Newform orbit 668.2.e.a

• Label: 668.2.e.a
• Level: $668$
• Weight: $2$
• Character orbit: 668.e
• Analytic conductor: $5.334$
• Analytic rank: $0$
• Dimension: $1148$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 668 = 2^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	668.e (of order \(83\), degree \(82\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.33400685502\)
Analytic rank:	\(0\)
Dimension:	\(1148\)
Relative dimension:	\(14\) over \(\Q(\zeta_{83})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{83}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 1148 q - 2 q^{5} - 14 q^{9}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
9.1		0		−0.426508	−	3.20063i		0		−3.18243	−	1.40730i		0		1.55798	−	3.35028i		0		−7.16683	+	1.94460i		0
9.2		0		−0.360076	−	2.70211i		0		2.66599	+	1.17892i		0		0.359959	−	0.774059i		0		−4.27644	+	1.16034i		0
9.3		0		−0.342745	−	2.57205i		0		−0.628586	−	0.277966i		0		−1.47454	+	3.17087i		0		−3.60268	+	0.977525i		0
9.4		0		−0.198520	−	1.48975i		0		−0.0282706	−	0.0125015i		0		2.06702	−	4.44493i		0		0.715369	−	0.194103i		0
9.5		0		−0.178407	−	1.33882i		0		−1.62557	−	0.718841i		0		−0.315447	+	0.678340i		0		1.13471	−	0.307883i		0
9.6		0		−0.0387535	−	0.290817i		0		−1.08122	−	0.478126i		0		−0.661531	+	1.42256i		0		2.81224	−	0.763053i		0
9.7		0		−0.0142693	−	0.107081i		0		2.37141	+	1.04866i		0		−0.709794	+	1.52635i		0		2.88405	−	0.782538i		0
9.8		0		0.0433273	+	0.325140i		0		3.41908	+	1.51194i		0		−1.67119	+	3.59374i		0		2.79147	−	0.757419i		0
9.9		0		0.174236	+	1.30752i		0		−3.98478	−	1.76210i		0		−0.546271	+	1.17470i		0		1.21607	−	0.329959i		0
9.10		0		0.189949	+	1.42543i		0		2.53109	+	1.11927i		0		0.927279	−	1.99403i		0		0.899535	−	0.244073i		0
9.11		0		0.200238	+	1.50264i		0		−0.247141	−	0.109288i		0		1.16135	−	2.49738i		0		0.677472	−	0.183820i		0
9.12		0		0.214009	+	1.60599i		0		−2.48108	−	1.09715i		0		1.06258	−	2.28498i		0		0.361919	−	0.0982005i		0
9.13		0		0.353640	+	2.65381i		0		−1.27217	−	0.562563i		0		−1.92428	+	4.13797i		0		−4.02235	+	1.09139i		0
9.14		0		0.383878	+	2.88073i		0		1.71454	+	0.758185i		0		0.166885	−	0.358871i		0		−5.25594	+	1.42611i		0
21.1		0		−3.30940	−	0.504914i		0		−2.26577	−	2.68874i		0		1.34830	−	1.87063i		0		7.83367	+	2.44733i		0
21.2		0		−2.93125	−	0.447219i		0		1.46272	+	1.73577i		0		−0.690896	+	0.958545i		0		5.52869	+	1.72722i		0
21.3		0		−1.90829	−	0.291147i		0		1.01478	+	1.20422i		0		2.44900	−	3.39773i		0		0.693279	+	0.216588i		0
21.4		0		−1.77371	−	0.270614i		0		−1.21708	−	1.44428i		0		−0.212405	+	0.294689i		0		0.209294	+	0.0653857i		0
21.5		0		−1.69992	−	0.259356i		0		−0.724427	−	0.859660i		0		−2.54038	+	3.52451i		0		−0.0410528	−	0.0128253i		0
21.6		0		−0.537235	−	0.0819657i		0		−1.19475	−	1.41779i		0		0.821536	−	1.13980i		0		−2.58161	−	0.806524i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
167.c	even	83	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	668.2.e.a	✓	1148
167.c	even	83	1	inner	668.2.e.a	✓	1148

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
668.2.e.a	✓	1148	1.a	even	1	1	trivial
668.2.e.a	✓	1148	167.c	even	83	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(668, [\chi])\).