Newform orbit 700.2.bh.a

• Label: 700.2.bh.a
• Level: $700$
• Weight: $2$
• Character orbit: 700.bh
• Analytic conductor: $5.590$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	700.bh (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.58952814149\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(20\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q + 2 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
13.1		0		−2.94578	−	1.50095i		0		−0.876158	+	2.05727i		0		−2.63954	−	0.181212i		0		4.66140	+	6.41586i		0
13.2		0		−2.65727	−	1.35395i		0		−1.95242	−	1.08997i		0		2.48964	+	0.895377i		0		3.46456	+	4.76856i		0
13.3		0		−2.19584	−	1.11884i		0		1.41574	−	1.73081i		0		−0.490779	−	2.59983i		0		1.80656	+	2.48651i		0
13.4		0		−2.01834	−	1.02839i		0		2.06081	+	0.867790i		0		1.95817	−	1.77920i		0		1.25274	+	1.72425i		0
13.5		0		−1.55612	−	0.792880i		0		−0.954449	−	2.02213i		0		−2.00091	+	1.73101i		0		0.0294804	+	0.0405763i		0
13.6		0		−1.50771	−	0.768215i		0		2.23479	+	0.0754702i		0		−1.41426	+	2.23604i		0		−0.0803288	−	0.110563i		0
13.7		0		−1.39158	−	0.709046i		0		−0.306430	+	2.21497i		0		2.05820	+	1.66247i		0		−0.329605	−	0.453663i		0
13.8		0		−0.684110	−	0.348571i		0		−2.23122	+	0.147165i		0		−0.447996	−	2.60755i		0		−1.41685	−	1.95013i		0
13.9		0		−0.147275	−	0.0750406i		0		−1.44837	+	1.70359i		0		−2.61341	−	0.412424i		0		−1.74730	−	2.40495i		0
13.10		0		−0.0273853	−	0.0139535i		0		−0.709500	−	2.12052i		0		2.59558	−	0.512826i		0		−1.76280	−	2.42629i		0
13.11		0		0.0273853	+	0.0139535i		0		0.709500	+	2.12052i		0		0.512826	−	2.59558i		0		−1.76280	−	2.42629i		0
13.12		0		0.147275	+	0.0750406i		0		1.44837	−	1.70359i		0		0.412424	+	2.61341i		0		−1.74730	−	2.40495i		0
13.13		0		0.684110	+	0.348571i		0		2.23122	−	0.147165i		0		2.60755	+	0.447996i		0		−1.41685	−	1.95013i		0
13.14		0		1.39158	+	0.709046i		0		0.306430	−	2.21497i		0		−1.66247	−	2.05820i		0		−0.329605	−	0.453663i		0
13.15		0		1.50771	+	0.768215i		0		−2.23479	−	0.0754702i		0		−2.23604	+	1.41426i		0		−0.0803288	−	0.110563i		0
13.16		0		1.55612	+	0.792880i		0		0.954449	+	2.02213i		0		−1.73101	+	2.00091i		0		0.0294804	+	0.0405763i		0
13.17		0		2.01834	+	1.02839i		0		−2.06081	−	0.867790i		0		1.77920	−	1.95817i		0		1.25274	+	1.72425i		0
13.18		0		2.19584	+	1.11884i		0		−1.41574	+	1.73081i		0		2.59983	+	0.490779i		0		1.80656	+	2.48651i		0
13.19		0		2.65727	+	1.35395i		0		1.95242	+	1.08997i		0		−0.895377	−	2.48964i		0		3.46456	+	4.76856i		0
13.20		0		2.94578	+	1.50095i		0		0.876158	−	2.05727i		0		0.181212	+	2.63954i		0		4.66140	+	6.41586i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
25.f	odd	20	1	inner
175.s	even	20	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	700.2.bh.a	✓	160
7.b	odd	2	1	inner	700.2.bh.a	✓	160
25.f	odd	20	1	inner	700.2.bh.a	✓	160
175.s	even	20	1	inner	700.2.bh.a	✓	160

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
700.2.bh.a	✓	160	1.a	even	1	1	trivial
700.2.bh.a	✓	160	7.b	odd	2	1	inner
700.2.bh.a	✓	160	25.f	odd	20	1	inner
700.2.bh.a	✓	160	175.s	even	20	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(700, [\chi])\).