Embedded newform 667.2.s.a.16.18

• Label: 667.2.s.a.16.18
• 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			16.18
Character	\(\chi\)	\(=\)	667.16
Dual form			667.2.s.a.542.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.32024 + 0.570680i) q^{2} +(-0.196646 + 1.92121i) q^{3} +(0.0470787 - 0.0500518i) q^{4} +(0.0871976 - 4.27381i) q^{5} +(-0.836777 - 2.64868i) q^{6} +(-1.16167 + 2.84549i) q^{7} +(0.944136 - 2.61240i) q^{8} +(-0.714590 - 0.147832i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3480 q - 49 q^{2} - 45 q^{3} + 7 q^{4} - 49 q^{5} - 39 q^{6} - 49 q^{7} - 50 q^{8} + q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.32024	+	0.570680i	−0.933552	+	0.403532i	−0.808419	−	0.588608i	\(-0.799676\pi\)
							−0.125133	+	0.992140i	\(0.539936\pi\)
\(3\)	−0.196646	+	1.92121i	−0.113533	+	1.10921i	0.769093	+	0.639137i	\(0.220708\pi\)
							−0.882626	+	0.470075i	\(0.844227\pi\)
\(5\)	0.0871976	−	4.27381i	0.0389959	−	1.91131i	−0.274291	−	0.961647i	\(-0.588443\pi\)
							0.313287	−	0.949659i	\(-0.398570\pi\)
\(7\)	−1.16167	+	2.84549i	−0.439072	+	1.07549i	0.534101	+	0.845420i	\(0.320650\pi\)
							−0.973173	+	0.230074i	\(0.926103\pi\)
\(11\)	2.33177	−	1.24254i	0.703055	−	0.374639i	−0.0798277	−	0.996809i	\(-0.525437\pi\)
							0.782882	+	0.622170i	\(0.213749\pi\)
\(13\)	−0.833488	+	1.49008i	−0.231168	+	0.413274i	−0.963514	−	0.267659i	\(-0.913750\pi\)
							0.732346	+	0.680933i	\(0.238425\pi\)
\(17\)	−0.503356	+	3.50091i	−0.122082	+	0.849097i	0.833110	+	0.553108i	\(0.186558\pi\)
							−0.955191	+	0.295989i	\(0.904351\pi\)
\(19\)	−4.41327	+	4.69198i	−1.01247	+	1.07641i	−0.0154066	+	0.999881i	\(0.504904\pi\)
							−0.997067	+	0.0765321i	\(0.975615\pi\)
\(23\)	−4.24815	−	2.22559i	−0.885800	−	0.464068i
							\(29\)	−3.45357	−	4.13193i	−0.641312	−	0.767280i
\(31\)	3.47697	+	5.66093i	0.624482	+	1.01673i								0.996090	+	0.0883431i	\(0.0281572\pi\)
							−0.371608	+	0.928390i	\(0.621193\pi\)
\(37\)	2.24446	+	0.464328i	0.368987	+	0.0763350i	0.383453	−	0.923560i	\(-0.374735\pi\)
							−0.0144662	+	0.999895i	\(0.504605\pi\)
\(41\)	−0.953567	−	2.08802i	−0.148922	−	0.326094i	0.820439	−	0.571734i	\(-0.193729\pi\)
							−0.969361	+	0.245640i	\(0.921002\pi\)
\(43\)	0.902753	−	1.46979i	0.137669	−	0.224141i	−0.775870	−	0.630892i	\(-0.782689\pi\)
							0.913539	+	0.406751i	\(0.133338\pi\)
\(47\)	−0.775734	+	3.39871i	−0.113152	+	0.495753i	0.886314	+	0.463085i	\(0.153258\pi\)
							−0.999466	+	0.0326682i	\(0.989600\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	667.2.s.a.16.18	✓	3480
23.13	even	11	inner	667.2.s.a.335.18	yes	3480
29.20	even	7	inner	667.2.s.a.223.18	yes	3480
667.542	even	77	inner	667.2.s.a.542.18	yes	3480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.s.a.16.18	✓	3480	1.1	even	1	trivial
667.2.s.a.223.18	yes	3480	29.20	even	7	inner
667.2.s.a.335.18	yes	3480	23.13	even	11	inner
667.2.s.a.542.18	yes	3480	667.542	even	77	inner