Embedded newform 385.2.o.b.54.16

• Label: 385.2.o.b.54.16
• 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			54.16
Character	\(\chi\)	\(=\)	385.54
Dual form			385.2.o.b.164.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.249992 + 0.433000i) q^{2} +(1.24153 + 2.15040i) q^{3} +(0.875008 + 1.51556i) q^{4} +(-2.10771 + 0.746701i) q^{5} -1.24149 q^{6} +(-2.20302 - 1.46517i) q^{7} -1.87495 q^{8} +(-1.58280 + 2.74149i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 28 q^{4} - 60 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.249992	+	0.433000i	−0.176771	+	0.306177i	−0.940773	−	0.339037i	\(-0.889899\pi\)
							0.764002	+	0.645214i	\(0.223232\pi\)
\(3\)	1.24153	+	2.15040i	0.716798	+	1.24153i	0.962262	+	0.272125i	\(0.0877266\pi\)
							−0.245463	+	0.969406i	\(0.578940\pi\)
\(5\)	−2.10771	+	0.746701i	−0.942596	+	0.333935i
							\(7\)	−2.20302	−	1.46517i	−0.832663	−	0.553780i
\(11\)	1.30072	−	3.05092i	0.392181	−	0.919888i
							\(13\)			5.78503i			1.60448i	0.597002	+	0.802240i	\(0.296359\pi\)
−0.597002	+	0.802240i	\(0.703641\pi\)
\(17\)	0.702087	−	0.405350i	0.170281	−	0.0983118i	−0.412437	−	0.910986i	\(-0.635322\pi\)
							0.582718	+	0.812674i	\(0.301989\pi\)
\(19\)	−1.16300	+	2.01438i	−0.266812	+	0.462131i	−0.968037	−	0.250809i	\(-0.919303\pi\)
							0.701225	+	0.712940i	\(0.252637\pi\)
\(23\)	3.13745	+	1.81141i	0.654203	+	0.377704i	0.790065	−	0.613023i	\(-0.210047\pi\)
							−0.135861	+	0.990728i	\(0.543380\pi\)
\(29\)		−	0.343508i		−	0.0637879i	−0.999491	−	0.0318940i	\(-0.989846\pi\)
							0.999491	−	0.0318940i	\(-0.0101539\pi\)
\(31\)	−0.312599	+	0.180479i	−0.0561445	+	0.0324150i	−0.527810	−	0.849363i	\(-0.676987\pi\)
							0.471665	+	0.881778i	\(0.343653\pi\)
\(37\)	2.53380	+	1.46289i	0.416554	+	0.240498i	0.693602	−	0.720359i	\(-0.256023\pi\)
							−0.277048	+	0.960856i	\(0.589356\pi\)
\(41\)	8.94254			1.39659			0.698295	−	0.715810i	\(-0.253942\pi\)
							0.698295	+	0.715810i	\(0.253942\pi\)
\(43\)	−9.31987			−1.42127			−0.710634	−	0.703562i	\(-0.751592\pi\)
							−0.710634	+	0.703562i	\(0.751592\pi\)
\(47\)	−3.27009	+	5.66395i	−0.476991	+	0.826173i	−0.999652	−	0.0263677i	\(-0.991606\pi\)
							0.522661	+	0.852540i	\(0.324939\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	385.2.o.b.54.16	yes	72
5.4	even	2	inner	385.2.o.b.54.21	yes	72
7.3	odd	6	inner	385.2.o.b.164.15	yes	72
11.10	odd	2	inner	385.2.o.b.54.22	yes	72
35.24	odd	6	inner	385.2.o.b.164.22	yes	72
55.54	odd	2	inner	385.2.o.b.54.15	✓	72
77.10	even	6	inner	385.2.o.b.164.21	yes	72
385.164	even	6	inner	385.2.o.b.164.16	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
385.2.o.b.54.15	✓	72	55.54	odd	2	inner
385.2.o.b.54.16	yes	72	1.1	even	1	trivial
385.2.o.b.54.21	yes	72	5.4	even	2	inner
385.2.o.b.54.22	yes	72	11.10	odd	2	inner
385.2.o.b.164.15	yes	72	7.3	odd	6	inner
385.2.o.b.164.16	yes	72	385.164	even	6	inner
385.2.o.b.164.21	yes	72	77.10	even	6	inner
385.2.o.b.164.22	yes	72	35.24	odd	6	inner