Embedded newform 385.2.o.b.54.3

• Label: 385.2.o.b.54.3
• 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			54.3
Character	\(\chi\)	\(=\)	385.54
Dual form			385.2.o.b.164.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.15617 + 2.00254i) q^{2} +(-1.18194 - 2.04718i) q^{3} +(-1.67346 - 2.89851i) q^{4} +(-2.07840 - 0.824775i) q^{5} +5.46608 q^{6} +(-2.64218 + 0.137378i) q^{7} +3.11452 q^{8} +(-1.29395 + 2.24119i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 28 q^{4} - 60 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.15617	+	2.00254i	−0.817535	+	1.41601i	0.0899577	+	0.995946i	\(0.471327\pi\)
							−0.907493	+	0.420067i	\(0.862007\pi\)
\(3\)	−1.18194	−	2.04718i	−0.682392	−	1.18194i	−0.974249	−	0.225475i	\(-0.927606\pi\)
							0.291857	−	0.956462i	\(-0.405727\pi\)
\(5\)	−2.07840	−	0.824775i	−0.929489	−	0.368850i
							\(7\)	−2.64218	+	0.137378i	−0.998651	+	0.0519238i
\(11\)	3.03078	−	1.34700i	0.913813	−	0.406135i
							\(13\)			0.917656i			0.254512i	0.991870	+	0.127256i	\(0.0406170\pi\)
−0.991870	+	0.127256i	\(0.959383\pi\)
\(17\)	−0.346682	+	0.200157i	−0.0840828	+	0.0485452i	−0.541452	−	0.840732i	\(-0.682125\pi\)
							0.457369	+	0.889277i	\(0.348792\pi\)
\(19\)	−3.02919	+	5.24671i	−0.694944	+	1.20368i	0.275255	+	0.961371i	\(0.411238\pi\)
							−0.970199	+	0.242308i	\(0.922096\pi\)
\(23\)	5.41901	+	3.12867i	1.12994	+	0.652372i	0.943920	−	0.330174i	\(-0.107107\pi\)
							0.186021	+	0.982546i	\(0.440441\pi\)
\(29\)			1.75471i			0.325841i	0.986639	+	0.162920i	\(0.0520913\pi\)
							−0.986639	+	0.162920i	\(0.947909\pi\)
\(31\)	−7.21517	+	4.16568i	−1.29588	+	0.748178i	−0.979690	−	0.200517i	\(-0.935738\pi\)
							−0.316193	+	0.948695i	\(0.602405\pi\)
\(37\)	−0.160587	−	0.0927151i	−0.0264004	−	0.0152423i	0.486742	−	0.873546i	\(-0.338185\pi\)
							−0.513142	+	0.858304i	\(0.671519\pi\)
\(41\)	9.12748			1.42547			0.712737	−	0.701432i	\(-0.247455\pi\)
							0.712737	+	0.701432i	\(0.247455\pi\)
\(43\)	−0.604899			−0.0922462			−0.0461231	−	0.998936i	\(-0.514687\pi\)
							−0.0461231	+	0.998936i	\(0.514687\pi\)
\(47\)	4.21439	−	7.29953i	0.614732	−	1.06475i	−0.375700	−	0.926741i	\(-0.622598\pi\)
							0.990432	−	0.138005i	\(-0.0440690\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	385.2.o.b.54.3	✓	72
5.4	even	2	inner	385.2.o.b.54.34	yes	72
7.3	odd	6	inner	385.2.o.b.164.4	yes	72
11.10	odd	2	inner	385.2.o.b.54.33	yes	72
35.24	odd	6	inner	385.2.o.b.164.33	yes	72
55.54	odd	2	inner	385.2.o.b.54.4	yes	72
77.10	even	6	inner	385.2.o.b.164.34	yes	72
385.164	even	6	inner	385.2.o.b.164.3	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
385.2.o.b.54.3	✓	72	1.1	even	1	trivial
385.2.o.b.54.4	yes	72	55.54	odd	2	inner
385.2.o.b.54.33	yes	72	11.10	odd	2	inner
385.2.o.b.54.34	yes	72	5.4	even	2	inner
385.2.o.b.164.3	yes	72	385.164	even	6	inner
385.2.o.b.164.4	yes	72	7.3	odd	6	inner
385.2.o.b.164.33	yes	72	35.24	odd	6	inner
385.2.o.b.164.34	yes	72	77.10	even	6	inner