Embedded newform 667.2.s.a.16.13

• Label: 667.2.s.a.16.13
• Level: $667$
• Weight: $2$
• Character: 667.16
• Analytic conductor: $5.326$
• Analytic rank: $0$
• Dimension: $3480$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 667 = 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	667.s (of order \(77\), degree \(60\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.32602181482\)
Analytic rank:	\(0\)
Dimension:	\(3480\)
Relative dimension:	\(58\) over \(\Q(\zeta_{77})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{77}]$

Embedding invariants

Embedding label			16.13
Character	\(\chi\)	\(=\)	667.16
Dual form			667.2.s.a.542.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.53669 + 0.664239i) q^{2} +(-0.0152959 + 0.149440i) q^{3} +(0.549907 - 0.584635i) q^{4} +(-0.00415421 + 0.203610i) q^{5} +(-0.0757589 - 0.239803i) q^{6} +(-0.850325 + 2.08285i) q^{7} +(0.681322 - 1.88520i) q^{8} +(2.91569 + 0.603191i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3480 q - 49 q^{2} - 45 q^{3} + 7 q^{4} - 49 q^{5} - 39 q^{6} - 49 q^{7} - 50 q^{8} + q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/667\mathbb{Z}\right)^\times\).

\(n\)	\(465\)	\(553\)
\(\chi(n)\)	\(e\left(\frac{4}{11}\right)\)	\(e\left(\frac{1}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.53669	+	0.664239i	−1.08660	+	0.469688i	−0.863109	−	0.505018i	\(-0.831486\pi\)
							−0.223492	+	0.974706i	\(0.571746\pi\)
\(3\)	−0.0152959	+	0.149440i	−0.00883111	+	0.0862793i	−0.998282	−	0.0585876i	\(-0.981340\pi\)
							0.989451	+	0.144867i	\(0.0462754\pi\)
\(5\)	−0.00415421	+	0.203610i	−0.00185782	+	0.0910572i	0.997826	+	0.0659060i	\(0.0209937\pi\)
							−0.999684	+	0.0251512i	\(0.991993\pi\)
\(7\)	−0.850325	+	2.08285i	−0.321393	+	0.787243i	0.677279	+	0.735726i	\(0.263159\pi\)
							−0.998672	+	0.0515170i	\(0.983594\pi\)
\(11\)	0.335698	−	0.178884i	0.101217	−	0.0539356i	−0.418890	−	0.908037i	\(-0.637581\pi\)
							0.520106	+	0.854102i	\(0.325892\pi\)
\(13\)	2.22092	−	3.97048i	0.615972	−	1.10121i	−0.369153	−	0.929369i	\(-0.620352\pi\)
							0.985124	−	0.171844i	\(-0.0549727\pi\)
\(17\)	0.458412	−	3.18832i	0.111181	−	0.773282i	−0.855593	−	0.517650i	\(-0.826807\pi\)
							0.966774	−	0.255633i	\(-0.0822838\pi\)
\(19\)	0.604705	−	0.642893i	0.138729	−	0.147490i	−0.655303	−	0.755366i	\(-0.727459\pi\)
							0.794032	+	0.607877i	\(0.207978\pi\)
\(23\)	−4.79178	+	0.197142i	−0.999155	+	0.0411070i
							\(29\)	4.65422	+	2.70892i	0.864267	+	0.503034i
\(31\)	−0.605946	−	0.986553i	−0.108831	−	0.177190i								0.793073	−	0.609127i	\(-0.208480\pi\)
							−0.901904	+	0.431936i	\(0.857831\pi\)
\(37\)	5.37968	+	1.11293i	0.884414	+	0.182965i	0.622963	−	0.782252i	\(-0.285929\pi\)
							0.261451	+	0.965217i	\(0.415799\pi\)
\(41\)	1.66636	+	3.64883i	0.260242	+	0.569851i	0.993978	−	0.109584i	\(-0.0349520\pi\)
							−0.733735	+	0.679436i	\(0.762225\pi\)
\(43\)	−4.63438	+	7.54534i	−0.706737	+	1.15065i	0.275190	+	0.961390i	\(0.411259\pi\)
							−0.981926	+	0.189263i	\(0.939390\pi\)
\(47\)	2.53780	−	11.1188i	0.370176	−	1.62185i	−0.356102	−	0.934447i	\(-0.615895\pi\)
							0.726279	−	0.687401i	\(-0.241248\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	667.2.s.a.16.13	✓	3480
23.13	even	11	inner	667.2.s.a.335.13	yes	3480
29.20	even	7	inner	667.2.s.a.223.13	yes	3480
667.542	even	77	inner	667.2.s.a.542.13	yes	3480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.s.a.16.13	✓	3480	1.1	even	1	trivial
667.2.s.a.223.13	yes	3480	29.20	even	7	inner
667.2.s.a.335.13	yes	3480	23.13	even	11	inner
667.2.s.a.542.13	yes	3480	667.542	even	77	inner