Embedded newform 667.2.s.a.16.11

• Label: 667.2.s.a.16.11
• 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			16.11
Character	\(\chi\)	\(=\)	667.16
Dual form			667.2.s.a.542.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.93014 + 0.834310i) q^{2} +(-0.255525 + 2.49645i) q^{3} +(1.65907 - 1.76385i) q^{4} +(-0.0578815 + 2.83694i) q^{5} +(-1.58962 - 5.03169i) q^{6} +(-0.855892 + 2.09648i) q^{7} +(-0.301249 + 0.833550i) q^{8} +(-3.22920 - 0.668048i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3480 q - 49 q^{2} - 45 q^{3} + 7 q^{4} - 49 q^{5} - 39 q^{6} - 49 q^{7} - 50 q^{8} + q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.93014	+	0.834310i	−1.36481	+	0.589947i	−0.947774	−	0.318942i	\(-0.896673\pi\)
							−0.417039	+	0.908888i	\(0.636932\pi\)
\(3\)	−0.255525	+	2.49645i	−0.147527	+	1.44133i	0.612002	+	0.790856i	\(0.290364\pi\)
							−0.759530	+	0.650473i	\(0.774571\pi\)
\(5\)	−0.0578815	+	2.83694i	−0.0258854	+	1.26872i	0.759826	+	0.650126i	\(0.225284\pi\)
							−0.785712	+	0.618593i	\(0.787703\pi\)
\(7\)	−0.855892	+	2.09648i	−0.323497	+	0.792396i	0.675012	+	0.737807i	\(0.264138\pi\)
							−0.998509	+	0.0545897i	\(0.982615\pi\)
\(11\)	−2.25350	+	1.20083i	−0.679457	+	0.362064i	−0.773760	−	0.633479i	\(-0.781626\pi\)
							0.0943026	+	0.995544i	\(0.469938\pi\)
\(13\)	0.488016	−	0.872458i	0.135351	−	0.241976i	−0.796643	−	0.604451i	\(-0.793393\pi\)
							0.931994	+	0.362474i	\(0.118068\pi\)
\(17\)	1.00732	−	7.00607i	0.244311	−	1.69922i	−0.385691	−	0.922628i	\(-0.626037\pi\)
							0.630002	−	0.776593i	\(-0.283054\pi\)
\(19\)	−0.178855	+	0.190150i	−0.0410320	+	0.0436233i	−0.748599	−	0.663023i	\(-0.769273\pi\)
							0.707567	+	0.706646i	\(0.249793\pi\)
\(23\)	−4.76940	+	0.502821i	−0.994489	+	0.104845i
							\(29\)	−5.06391	+	1.83216i	−0.940345	+	0.340223i
\(31\)	2.17301	+	3.53792i	0.390284	+	0.635430i								0.985802	−	0.167912i	\(-0.0537025\pi\)
							−0.595518	+	0.803342i	\(0.703053\pi\)
\(37\)	−3.95512	−	0.818225i	−0.650218	−	0.134515i	−0.134012	−	0.990980i	\(-0.542786\pi\)
							−0.516206	+	0.856464i	\(0.672656\pi\)
\(41\)	1.79464	+	3.92972i	0.280276	+	0.613719i	0.996448	−	0.0842046i	\(-0.0268349\pi\)
							−0.716173	+	0.697923i	\(0.754108\pi\)
\(43\)	5.60236	−	9.12132i	0.854352	−	1.39099i	−0.0651390	−	0.997876i	\(-0.520749\pi\)
							0.919491	−	0.393112i	\(-0.128602\pi\)
\(47\)	−2.44210	+	10.6995i	−0.356217	+	1.56069i	0.406313	+	0.913734i	\(0.366814\pi\)
							−0.762530	+	0.646953i	\(0.776043\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	667.2.s.a.16.11	✓	3480
23.13	even	11	inner	667.2.s.a.335.11	yes	3480
29.20	even	7	inner	667.2.s.a.223.11	yes	3480
667.542	even	77	inner	667.2.s.a.542.11	yes	3480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.s.a.16.11	✓	3480	1.1	even	1	trivial
667.2.s.a.223.11	yes	3480	29.20	even	7	inner
667.2.s.a.335.11	yes	3480	23.13	even	11	inner
667.2.s.a.542.11	yes	3480	667.542	even	77	inner