Embedded newform 385.2.o.b.54.1

• Label: 385.2.o.b.54.1
• 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			54.1
Character	\(\chi\)	\(=\)	385.54
Dual form			385.2.o.b.164.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30077 + 2.25300i) q^{2} +(-0.878341 - 1.52133i) q^{3} +(-2.38400 - 4.12920i) q^{4} +(-1.24339 + 1.85849i) q^{5} +4.57007 q^{6} +(2.30616 + 1.29678i) q^{7} +7.20103 q^{8} +(-0.0429672 + 0.0744215i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 28 q^{4} - 60 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.30077	+	2.25300i	−0.919782	+	1.59311i	−0.120037	+	0.992769i	\(0.538301\pi\)
							−0.799745	+	0.600340i	\(0.795032\pi\)
\(3\)	−0.878341	−	1.52133i	−0.507111	−	0.878341i	−0.999966	−	0.00823022i	\(-0.997380\pi\)
							0.492855	−	0.870111i	\(-0.335953\pi\)
\(5\)	−1.24339	+	1.85849i	−0.556060	+	0.831142i
							\(7\)	2.30616	+	1.29678i	0.871645	+	0.490138i
\(11\)	−0.676475	+	3.24690i	−0.203965	+	0.978978i
							\(13\)		−	0.0378692i		−	0.0105030i	−0.999986	−	0.00525151i	\(-0.998328\pi\)
0.999986	−	0.00525151i	\(-0.00167161\pi\)
\(17\)	−3.77176	+	2.17762i	−0.914785	+	0.528151i	−0.881968	−	0.471310i	\(-0.843781\pi\)
							−0.0328175	+	0.999461i	\(0.510448\pi\)
\(19\)	0.906971	−	1.57092i	0.208074	−	0.360394i	−0.743034	−	0.669254i	\(-0.766614\pi\)
							0.951108	+	0.308860i	\(0.0999473\pi\)
\(23\)	−7.85846	−	4.53709i	−1.63860	−	0.946048i	−0.981315	−	0.192408i	\(-0.938370\pi\)
							−0.657288	−	0.753640i	\(-0.728296\pi\)
\(29\)			7.24768i			1.34586i	0.739706	+	0.672930i	\(0.234964\pi\)
							−0.739706	+	0.672930i	\(0.765036\pi\)
\(31\)	−4.81506	+	2.77998i	−0.864811	+	0.499299i	−0.865620	−	0.500701i	\(-0.833076\pi\)
							0.000809476		1.00000i	\(0.499742\pi\)
\(37\)	2.58618	+	1.49313i	0.425165	+	0.245469i	0.697285	−	0.716794i	\(-0.254391\pi\)
							−0.272120	+	0.962263i	\(0.587725\pi\)
\(41\)	−5.22208			−0.815552			−0.407776	−	0.913082i	\(-0.633696\pi\)
							−0.407776	+	0.913082i	\(0.633696\pi\)
\(43\)	−6.82145			−1.04026			−0.520131	−	0.854087i	\(-0.674117\pi\)
							−0.520131	+	0.854087i	\(0.674117\pi\)
\(47\)	−1.51540	+	2.62474i	−0.221043	+	0.382858i	−0.955125	−	0.296203i	\(-0.904280\pi\)
							0.734082	+	0.679061i	\(0.237613\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	385.2.o.b.54.1	✓	72
5.4	even	2	inner	385.2.o.b.54.36	yes	72
7.3	odd	6	inner	385.2.o.b.164.2	yes	72
11.10	odd	2	inner	385.2.o.b.54.35	yes	72
35.24	odd	6	inner	385.2.o.b.164.35	yes	72
55.54	odd	2	inner	385.2.o.b.54.2	yes	72
77.10	even	6	inner	385.2.o.b.164.36	yes	72
385.164	even	6	inner	385.2.o.b.164.1	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
385.2.o.b.54.1	✓	72	1.1	even	1	trivial
385.2.o.b.54.2	yes	72	55.54	odd	2	inner
385.2.o.b.54.35	yes	72	11.10	odd	2	inner
385.2.o.b.54.36	yes	72	5.4	even	2	inner
385.2.o.b.164.1	yes	72	385.164	even	6	inner
385.2.o.b.164.2	yes	72	7.3	odd	6	inner
385.2.o.b.164.35	yes	72	35.24	odd	6	inner
385.2.o.b.164.36	yes	72	77.10	even	6	inner