Embedded newform 966.2.bd.a.59.61

• Label: 966.2.bd.a.59.61
• Level: $966$
• Weight: $2$
• Character: 966.59
• Analytic conductor: $7.714$
• Analytic rank: $0$
• Dimension: $1280$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	966.bd (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.71354883526\)
Analytic rank:	\(0\)
Dimension:	\(1280\)
Relative dimension:	\(64\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			59.61
Character	\(\chi\)	\(=\)	966.59
Dual form			966.2.bd.a.131.61

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.945001 + 0.327068i) q^{2} +(1.57268 + 0.725726i) q^{3} +(0.786053 + 0.618159i) q^{4} +(2.90323 - 0.277225i) q^{5} +(1.24882 + 1.20018i) q^{6} +(-2.41573 + 1.07900i) q^{7} +(0.540641 + 0.841254i) q^{8} +(1.94664 + 2.28267i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1280 q - 64 q^{4} + 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/966\mathbb{Z}\right)^\times\).

\(n\)	\(323\)	\(829\)	\(925\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.945001	+	0.327068i	0.668216	+	0.231272i
							\(3\)	1.57268	+	0.725726i	0.907987	+	0.418998i
\(5\)	2.90323	−	0.277225i	1.29836	−	0.123979i								0.577120	−	0.816660i	\(-0.304177\pi\)
							0.721244	+	0.692681i	\(0.243571\pi\)
\(7\)	−2.41573	+	1.07900i	−0.913061	+	0.407822i
							\(11\)	−4.25615	+	1.47307i	−1.28328	+	0.444146i	−0.881757	−	0.471704i	\(-0.843639\pi\)
−0.401519	+	0.915851i	\(0.631518\pi\)
\(13\)	−0.966377	−	3.29118i	−0.268025	−	0.912808i	−0.978004	−	0.208586i	\(-0.933114\pi\)
							0.709979	−	0.704223i	\(-0.248704\pi\)
\(17\)	6.79098	+	2.71870i	1.64706	+	0.659382i	0.995100	−	0.0988752i	\(-0.0315245\pi\)
							0.651956	+	0.758257i	\(0.273949\pi\)
\(19\)	1.36345	+	3.40572i	0.312796	+	0.781326i	0.998567	+	0.0535093i	\(0.0170407\pi\)
							−0.685772	+	0.727817i	\(0.740535\pi\)
\(23\)	−4.05460	−	2.56129i	−0.845443	−	0.534065i
							\(29\)	2.30599	−	0.331552i	0.428212	−	0.0615676i	0.0751604	−	0.997171i	\(-0.476053\pi\)
0.353052	+	0.935604i	\(0.385144\pi\)
\(31\)	4.56750	−	0.217577i	0.820348	−	0.0390780i	0.366789	−	0.930304i	\(-0.380457\pi\)
							0.453559	+	0.891226i	\(0.350154\pi\)
\(37\)	3.55943	−	4.99851i	0.585166	−	0.821751i	−0.410777	−	0.911736i	\(-0.634743\pi\)
							0.995943	+	0.0899851i	\(0.0286819\pi\)
\(41\)	0.683268	−	1.49615i	0.106709	−	0.233659i	−0.848744	−	0.528804i	\(-0.822641\pi\)
							0.955452	+	0.295145i	\(0.0953679\pi\)
\(43\)	−4.36957	−	2.80815i	−0.666353	−	0.428239i	0.163256	−	0.986584i	\(-0.447800\pi\)
							−0.829609	+	0.558345i	\(0.811437\pi\)
\(47\)	−4.81816	−	8.34530i	−0.702801	−	1.21729i	−0.967479	−	0.252951i	\(-0.918599\pi\)
							0.264678	−	0.964337i	\(-0.414734\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	966.2.bd.a.59.61	yes	1280
3.2	odd	2	inner	966.2.bd.a.59.13	✓	1280
7.5	odd	6	inner	966.2.bd.a.887.25	yes	1280
21.5	even	6	inner	966.2.bd.a.887.57	yes	1280
23.16	even	11	inner	966.2.bd.a.269.57	yes	1280
69.62	odd	22	inner	966.2.bd.a.269.25	yes	1280
161.131	odd	66	inner	966.2.bd.a.131.13	yes	1280
483.131	even	66	inner	966.2.bd.a.131.61	yes	1280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.bd.a.59.13	✓	1280	3.2	odd	2	inner
966.2.bd.a.59.61	yes	1280	1.1	even	1	trivial
966.2.bd.a.131.13	yes	1280	161.131	odd	66	inner
966.2.bd.a.131.61	yes	1280	483.131	even	66	inner
966.2.bd.a.269.25	yes	1280	69.62	odd	22	inner
966.2.bd.a.269.57	yes	1280	23.16	even	11	inner
966.2.bd.a.887.25	yes	1280	7.5	odd	6	inner
966.2.bd.a.887.57	yes	1280	21.5	even	6	inner