Embedded newform 667.2.s.a.16.3

• Label: 667.2.s.a.16.3
• 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			16.3
Character	\(\chi\)	\(=\)	667.16
Dual form			667.2.s.a.542.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.50540 + 1.08297i) q^{2} +(0.310483 - 3.03340i) q^{3} +(3.73393 - 3.96973i) q^{4} +(0.0136512 - 0.669086i) q^{5} +(2.50719 + 7.93612i) q^{6} +(1.55184 - 3.80118i) q^{7} +(-3.20047 + 8.85561i) q^{8} +(-6.16731 - 1.27588i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3480 q - 49 q^{2} - 45 q^{3} + 7 q^{4} - 49 q^{5} - 39 q^{6} - 49 q^{7} - 50 q^{8} + q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.50540	+	1.08297i	−1.77159	+	0.765776i	−0.781745	+	0.623598i	\(0.785670\pi\)
							−0.989841	+	0.142178i	\(0.954589\pi\)
\(3\)	0.310483	−	3.03340i	0.179258	−	1.75133i	−0.382431	−	0.923984i	\(-0.624913\pi\)
							0.561689	−	0.827349i	\(-0.310152\pi\)
\(5\)	0.0136512	−	0.669086i	0.00610501	−	0.299224i	−0.985482	−	0.169781i	\(-0.945694\pi\)
							0.991587	−	0.129443i	\(-0.0413190\pi\)
\(7\)	1.55184	−	3.80118i	0.586539	−	1.43671i	−0.290774	−	0.956792i	\(-0.593913\pi\)
							0.877313	−	0.479919i	\(-0.159334\pi\)
\(11\)	0.102636	−	0.0546919i	0.0309459	−	0.0164902i	−0.454727	−	0.890631i	\(-0.650263\pi\)
							0.485673	+	0.874141i	\(0.338575\pi\)
\(13\)	1.50700	−	2.69417i	0.417968	−	0.747228i	−0.579747	−	0.814796i	\(-0.696849\pi\)
							0.997715	+	0.0675680i	\(0.0215240\pi\)
\(17\)	0.756899	−	5.26435i	0.183575	−	1.27679i	−0.664649	−	0.747155i	\(-0.731419\pi\)
							0.848224	−	0.529637i	\(-0.177672\pi\)
\(19\)	−0.774142	+	0.823031i	−0.177600	+	0.188816i	−0.811066	−	0.584955i	\(-0.801112\pi\)
							0.633466	+	0.773771i	\(0.281632\pi\)
\(23\)	4.27002	−	2.18332i	0.890362	−	0.455254i
							\(29\)	0.423927	+	5.36845i	0.0787213	+	0.996897i
\(31\)	−3.68969	−	6.00726i	−0.662688	−	1.07894i								−0.990935	−	0.134345i	\(-0.957107\pi\)
							0.328247	−	0.944592i	\(-0.393542\pi\)
\(37\)	8.55200	+	1.76922i	1.40594	+	0.290857i	0.844010	−	0.536327i	\(-0.180189\pi\)
							0.561930	+	0.827184i	\(0.310059\pi\)
\(41\)	−0.828821	−	1.81486i	−0.129440	−	0.283434i	0.833805	−	0.552060i	\(-0.186158\pi\)
							−0.963245	+	0.268625i	\(0.913431\pi\)
\(43\)	−4.43090	+	7.21405i	−0.675706	+	1.10013i	0.312926	+	0.949778i	\(0.398691\pi\)
							−0.988632	+	0.150354i	\(0.951959\pi\)
\(47\)	−2.25871	+	9.89606i	−0.329467	+	1.44349i	0.490682	+	0.871339i	\(0.336748\pi\)
							−0.820149	+	0.572150i	\(0.806110\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	667.2.s.a.16.3	✓	3480
23.13	even	11	inner	667.2.s.a.335.3	yes	3480
29.20	even	7	inner	667.2.s.a.223.3	yes	3480
667.542	even	77	inner	667.2.s.a.542.3	yes	3480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.s.a.16.3	✓	3480	1.1	even	1	trivial
667.2.s.a.223.3	yes	3480	29.20	even	7	inner
667.2.s.a.335.3	yes	3480	23.13	even	11	inner
667.2.s.a.542.3	yes	3480	667.542	even	77	inner