Embedded newform 301.2.ba.a.100.25

• Label: 301.2.ba.a.100.25
• 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			100.25
Character	\(\chi\)	\(=\)	301.100
Dual form			301.2.ba.a.298.25

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.179276 - 2.39227i) q^{2} +(-0.835229 + 0.569449i) q^{3} +(-3.71314 - 0.559666i) q^{4} +(-0.210856 + 0.923820i) q^{5} +(1.21254 + 2.10018i) q^{6} +(-1.16252 + 2.37667i) q^{7} +(-0.936903 + 4.10484i) q^{8} +(-0.722688 + 1.84138i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q - 4 q^{2} - 11 q^{3} + 20 q^{4} - q^{5} - 16 q^{6} - 7 q^{7} - 14 q^{8} + 40 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.179276	−	2.39227i	0.126767	−	1.69159i	−0.464866	−	0.885381i	\(-0.653897\pi\)
							0.591633	−	0.806207i	\(-0.298483\pi\)
\(3\)	−0.835229	+	0.569449i	−0.482220	+	0.328772i	−0.779911	−	0.625890i	\(-0.784736\pi\)
							0.297691	+	0.954662i	\(0.403783\pi\)
\(5\)	−0.210856	+	0.923820i	−0.0942976	+	0.413145i	−0.999941	−	0.0108960i	\(-0.996532\pi\)
							0.905643	+	0.424041i	\(0.139389\pi\)
\(7\)	−1.16252	+	2.37667i	−0.439393	+	0.898295i
							\(11\)	−2.00220	+	5.10152i	−0.603685	+	1.53817i	0.222598	+	0.974910i	\(0.428546\pi\)
−0.826283	+	0.563255i	\(0.809549\pi\)
\(13\)	−2.48715	−	2.30774i	−0.689812	−	0.640052i	0.255276	−	0.966868i	\(-0.417834\pi\)
							−0.945088	+	0.326816i	\(0.894024\pi\)
\(17\)	−0.682779	−	2.99145i	−0.165598	−	0.725533i	−0.987722	−	0.156224i	\(-0.950068\pi\)
							0.822124	−	0.569309i	\(-0.192789\pi\)
\(19\)	0.0973710	−	0.122099i	0.0223384	−	0.0280115i	−0.770537	−	0.637396i	\(-0.780012\pi\)
							0.792875	+	0.609384i	\(0.208583\pi\)
\(23\)	−0.956920	−	1.19994i	−0.199532	−	0.250205i	0.671992	−	0.740559i	\(-0.265439\pi\)
							−0.871523	+	0.490354i	\(0.836868\pi\)
\(29\)	0.0367346	−	0.490190i	0.00682145	−	0.0910259i	−0.992754	−	0.120165i	\(-0.961658\pi\)
							0.999575	+	0.0291393i	\(0.00927663\pi\)
\(31\)	−0.351586	+	4.69158i	−0.0631466	+	0.842633i	0.872256	+	0.489049i	\(0.162656\pi\)
							−0.935403	+	0.353584i	\(0.884963\pi\)
\(37\)	5.98178			0.983398			0.491699	−	0.870765i	\(-0.336376\pi\)
							0.491699	+	0.870765i	\(0.336376\pi\)
\(41\)	−7.02655	+	3.38381i	−1.09736	+	0.528462i	−0.892828	−	0.450397i	\(-0.851283\pi\)
							−0.204535	+	0.978859i	\(0.565568\pi\)
\(43\)	−5.38371	+	3.74375i	−0.821008	+	0.570916i
							\(47\)	0.0518021	+	0.131990i	0.00755612	+	0.0192527i	0.934602	−	0.355695i	\(-0.115756\pi\)
−0.927046	+	0.374948i	\(0.877661\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	301.2.ba.a.100.25	yes	324
7.4	even	3		301.2.z.a.186.25	✓	324
43.40	even	21		301.2.z.a.212.25	yes	324
301.298	even	21	inner	301.2.ba.a.298.25	yes	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
301.2.z.a.186.25	✓	324	7.4	even	3
301.2.z.a.212.25	yes	324	43.40	even	21
301.2.ba.a.100.25	yes	324	1.1	even	1	trivial
301.2.ba.a.298.25	yes	324	301.298	even	21	inner