Embedded newform 667.2.s.a.16.16

• Label: 667.2.s.a.16.16
• 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			16.16
Character	\(\chi\)	\(=\)	667.16
Dual form			667.2.s.a.542.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34692 + 0.582212i) q^{2} +(-0.226539 + 2.21327i) q^{3} +(0.104937 - 0.111564i) q^{4} +(0.0257547 - 1.26231i) q^{5} +(-0.983460 - 3.11298i) q^{6} +(1.79332 - 4.39269i) q^{7} +(0.921096 - 2.54865i) q^{8} +(-1.90943 - 0.395018i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3480 q - 49 q^{2} - 45 q^{3} + 7 q^{4} - 49 q^{5} - 39 q^{6} - 49 q^{7} - 50 q^{8} + q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34692	+	0.582212i	−0.952416	+	0.411686i	−0.815410	−	0.578884i	\(-0.803488\pi\)
							−0.137006	+	0.990570i	\(0.543748\pi\)
\(3\)	−0.226539	+	2.21327i	−0.130792	+	1.27783i	0.697113	+	0.716961i	\(0.254468\pi\)
							−0.827905	+	0.560868i	\(0.810467\pi\)
\(5\)	0.0257547	−	1.26231i	0.0115179	−	0.564524i	−0.953361	−	0.301831i	\(-0.902402\pi\)
							0.964879	−	0.262693i	\(-0.0846108\pi\)
\(7\)	1.79332	−	4.39269i	0.677811	−	1.66028i	−0.0709821	−	0.997478i	\(-0.522613\pi\)
							0.748794	−	0.662803i	\(-0.230633\pi\)
\(11\)	−1.12802	+	0.601090i	−0.340110	+	0.181235i	−0.631514	−	0.775365i	\(-0.717566\pi\)
							0.291404	+	0.956600i	\(0.405878\pi\)
\(13\)	−3.07661	+	5.50026i	−0.853299	+	1.52550i	−0.00250627	+	0.999997i	\(0.500798\pi\)
							−0.850792	+	0.525502i	\(0.823878\pi\)
\(17\)	0.428805	−	2.98241i	0.104001	−	0.723340i	−0.869380	−	0.494144i	\(-0.835481\pi\)
							0.973380	−	0.229195i	\(-0.0736095\pi\)
\(19\)	−4.36186	+	4.63732i	−1.00068	+	1.06387i	−0.00272179	+	0.999996i	\(0.500866\pi\)
							−0.997958	+	0.0638781i	\(0.979653\pi\)
\(23\)	2.80350	+	3.89106i	0.584571	+	0.811342i
							\(29\)	1.49272	+	5.17415i	0.277191	+	0.960815i
\(31\)	2.21593	+	3.60781i	0.397993	+	0.647981i								0.987106	−	0.160070i	\(-0.0511719\pi\)
							−0.589113	+	0.808051i	\(0.700523\pi\)
\(37\)	−7.45085	−	1.54141i	−1.22491	−	0.253407i	−0.454381	−	0.890808i	\(-0.650139\pi\)
							−0.770532	+	0.637401i	\(0.780009\pi\)
\(41\)	5.19543	+	11.3764i	0.811391	+	1.77670i	0.601315	+	0.799012i	\(0.294644\pi\)
							0.210076	+	0.977685i	\(0.432629\pi\)
\(43\)	0.136642	−	0.222470i	0.0208377	−	0.0339263i	−0.841519	−	0.540227i	\(-0.818338\pi\)
							0.862357	+	0.506301i	\(0.168988\pi\)
\(47\)	−0.0101832	+	0.0446157i	−0.00148538	+	0.00650787i	−0.975665	−	0.219266i	\(-0.929634\pi\)
							0.974180	+	0.225774i	\(0.0724910\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	667.2.s.a.16.16	✓	3480
23.13	even	11	inner	667.2.s.a.335.16	yes	3480
29.20	even	7	inner	667.2.s.a.223.16	yes	3480
667.542	even	77	inner	667.2.s.a.542.16	yes	3480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.s.a.16.16	✓	3480	1.1	even	1	trivial
667.2.s.a.223.16	yes	3480	29.20	even	7	inner
667.2.s.a.335.16	yes	3480	23.13	even	11	inner
667.2.s.a.542.16	yes	3480	667.542	even	77	inner