Newform orbit 668.2.b.a

• Label: 668.2.b.a
• Level: $668$
• Weight: $2$
• Character orbit: 668.b
• Analytic conductor: $5.334$
• Analytic rank: $0$
• Dimension: $22$
• CM discriminant: -167
• 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:	\(22\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 66 q^{9}+O(q^{10}) \)

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	−1.40042	−	0.197044i			2.07563i	1.92235	+	0.551888i		0		0.408990	−	2.90675i		−	4.81975i	−2.58335	−	1.15166i	−1.30824				0
667.2	−1.40042	+	0.197044i		−	2.07563i	1.92235	−	0.551888i		0		0.408990	+	2.90675i			4.81975i	−2.58335	+	1.15166i	−1.30824				0
667.3	−1.28464	−	0.591360i		−	0.736748i	1.30059	+	1.51937i		0		−0.435683	+	0.946454i		−	5.23983i	−0.772291	−	2.72095i	2.45720				0
667.4	−1.28464	+	0.591360i			0.736748i	1.30059	−	1.51937i		0		−0.435683	−	0.946454i			5.23983i	−0.772291	+	2.72095i	2.45720				0
667.5	−1.07158	−	0.922888i		−	3.45525i	0.296557	+	1.97789i		0		−3.18880	+	3.70256i			4.00920i	1.50759	−	2.39315i	−8.93873				0
667.6	−1.07158	+	0.922888i			3.45525i	0.296557	−	1.97789i		0		−3.18880	−	3.70256i		−	4.00920i	1.50759	+	2.39315i	−8.93873				0
667.7	−0.760993	−	1.19201i			3.04057i	−0.841781	+	1.81422i		0		3.62439	−	2.31385i			5.23541i	2.80316	−	0.377198i	−6.24504				0
667.8	−0.760993	+	1.19201i		−	3.04057i	−0.841781	−	1.81422i		0		3.62439	+	2.31385i		−	5.23541i	2.80316	+	0.377198i	−6.24504				0
667.9	−0.402518	−	1.35572i			2.44978i	−1.67596	+	1.09140i		0		3.32122	−	0.986080i		−	2.87385i	2.15424	+	1.83282i	−3.00142				0
667.10	−0.402518	+	1.35572i		−	2.44978i	−1.67596	−	1.09140i		0		3.32122	+	0.986080i			2.87385i	2.15424	−	1.83282i	−3.00142				0
667.11	0.00426214	−	1.41421i		−	3.24555i	−1.99996	−	0.0120551i		0		−4.58987	−	0.0138330i		−	4.80685i	−0.0255725	+	2.82831i	−7.53357				0
667.12	0.00426214	+	1.41421i			3.24555i	−1.99996	+	0.0120551i		0		−4.58987	+	0.0138330i			4.80685i	−0.0255725	−	2.82831i	−7.53357				0
667.13	0.394339	−	1.35812i			0.246717i	−1.68899	−	1.07112i		0		0.335073	+	0.0972902i			1.50567i	−2.12075	+	1.87148i	2.93913				0
667.14	0.394339	+	1.35812i		−	0.246717i	−1.68899	+	1.07112i		0		0.335073	−	0.0972902i		−	1.50567i	−2.12075	−	1.87148i	2.93913				0
667.15	0.768164	−	1.18740i			1.21020i	−0.819849	−	1.82424i		0		1.43699	+	0.929628i			3.98887i	−2.79588	−	0.427823i	1.53543				0
667.16	0.768164	+	1.18740i		−	1.21020i	−0.819849	+	1.82424i		0		1.43699	−	0.929628i		−	3.98887i	−2.79588	+	0.427823i	1.53543				0
667.17	1.06600	−	0.929330i		−	2.77291i	0.272692	−	1.98132i		0		−2.57695	−	2.95591i		−	0.0155204i	−1.55061	−	2.36550i	−4.68903				0
667.18	1.06600	+	0.929330i			2.77291i	0.272692	+	1.98132i		0		−2.57695	+	2.95591i			0.0155204i	−1.55061	+	2.36550i	−4.68903				0
667.19	1.28818	−	0.583606i			1.66053i	1.31881	−	1.50358i		0		0.969093	+	2.13906i		−	2.84773i	0.821364	−	2.70654i	0.242651				0
667.20	1.28818	+	0.583606i		−	1.66053i	1.31881	+	1.50358i		0		0.969093	−	2.13906i			2.84773i	0.821364	+	2.70654i	0.242651				0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
167.b	odd	2	1	CM by \(\Q(\sqrt{-167}) \)
4.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.a	✓	22
4.b	odd	2	1	inner	668.2.b.a	✓	22
167.b	odd	2	1	CM	668.2.b.a	✓	22
668.b	even	2	1	inner	668.2.b.a	✓	22

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
668.2.b.a	✓	22	1.a	even	1	1	trivial
668.2.b.a	✓	22	4.b	odd	2	1	inner
668.2.b.a	✓	22	167.b	odd	2	1	CM
668.2.b.a	✓	22	668.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3^22 + 66*T3^20 + 1881*T3^18 + 30294*T3^16 + 302940*T3^14 + 1945944*T3^12 + 8027019*T3^10 + 20640906*T3^8 + 30961359*T3^6 + 23816430*T3^4 + 7144929*T3^2 + 353372