Embedded newform 301.2.bb.a.102.19

• Label: 301.2.bb.a.102.19
• Level: $301$
• Weight: $2$
• Character: 301.102
• Analytic conductor: $2.403$
• Analytic rank: $0$
• Dimension: $336$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 301 = 7 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	301.bb (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.40349710084\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(28\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			102.19
Character	\(\chi\)	\(=\)	301.102
Dual form			301.2.bb.a.121.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.814776 - 0.756002i) q^{2} +(-1.92124 - 1.78265i) q^{3} +(-0.0571387 + 0.762463i) q^{4} +(-0.646797 + 1.64801i) q^{5} -2.91306 q^{6} +(-1.77289 + 1.96389i) q^{7} +(1.91587 + 2.40242i) q^{8} +(0.289129 + 3.85816i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 7 q^{2} - 7 q^{3} + 21 q^{4} - 7 q^{5} - 64 q^{6} - 24 q^{7} - 44 q^{8} + 11 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/301\mathbb{Z}\right)^\times\).

\(n\)	\(87\)	\(218\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{4}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.814776	−	0.756002i	0.576134	−	0.534574i	−0.337387	−	0.941366i	\(-0.609543\pi\)
							0.913521	+	0.406792i	\(0.133353\pi\)
\(3\)	−1.92124	−	1.78265i	−1.10923	−	1.02921i	−0.999378	−	0.0352630i	\(-0.988773\pi\)
							−0.109848	−	0.993948i	\(-0.535036\pi\)
\(5\)	−0.646797	+	1.64801i	−0.289257	+	0.737014i	0.710198	+	0.704002i	\(0.248605\pi\)
							−0.999455	+	0.0330122i	\(0.989490\pi\)
\(7\)	−1.77289	+	1.96389i	−0.670090	+	0.742280i
							\(11\)	−2.46862	+	1.68307i	−0.744316	+	0.507466i	−0.875098	−	0.483945i	\(-0.839203\pi\)
0.130782	+	0.991411i	\(0.458251\pi\)
\(13\)	0.449650	+	0.563844i	0.124711	+	0.156382i	0.840267	−	0.542173i	\(-0.182398\pi\)
							−0.715556	+	0.698555i	\(0.753827\pi\)
\(17\)	−0.325606	−	0.829630i	−0.0789710	−	0.201215i	0.885936	−	0.463808i	\(-0.153517\pi\)
							−0.964907	+	0.262594i	\(0.915422\pi\)
\(19\)	−0.548243	−	0.373786i	−0.125775	−	0.0857523i	0.498798	−	0.866718i	\(-0.333775\pi\)
							−0.624574	+	0.780966i	\(0.714727\pi\)
\(23\)	−0.0181738	−	0.242512i	−0.00378950	−	0.0505673i	0.994988	−	0.0999897i	\(-0.0318810\pi\)
							−0.998778	+	0.0494224i	\(0.984262\pi\)
\(29\)	1.31761	+	5.77285i	0.244675	+	1.07199i	0.936704	+	0.350122i	\(0.113860\pi\)
							−0.692029	+	0.721869i	\(0.743283\pi\)
\(31\)	−7.75797	−	2.39302i	−1.39337	−	0.429798i	−0.494946	−	0.868924i	\(-0.664812\pi\)
							−0.898426	+	0.439126i	\(0.855288\pi\)
\(37\)	−0.965281	−	1.67191i	−0.158691	−	0.274861i	0.775706	−	0.631095i	\(-0.217394\pi\)
							−0.934397	+	0.356234i	\(0.884061\pi\)
\(41\)	−0.576047	−	2.52383i	−0.0899635	−	0.394156i	0.909819	−	0.415005i	\(-0.136220\pi\)
							−0.999783	+	0.0208491i	\(0.993363\pi\)
\(43\)	3.30097	+	5.66600i	0.503394	+	0.864057i
							\(47\)	6.19440	+	4.22327i	0.903546	+	0.616027i	0.923426	−	0.383775i	\(-0.125376\pi\)
−0.0198809	+	0.999802i	\(0.506329\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	301.2.bb.a.102.19	yes	336
7.2	even	3	inner	301.2.bb.a.16.10	✓	336
43.35	even	7	inner	301.2.bb.a.207.10	yes	336
301.121	even	21	inner	301.2.bb.a.121.19	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
301.2.bb.a.16.10	✓	336	7.2	even	3	inner
301.2.bb.a.102.19	yes	336	1.1	even	1	trivial
301.2.bb.a.121.19	yes	336	301.121	even	21	inner
301.2.bb.a.207.10	yes	336	43.35	even	7	inner