Newform orbit 668.2.b.b

• Label: 668.2.b.b
• Level: $668$
• Weight: $2$
• Character orbit: 668.b
• Analytic conductor: $5.334$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 668 = 2^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	668.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.33400685502\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 2 q^{2} + 2 q^{4} - 8 q^{6} - 8 q^{8} - 16 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
667.1	−1.37801	−	0.317929i			2.42728i	1.79784	+	0.876221i		−	2.41022i	0.771702	−	3.34482i			1.33262i	−2.19887	−	1.77903i	−2.89167			−0.766278	+	3.32131i
667.2	−1.37801	−	0.317929i			2.42728i	1.79784	+	0.876221i			2.41022i	0.771702	−	3.34482i			1.33262i	−2.19887	−	1.77903i	−2.89167			0.766278	−	3.32131i
667.3	−1.37801	+	0.317929i		−	2.42728i	1.79784	−	0.876221i		−	2.41022i	0.771702	+	3.34482i		−	1.33262i	−2.19887	+	1.77903i	−2.89167			0.766278	+	3.32131i
667.4	−1.37801	+	0.317929i		−	2.42728i	1.79784	−	0.876221i			2.41022i	0.771702	+	3.34482i		−	1.33262i	−2.19887	+	1.77903i	−2.89167			−0.766278	−	3.32131i
667.5	−1.37670	−	0.323577i		−	0.453678i	1.79060	+	0.890936i		−	4.09468i	−0.146800	+	0.624578i			3.11488i	−2.17682	−	1.80595i	2.79418			−1.32494	+	5.63714i
667.6	−1.37670	−	0.323577i		−	0.453678i	1.79060	+	0.890936i			4.09468i	−0.146800	+	0.624578i			3.11488i	−2.17682	−	1.80595i	2.79418			1.32494	−	5.63714i
667.7	−1.37670	+	0.323577i			0.453678i	1.79060	−	0.890936i		−	4.09468i	−0.146800	−	0.624578i		−	3.11488i	−2.17682	+	1.80595i	2.79418			1.32494	+	5.63714i
667.8	−1.37670	+	0.323577i			0.453678i	1.79060	−	0.890936i			4.09468i	−0.146800	−	0.624578i		−	3.11488i	−2.17682	+	1.80595i	2.79418			−1.32494	−	5.63714i
667.9	−1.30799	−	0.537726i		−	2.06566i	1.42170	+	1.40669i		−	1.78350i	−1.11076	+	2.70187i		−	0.540270i	−1.10317	−	2.60442i	−1.26695			−0.959034	+	2.33281i
667.10	−1.30799	−	0.537726i		−	2.06566i	1.42170	+	1.40669i			1.78350i	−1.11076	+	2.70187i		−	0.540270i	−1.10317	−	2.60442i	−1.26695			0.959034	−	2.33281i
667.11	−1.30799	+	0.537726i			2.06566i	1.42170	−	1.40669i		−	1.78350i	−1.11076	−	2.70187i			0.540270i	−1.10317	+	2.60442i	−1.26695			0.959034	+	2.33281i
667.12	−1.30799	+	0.537726i			2.06566i	1.42170	−	1.40669i			1.78350i	−1.11076	−	2.70187i			0.540270i	−1.10317	+	2.60442i	−1.26695			−0.959034	−	2.33281i
667.13	−1.12623	−	0.855338i			0.530251i	0.536794	+	1.92662i		−	0.936151i	0.453544	−	0.597186i			2.10467i	1.04335	−	2.62896i	2.71883			−0.800726	+	1.05432i
667.14	−1.12623	−	0.855338i			0.530251i	0.536794	+	1.92662i			0.936151i	0.453544	−	0.597186i			2.10467i	1.04335	−	2.62896i	2.71883			0.800726	−	1.05432i
667.15	−1.12623	+	0.855338i		−	0.530251i	0.536794	−	1.92662i		−	0.936151i	0.453544	+	0.597186i		−	2.10467i	1.04335	+	2.62896i	2.71883			0.800726	+	1.05432i
667.16	−1.12623	+	0.855338i		−	0.530251i	0.536794	−	1.92662i			0.936151i	0.453544	+	0.597186i		−	2.10467i	1.04335	+	2.62896i	2.71883			−0.800726	−	1.05432i
667.17	−0.844041	−	1.13472i			0.873998i	−0.575190	+	1.91550i		−	2.77771i	0.991745	−	0.737690i		−	1.63351i	2.65905	−	0.964083i	2.23613			−3.15193	+	2.34450i
667.18	−0.844041	−	1.13472i			0.873998i	−0.575190	+	1.91550i			2.77771i	0.991745	−	0.737690i		−	1.63351i	2.65905	−	0.964083i	2.23613			3.15193	−	2.34450i
667.19	−0.844041	+	1.13472i		−	0.873998i	−0.575190	−	1.91550i		−	2.77771i	0.991745	+	0.737690i			1.63351i	2.65905	+	0.964083i	2.23613			3.15193	+	2.34450i
667.20	−0.844041	+	1.13472i		−	0.873998i	−0.575190	−	1.91550i			2.77771i	0.991745	+	0.737690i			1.63351i	2.65905	+	0.964083i	2.23613			−3.15193	−	2.34450i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
167.b	odd	2	1	inner
668.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	668.2.b.b	✓	60
4.b	odd	2	1	inner	668.2.b.b	✓	60
167.b	odd	2	1	inner	668.2.b.b	✓	60
668.b	even	2	1	inner	668.2.b.b	✓	60

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
668.2.b.b	✓	60	1.a	even	1	1	trivial
668.2.b.b	✓	60	4.b	odd	2	1	inner
668.2.b.b	✓	60	167.b	odd	2	1	inner
668.2.b.b	✓	60	668.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3^30 + 49*T3^28 + 1058*T3^26 + 13271*T3^24 + 107334*T3^22 + 587566*T3^20 + 2225450*T3^18 + 5869273*T3^16 + 10731362*T3^14 + 13423388*T3^12 + 11219086*T3^10 + 6029013*T3^8 + 1956649*T3^6 + 343401*T3^4 + 25696*T3^2 + 256