Embedded newform 667.2.s.a.16.10

• Label: 667.2.s.a.16.10
• 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			16.10
Character	\(\chi\)	\(=\)	667.16
Dual form			667.2.s.a.542.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.94761 + 0.841863i) q^{2} +(-0.0985846 + 0.963163i) q^{3} +(1.71417 - 1.82242i) q^{4} +(-0.0626912 + 3.07268i) q^{5} +(-0.618847 - 1.95886i) q^{6} +(0.801093 - 1.96226i) q^{7} +(-0.361971 + 1.00156i) q^{8} +(2.01983 + 0.417857i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3480 q - 49 q^{2} - 45 q^{3} + 7 q^{4} - 49 q^{5} - 39 q^{6} - 49 q^{7} - 50 q^{8} + q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.94761	+	0.841863i	−1.37717	+	0.595287i	−0.950953	−	0.309334i	\(-0.899894\pi\)
							−0.426216	+	0.904622i	\(0.640154\pi\)
\(3\)	−0.0985846	+	0.963163i	−0.0569178	+	0.556083i	0.926698	+	0.375806i	\(0.122634\pi\)
							−0.983616	+	0.180276i	\(0.942301\pi\)
\(5\)	−0.0626912	+	3.07268i	−0.0280363	+	1.37414i	0.712285	+	0.701890i	\(0.247660\pi\)
							−0.740322	+	0.672253i	\(0.765327\pi\)
\(7\)	0.801093	−	1.96226i	0.302785	−	0.741663i	−0.696915	−	0.717154i	\(-0.745444\pi\)
							0.999700	−	0.0245092i	\(-0.00780230\pi\)
\(11\)	−3.03932	+	1.61957i	−0.916389	+	0.488319i	−0.860117	−	0.510096i	\(-0.829610\pi\)
							−0.0562721	+	0.998415i	\(0.517921\pi\)
\(13\)	1.25113	−	2.23674i	0.347002	−	0.620359i	−0.642264	−	0.766483i	\(-0.722005\pi\)
							0.989266	+	0.146125i	\(0.0466801\pi\)
\(17\)	−1.10196	+	7.66432i	−0.267265	+	1.85887i	0.206803	+	0.978383i	\(0.433694\pi\)
							−0.474068	+	0.880488i	\(0.657215\pi\)
\(19\)	−1.60633	+	1.70777i	−0.368516	+	0.391789i	−0.885832	−	0.464006i	\(-0.846412\pi\)
							0.517316	+	0.855795i	\(0.326931\pi\)
\(23\)	4.48926	−	1.68716i	0.936076	−	0.351797i
							\(29\)	−0.992006	+	5.29301i	−0.184211	+	0.982887i
\(31\)	−1.66131	−	2.70482i	−0.298380	−	0.485800i								0.667563	−	0.744553i	\(-0.267338\pi\)
							−0.965943	+	0.258754i	\(0.916688\pi\)
\(37\)	3.00008	+	0.620648i	0.493209	+	0.102034i	0.442663	−	0.896688i	\(-0.354034\pi\)
							0.0505461	+	0.998722i	\(0.483904\pi\)
\(41\)	−3.15351	−	6.90522i	−0.492495	−	1.07841i	−0.978836	−	0.204645i	\(-0.934396\pi\)
							0.486341	−	0.873769i	\(-0.338331\pi\)
\(43\)	−3.74112	+	6.09100i	−0.570516	+	0.928869i	0.429156	+	0.903231i	\(0.358811\pi\)
							−0.999671	+	0.0256385i	\(0.991838\pi\)
\(47\)	−0.816700	+	3.57820i	−0.119128	+	0.521934i	0.879787	+	0.475368i	\(0.157685\pi\)
							−0.998915	+	0.0465660i	\(0.985172\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	667.2.s.a.16.10	✓	3480
23.13	even	11	inner	667.2.s.a.335.10	yes	3480
29.20	even	7	inner	667.2.s.a.223.10	yes	3480
667.542	even	77	inner	667.2.s.a.542.10	yes	3480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.s.a.16.10	✓	3480	1.1	even	1	trivial
667.2.s.a.223.10	yes	3480	29.20	even	7	inner
667.2.s.a.335.10	yes	3480	23.13	even	11	inner
667.2.s.a.542.10	yes	3480	667.542	even	77	inner