Embedded newform 385.2.o.b.54.7

• Label: 385.2.o.b.54.7
• 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			54.7
Character	\(\chi\)	\(=\)	385.54
Dual form			385.2.o.b.164.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.901007 + 1.56059i) q^{2} +(-0.984554 - 1.70530i) q^{3} +(-0.623628 - 1.08016i) q^{4} +(1.31996 - 1.80491i) q^{5} +3.54836 q^{6} +(-0.686066 + 2.55525i) q^{7} -1.35646 q^{8} +(-0.438692 + 0.759836i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 28 q^{4} - 60 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.901007	+	1.56059i	−0.637108	+	1.10350i	0.348956	+	0.937139i	\(0.386536\pi\)
							−0.986064	+	0.166365i	\(0.946797\pi\)
\(3\)	−0.984554	−	1.70530i	−0.568432	−	0.984554i	−0.996721	−	0.0809118i	\(-0.974217\pi\)
							0.428289	−	0.903642i	\(-0.359117\pi\)
\(5\)	1.31996	−	1.80491i	0.590306	−	0.807179i
							\(7\)	−0.686066	+	2.55525i	−0.259308	+	0.965795i
\(11\)	0.607326	+	3.26055i	0.183116	+	0.983091i
							\(13\)		−	3.94782i		−	1.09493i	−0.836829	−	0.547465i	\(-0.815593\pi\)
0.836829	−	0.547465i	\(-0.184407\pi\)
\(17\)	4.58957	−	2.64979i	1.11313	−	0.642668i	0.173495	−	0.984835i	\(-0.444494\pi\)
							0.939639	+	0.342166i	\(0.111161\pi\)
\(19\)	2.73899	−	4.74407i	0.628368	−	1.08836i	−0.359512	−	0.933141i	\(-0.617057\pi\)
							0.987879	−	0.155224i	\(-0.0496099\pi\)
\(23\)	−0.776241	−	0.448163i	−0.161857	−	0.0934485i	0.416883	−	0.908960i	\(-0.363122\pi\)
							−0.578741	+	0.815512i	\(0.696456\pi\)
\(29\)		−	5.69188i		−	1.05696i	−0.848947	−	0.528478i	\(-0.822763\pi\)
							0.848947	−	0.528478i	\(-0.177237\pi\)
\(31\)	3.94157	−	2.27567i	0.707926	−	0.408722i	−0.102366	−	0.994747i	\(-0.532641\pi\)
							0.810293	+	0.586025i	\(0.199308\pi\)
\(37\)	9.43493	+	5.44726i	1.55109	+	0.895524i	0.998053	+	0.0623701i	\(0.0198659\pi\)
							0.553041	+	0.833154i	\(0.313467\pi\)
\(41\)	−3.01681			−0.471147			−0.235573	−	0.971857i	\(-0.575697\pi\)
							−0.235573	+	0.971857i	\(0.575697\pi\)
\(43\)	2.03128			0.309767			0.154883	−	0.987933i	\(-0.450500\pi\)
							0.154883	+	0.987933i	\(0.450500\pi\)
\(47\)	2.92950	−	5.07404i	0.427311	−	0.740125i	−0.569322	−	0.822115i	\(-0.692794\pi\)
							0.996633	+	0.0819899i	\(0.0261275\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	385.2.o.b.54.7	✓	72
5.4	even	2	inner	385.2.o.b.54.30	yes	72
7.3	odd	6	inner	385.2.o.b.164.8	yes	72
11.10	odd	2	inner	385.2.o.b.54.29	yes	72
35.24	odd	6	inner	385.2.o.b.164.29	yes	72
55.54	odd	2	inner	385.2.o.b.54.8	yes	72
77.10	even	6	inner	385.2.o.b.164.30	yes	72
385.164	even	6	inner	385.2.o.b.164.7	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
385.2.o.b.54.7	✓	72	1.1	even	1	trivial
385.2.o.b.54.8	yes	72	55.54	odd	2	inner
385.2.o.b.54.29	yes	72	11.10	odd	2	inner
385.2.o.b.54.30	yes	72	5.4	even	2	inner
385.2.o.b.164.7	yes	72	385.164	even	6	inner
385.2.o.b.164.8	yes	72	7.3	odd	6	inner
385.2.o.b.164.29	yes	72	35.24	odd	6	inner
385.2.o.b.164.30	yes	72	77.10	even	6	inner