Embedded newform 667.2.s.a.16.7

• Label: 667.2.s.a.16.7
• 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			16.7
Character	\(\chi\)	\(=\)	667.16
Dual form			667.2.s.a.542.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.10118 + 0.908245i) q^{2} +(0.241745 - 2.36183i) q^{3} +(2.21977 - 2.35996i) q^{4} +(0.00893255 - 0.437810i) q^{5} +(1.63717 + 5.18221i) q^{6} +(-1.25928 + 3.08457i) q^{7} +(-0.964662 + 2.66920i) q^{8} +(-2.58203 - 0.534163i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3480 q - 49 q^{2} - 45 q^{3} + 7 q^{4} - 49 q^{5} - 39 q^{6} - 49 q^{7} - 50 q^{8} + q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.10118	+	0.908245i	−1.48576	+	0.642226i	−0.975937	−	0.218053i	\(-0.930029\pi\)
							−0.509823	+	0.860279i	\(0.670289\pi\)
\(3\)	0.241745	−	2.36183i	0.139572	−	1.36361i	−0.654629	−	0.755950i	\(-0.727175\pi\)
							0.794201	−	0.607655i	\(-0.207890\pi\)
\(5\)	0.00893255	−	0.437810i	0.00399476	−	0.195795i	−0.992990	−	0.118198i	\(-0.962288\pi\)
							0.996985	−	0.0775968i	\(-0.0247247\pi\)
\(7\)	−1.25928	+	3.08457i	−0.475962	+	1.16586i	0.481279	+	0.876567i	\(0.340172\pi\)
							−0.957242	+	0.289289i	\(0.906581\pi\)
\(11\)	3.28383	−	1.74987i	0.990113	−	0.527605i	0.105744	−	0.994393i	\(-0.466277\pi\)
							0.884369	+	0.466789i	\(0.154589\pi\)
\(13\)	2.91621	−	5.21350i	0.808811	−	1.44597i	−0.0844502	−	0.996428i	\(-0.526913\pi\)
							0.893261	−	0.449538i	\(-0.148411\pi\)
\(17\)	−0.965063	+	6.71216i	−0.234062	+	1.62794i	0.446175	+	0.894946i	\(0.352786\pi\)
							−0.680237	+	0.732992i	\(0.738123\pi\)
\(19\)	1.19178	−	1.26704i	0.273413	−	0.290679i	−0.577056	−	0.816704i	\(-0.695799\pi\)
							0.850469	+	0.526025i	\(0.176318\pi\)
\(23\)	1.31352	+	4.61245i	0.273888	+	0.961762i
							\(29\)	−2.58715	−	4.72299i	−0.480421	−	0.877038i
\(31\)	2.25619	+	3.67336i	0.405224	+	0.659755i								0.988276	−	0.152681i	\(-0.0487906\pi\)
							−0.583051	+	0.812435i	\(0.698141\pi\)
\(37\)	2.98934	+	0.618426i	0.491444	+	0.101669i	0.441828	−	0.897100i	\(-0.354330\pi\)
							0.0496160	+	0.998768i	\(0.484200\pi\)
\(41\)	−2.28685	−	5.00750i	−0.357146	−	0.782041i	−0.999873	−	0.0159454i	\(-0.994924\pi\)
							0.642727	−	0.766095i	\(-0.277803\pi\)
\(43\)	2.11830	−	3.44885i	0.323038	−	0.525945i	−0.648994	−	0.760793i	\(-0.724810\pi\)
							0.972032	+	0.234849i	\(0.0754594\pi\)
\(47\)	1.36970	−	6.00105i	0.199791	−	0.875343i	−0.771269	−	0.636509i	\(-0.780378\pi\)
							0.971060	−	0.238834i	\(-0.0767652\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	667.2.s.a.16.7	✓	3480
23.13	even	11	inner	667.2.s.a.335.7	yes	3480
29.20	even	7	inner	667.2.s.a.223.7	yes	3480
667.542	even	77	inner	667.2.s.a.542.7	yes	3480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.s.a.16.7	✓	3480	1.1	even	1	trivial
667.2.s.a.223.7	yes	3480	29.20	even	7	inner
667.2.s.a.335.7	yes	3480	23.13	even	11	inner
667.2.s.a.542.7	yes	3480	667.542	even	77	inner