{5} +(2.67902 + 4.64019i) q^{6} +(2.06208 - 1.65765i) q^{7} +(1.40870 - 6.17192i) q^{8} +(0.521110 - 1.32777i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q - 4 q^{2} - 11 q^{3} + 20 q^{4} - q^{5} - 16 q^{6} - 7 q^{7} - 14 q^{8} + 40 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.190317	+	2.53960i	−0.134574	+	1.79577i	0.366554	+	0.930397i	\(0.380538\pi\)
							−0.501128	+	0.865373i	\(0.667081\pi\)
\(3\)	1.73832	−	1.18517i	1.00362	−	0.684255i	0.0542848	−	0.998525i	\(-0.482712\pi\)
							0.949334	+	0.314270i	\(0.101760\pi\)
\(5\)	0.544306	−	2.38476i	0.243421	−	1.06650i	−0.694458	−	0.719534i	\(-0.744356\pi\)
							0.937879	−	0.346964i	\(-0.112787\pi\)
\(7\)	2.06208	−	1.65765i	0.779394	−	0.626535i
							\(11\)	1.38342	−	3.52489i	0.417116	−	1.06280i	−0.555734	−	0.831360i	\(-0.687563\pi\)
0.972850	−	0.231435i	\(-0.0743421\pi\)
\(13\)	−0.124613	−	0.115624i	−0.0345615	−	0.0320684i	0.662703	−	0.748882i	\(-0.269409\pi\)
							−0.697264	+	0.716814i	\(0.745600\pi\)
\(17\)	0.328055	+	1.43730i	0.0795650	+	0.348597i	0.999003	−	0.0446374i	\(-0.0142133\pi\)
							−0.919438	+	0.393234i	\(0.871356\pi\)
\(19\)	−4.21005	+	5.27924i	−0.965853	+	1.21114i	0.0115883	+	0.999933i	\(0.496311\pi\)
							−0.977441	+	0.211208i	\(0.932260\pi\)
\(23\)	1.23511	+	1.54877i	0.257537	+	0.322942i	0.893744	−	0.448577i	\(-0.148069\pi\)
							−0.636207	+	0.771519i	\(0.719498\pi\)
\(29\)	−0.501538	+	6.69256i	−0.0931333	+	1.24278i	0.733364	+	0.679836i	\(0.237949\pi\)
							−0.826497	+	0.562941i	\(0.809670\pi\)
\(31\)	0.471768	−	6.29530i	0.0847320	−	1.13067i	−0.779103	−	0.626896i	\(-0.784325\pi\)
							0.863835	−	0.503774i	\(-0.168056\pi\)
\(37\)	−2.92208			−0.480388			−0.240194	−	0.970725i	\(-0.577211\pi\)
							−0.240194	+	0.970725i	\(0.577211\pi\)
\(41\)	−10.4293	+	5.02248i	−1.62878	+	0.784379i	−0.628804	+	0.777564i	\(0.716455\pi\)
							−0.999977	+	0.00681509i	\(0.997831\pi\)
\(43\)	3.37252	−	5.62371i	0.514304	−	0.857608i
							\(47\)	4.14385	+	10.5584i	0.604443	+	1.54010i	0.825243	+	0.564777i	\(0.191038\pi\)
−0.220800	+	0.975319i	\(0.570867\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	301.2.ba.a.100.2	yes	324
7.4	even	3		301.2.z.a.186.2	✓	324
43.40	even	21		301.2.z.a.212.2	yes	324
301.298	even	21	inner	301.2.ba.a.298.2	yes	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
301.2.z.a.186.2	✓	324	7.4	even	3
301.2.z.a.212.2	yes	324	43.40	even	21
301.2.ba.a.100.2	yes	324	1.1	even	1	trivial
301.2.ba.a.298.2	yes	324	301.298	even	21	inner