Embedded newform 385.2.o.b.54.13

• Label: 385.2.o.b.54.13
• Level: $385$
• Weight: $2$
• Character: 385.54
• Analytic conductor: $3.074$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 385 = 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	385.o (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.07424047782\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(36\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			54.13
Character	\(\chi\)	\(=\)	385.54
Dual form			385.2.o.b.164.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.408979 + 0.708373i) q^{2} +(-1.44107 - 2.49601i) q^{3} +(0.665472 + 1.15263i) q^{4} +(-1.82905 + 1.28630i) q^{5} +2.35748 q^{6} +(2.08705 - 1.62611i) q^{7} -2.72457 q^{8} +(-2.65339 + 4.59580i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 28 q^{4} - 60 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/385\mathbb{Z}\right)^\times\).

\(n\)	\(211\)	\(232\)	\(276\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.408979	+	0.708373i	−0.289192	+	0.500895i	−0.973617	−	0.228188i	\(-0.926720\pi\)
							0.684425	+	0.729083i	\(0.260053\pi\)
\(3\)	−1.44107	−	2.49601i	−0.832004	−	1.44107i	−0.896446	−	0.443152i	\(-0.853860\pi\)
							0.0644421	−	0.997921i	\(-0.479473\pi\)
\(5\)	−1.82905	+	1.28630i	−0.817978	+	0.575250i
							\(7\)	2.08705	−	1.62611i	0.788830	−	0.614612i
\(11\)	3.29882	+	0.343153i	0.994633	+	0.103465i
							\(13\)			2.50240i			0.694040i	0.937858	+	0.347020i	\(0.112806\pi\)
−0.937858	+	0.347020i	\(0.887194\pi\)
\(17\)	4.73654	−	2.73464i	1.14878	−	0.663249i	0.200190	−	0.979757i	\(-0.435844\pi\)
							0.948590	+	0.316508i	\(0.102511\pi\)
\(19\)	2.68808	−	4.65588i	0.616687	−	1.06813i	−0.373399	−	0.927671i	\(-0.621808\pi\)
							0.990086	−	0.140462i	\(-0.0448588\pi\)
\(23\)	2.49166	+	1.43856i	0.519548	+	0.299961i	0.736750	−	0.676166i	\(-0.236360\pi\)
							−0.217202	+	0.976127i	\(0.569693\pi\)
\(29\)			4.76380i			0.884616i	0.896863	+	0.442308i	\(0.145840\pi\)
							−0.896863	+	0.442308i	\(0.854160\pi\)
\(31\)	6.61332	−	3.81820i	1.18779	−	0.685769i	0.229984	−	0.973194i	\(-0.426133\pi\)
							0.957803	+	0.287425i	\(0.0927993\pi\)
\(37\)	3.33285	+	1.92422i	0.547917	+	0.316340i	0.748281	−	0.663382i	\(-0.230879\pi\)
							−0.200365	+	0.979721i	\(0.564213\pi\)
\(41\)	2.69547			0.420962			0.210481	−	0.977598i	\(-0.432497\pi\)
							0.210481	+	0.977598i	\(0.432497\pi\)
\(43\)	12.0332			1.83505			0.917524	−	0.397681i	\(-0.130185\pi\)
							0.917524	+	0.397681i	\(0.130185\pi\)
\(47\)	−3.99406	+	6.91792i	−0.582594	+	1.00908i	0.412577	+	0.910923i	\(0.364629\pi\)
							−0.995171	+	0.0981596i	\(0.968704\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	385.2.o.b.54.13	✓	72
5.4	even	2	inner	385.2.o.b.54.24	yes	72
7.3	odd	6	inner	385.2.o.b.164.14	yes	72
11.10	odd	2	inner	385.2.o.b.54.23	yes	72
35.24	odd	6	inner	385.2.o.b.164.23	yes	72
55.54	odd	2	inner	385.2.o.b.54.14	yes	72
77.10	even	6	inner	385.2.o.b.164.24	yes	72
385.164	even	6	inner	385.2.o.b.164.13	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
385.2.o.b.54.13	✓	72	1.1	even	1	trivial
385.2.o.b.54.14	yes	72	55.54	odd	2	inner
385.2.o.b.54.23	yes	72	11.10	odd	2	inner
385.2.o.b.54.24	yes	72	5.4	even	2	inner
385.2.o.b.164.13	yes	72	385.164	even	6	inner
385.2.o.b.164.14	yes	72	7.3	odd	6	inner
385.2.o.b.164.23	yes	72	35.24	odd	6	inner
385.2.o.b.164.24	yes	72	77.10	even	6	inner