Embedded newform 301.2.ba.a.100.1

• Label: 301.2.ba.a.100.1
• 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			100.1
Character	\(\chi\)	\(=\)	301.100
Dual form			301.2.ba.a.298.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.208649 + 2.78423i) q^{2} +(-1.95773 + 1.33476i) q^{3} +(-5.73075 - 0.863772i) q^{4} +(0.712114 - 3.11998i) q^{5} +(-3.30780 - 5.72928i) q^{6} +(-0.851003 + 2.50515i) q^{7} +(2.35808 - 10.3314i) q^{8} +(0.955111 - 2.43358i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q - 4 q^{2} - 11 q^{3} + 20 q^{4} - q^{5} - 16 q^{6} - 7 q^{7} - 14 q^{8} + 40 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.208649	+	2.78423i	−0.147537	+	1.96875i	0.0856746	+	0.996323i	\(0.472695\pi\)
							−0.233212	+	0.972426i	\(0.574924\pi\)
\(3\)	−1.95773	+	1.33476i	−1.13030	+	0.770624i	−0.976080	−	0.217413i	\(-0.930238\pi\)
							−0.154218	+	0.988037i	\(0.549286\pi\)
\(5\)	0.712114	−	3.11998i	0.318467	−	1.39530i	−0.521774	−	0.853084i	\(-0.674730\pi\)
							0.840241	−	0.542212i	\(-0.182413\pi\)
\(7\)	−0.851003	+	2.50515i	−0.321649	+	0.946859i
							\(11\)	−0.738202	+	1.88091i	−0.222576	+	0.567115i	−0.997923	−	0.0644208i	\(-0.979480\pi\)
0.775346	+	0.631536i	\(0.217575\pi\)
\(13\)	−1.80680	−	1.67647i	−0.501117	−	0.464969i	0.388649	−	0.921386i	\(-0.372942\pi\)
							−0.889766	+	0.456417i	\(0.849132\pi\)
\(17\)	−1.01308	−	4.43858i	−0.245707	−	1.07651i	−0.935728	−	0.352723i	\(-0.885256\pi\)
							0.690021	−	0.723790i	\(-0.257601\pi\)
\(19\)	−0.118996	+	0.149216i	−0.0272995	+	0.0342324i	−0.795294	−	0.606224i	\(-0.792683\pi\)
							0.767994	+	0.640457i	\(0.221255\pi\)
\(23\)	−2.66064	−	3.33634i	−0.554782	−	0.695675i	0.422801	−	0.906222i	\(-0.361047\pi\)
							−0.977584	+	0.210547i	\(0.932475\pi\)
\(29\)	−0.156910	+	2.09382i	−0.0291374	+	0.388812i	0.963502	+	0.267702i	\(0.0862644\pi\)
							−0.992639	+	0.121110i	\(0.961355\pi\)
\(31\)	0.112349	−	1.49919i	0.0201785	−	0.269263i	−0.977985	−	0.208678i	\(-0.933084\pi\)
							0.998163	−	0.0605857i	\(-0.0192969\pi\)
\(37\)	−5.17032			−0.849996			−0.424998	−	0.905194i	\(-0.639725\pi\)
							−0.424998	+	0.905194i	\(0.639725\pi\)
\(41\)	8.00131	−	3.85323i	1.24959	−	0.601773i	0.312192	−	0.950019i	\(-0.398937\pi\)
							0.937403	+	0.348246i	\(0.113223\pi\)
\(43\)	−6.23279	−	2.03773i	−0.950492	−	0.310750i
							\(47\)	0.0213166	+	0.0543139i	0.00310935	+	0.00792250i	0.932420	−	0.361376i	\(-0.117693\pi\)
−0.929311	+	0.369299i	\(0.879598\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	301.2.ba.a.100.1	yes	324
7.4	even	3		301.2.z.a.186.1	✓	324
43.40	even	21		301.2.z.a.212.1	yes	324
301.298	even	21	inner	301.2.ba.a.298.1	yes	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
301.2.z.a.186.1	✓	324	7.4	even	3
301.2.z.a.212.1	yes	324	43.40	even	21
301.2.ba.a.100.1	yes	324	1.1	even	1	trivial
301.2.ba.a.298.1	yes	324	301.298	even	21	inner