Embedded newform 385.2.bb.a.64.3

• Label: 385.2.bb.a.64.3
• Level: $385$
• Weight: $2$
• Character: 385.64
• Analytic conductor: $3.074$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 385 = 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	385.bb (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.07424047782\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(36\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			64.3
Character	\(\chi\)	\(=\)	385.64
Dual form			385.2.bb.a.379.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.40348 - 0.780938i) q^{2} +(1.66984 + 2.29834i) q^{3} +(3.54881 + 2.57836i) q^{4} +(-0.646153 + 2.14067i) q^{5} +(-2.21857 - 6.82805i) q^{6} +(-0.587785 + 0.809017i) q^{7} +(-3.54509 - 4.87940i) q^{8} +(-1.56694 + 4.82255i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q + 36 q^{4} + 2 q^{5} + 26 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(211\)	\(232\)	\(276\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)	\(-1\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.40348	−	0.780938i	−1.69952	−	0.552206i	−0.710983	−	0.703209i	\(-0.751750\pi\)
							−0.988533	+	0.151003i	\(0.951750\pi\)
\(3\)	1.66984	+	2.29834i	0.964083	+	1.32695i	0.944980	+	0.327129i	\(0.106081\pi\)
							0.0191031	+	0.999818i	\(0.493919\pi\)
\(5\)	−0.646153	+	2.14067i	−0.288968	+	0.957339i
							\(7\)	−0.587785	+	0.809017i	−0.222162	+	0.305780i
\(11\)	3.31412	+	0.128851i	0.999245	+	0.0388501i
							\(13\)	5.83760	+	1.89675i	1.61906	+	0.526064i	0.971719	−	0.236139i	\(-0.0758820\pi\)
0.647339	+	0.762203i	\(0.275882\pi\)
\(17\)	−2.62367	+	0.852482i	−0.636333	+	0.206757i	−0.609379	−	0.792879i	\(-0.708581\pi\)
							−0.0269548	+	0.999637i	\(0.508581\pi\)
\(19\)	1.76304	−	1.28092i	0.404469	−	0.293864i	−0.366890	−	0.930264i	\(-0.619577\pi\)
							0.771359	+	0.636401i	\(0.219577\pi\)
\(23\)		−	0.204416i		−	0.0426237i	−0.999773	−	0.0213119i	\(-0.993216\pi\)
							0.999773	−	0.0213119i	\(-0.00678429\pi\)
\(29\)	−8.49799	−	6.17415i	−1.57804	−	1.14651i	−0.918902	−	0.394486i	\(-0.870923\pi\)
							−0.659135	−	0.752025i	\(-0.729077\pi\)
\(31\)	−2.42636	+	7.46757i	−0.435787	+	1.34121i	0.456491	+	0.889728i	\(0.349106\pi\)
							−0.892278	+	0.451487i	\(0.850894\pi\)
\(37\)	4.98073	−	6.85539i	0.818827	−	1.12702i	−0.171073	−	0.985258i	\(-0.554724\pi\)
							0.989901	−	0.141761i	\(-0.0452764\pi\)
\(41\)	−3.10571	+	2.25643i	−0.485030	+	0.352395i	−0.803270	−	0.595615i	\(-0.796908\pi\)
							0.318240	+	0.948010i	\(0.396908\pi\)
\(43\)		−	3.09528i		−	0.472026i	−0.971750	−	0.236013i	\(-0.924159\pi\)
							0.971750	−	0.236013i	\(-0.0758407\pi\)
\(47\)	−0.566594	−	0.779849i	−0.0826462	−	0.113753i	0.765692	−	0.643207i	\(-0.222397\pi\)
							−0.848338	+	0.529455i	\(0.822397\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	385.2.bb.a.64.3	✓	144
5.4	even	2	inner	385.2.bb.a.64.34	yes	144
11.5	even	5	inner	385.2.bb.a.379.34	yes	144
55.49	even	10	inner	385.2.bb.a.379.3	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
385.2.bb.a.64.3	✓	144	1.1	even	1	trivial
385.2.bb.a.64.34	yes	144	5.4	even	2	inner
385.2.bb.a.379.3	yes	144	55.49	even	10	inner
385.2.bb.a.379.34	yes	144	11.5	even	5	inner