Newform orbit 440.2.bn.a

• Label: 440.2.bn.a
• Level: $440$
• Weight: $2$
• Character orbit: 440.bn
• Analytic conductor: $3.513$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	440.bn (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.51341768894\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(18\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 18 q^{9}+O(q^{10}) \)

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		−1.97047	−	2.71211i		0		2.19738	+	0.414130i		0		−2.38609	+	3.28417i		0		−2.54578	+	7.83509i		0
9.2		0		−1.51545	−	2.08584i		0		−1.46579	+	1.68863i		0		2.00676	−	2.76207i		0		−1.12708	+	3.46880i		0
9.3		0		−1.35923	−	1.87082i		0		0.939384	+	2.02918i		0		0.618827	−	0.851742i		0		−0.725409	+	2.23258i		0
9.4		0		−1.28339	−	1.76644i		0		−0.244412	−	2.22267i		0		−2.55269	+	3.51348i		0		−0.546162	+	1.68091i		0
9.5		0		−1.27634	−	1.75674i		0		1.23252	−	1.86572i		0		2.56608	−	3.53191i		0		−0.530019	+	1.63123i		0
9.6		0		−0.824263	−	1.13450i		0		−0.928972	−	2.03396i		0		0.697744	−	0.960362i		0		0.319368	−	0.982913i		0
9.7		0		−0.662675	−	0.912094i		0		−2.08999	+	0.794950i		0		−1.59515	+	2.19553i		0		0.534274	−	1.64433i		0
9.8		0		−0.126982	−	0.174776i		0		1.82534	−	1.29157i		0		−0.564009	+	0.776292i		0		0.912629	−	2.80878i		0
9.9		0		−0.00529914	−	0.00729365i		0		−0.510973	+	2.17690i		0		−1.52189	+	2.09471i		0		0.927026	−	2.85309i		0
9.10		0		0.00529914	+	0.00729365i		0		1.69294	+	1.46081i		0		1.52189	−	2.09471i		0		0.927026	−	2.85309i		0
9.11		0		0.126982	+	0.174776i		0		−2.23589	+	0.0280054i		0		0.564009	−	0.776292i		0		0.912629	−	2.80878i		0
9.12		0		0.662675	+	0.912094i		0		2.15810	−	0.585337i		0		1.59515	−	2.19553i		0		0.534274	−	1.64433i		0
9.13		0		0.824263	+	1.13450i		0		−0.443981	−	2.19155i		0		−0.697744	+	0.960362i		0		0.319368	−	0.982913i		0
9.14		0		1.27634	+	1.75674i		0		−2.09377	−	0.784938i		0		−2.56608	+	3.53191i		0		−0.530019	+	1.63123i		0
9.15		0		1.28339	+	1.76644i		0		−1.10872	−	1.94184i		0		2.55269	−	3.51348i		0		−0.546162	+	1.68091i		0
9.16		0		1.35923	+	1.87082i		0		0.432742	+	2.19379i		0		−0.618827	+	0.851742i		0		−0.725409	+	2.23258i		0
9.17		0		1.51545	+	2.08584i		0		2.17840	+	0.504557i		0		−2.00676	+	2.76207i		0		−1.12708	+	3.46880i		0
9.18		0		1.97047	+	2.71211i		0		−1.53430	+	1.62663i		0		2.38609	−	3.28417i		0		−2.54578	+	7.83509i		0
49.1		0		−1.97047	+	2.71211i		0		2.19738	−	0.414130i		0		−2.38609	−	3.28417i		0		−2.54578	−	7.83509i		0
49.2		0		−1.51545	+	2.08584i		0		−1.46579	−	1.68863i		0		2.00676	+	2.76207i		0		−1.12708	−	3.46880i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
11.c	even	5	1	inner
55.j	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	440.2.bn.a	✓	72
4.b	odd	2	1		880.2.cd.e		72
5.b	even	2	1	inner	440.2.bn.a	✓	72
11.c	even	5	1	inner	440.2.bn.a	✓	72
20.d	odd	2	1		880.2.cd.e		72
44.h	odd	10	1		880.2.cd.e		72
55.j	even	10	1	inner	440.2.bn.a	✓	72
220.n	odd	10	1		880.2.cd.e		72

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
440.2.bn.a	✓	72	1.a	even	1	1	trivial
440.2.bn.a	✓	72	5.b	even	2	1	inner
440.2.bn.a	✓	72	11.c	even	5	1	inner
440.2.bn.a	✓	72	55.j	even	10	1	inner
880.2.cd.e		72	4.b	odd	2	1
880.2.cd.e		72	20.d	odd	2	1
880.2.cd.e		72	44.h	odd	10	1
880.2.cd.e		72	220.n	odd	10	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(440, [\chi])\).