Newform orbit 252.9.bk.a

• Label: 252.9.bk.a
• Level: $252$
• Weight: $9$
• Character orbit: 252.bk
• Analytic conductor: $102.659$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	252.bk (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(102.659409735\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(22\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 1230 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
53.1		0			0			0		−1025.51	+	592.080i		0		2369.75	+	386.132i		0			0			0
53.2		0			0			0		−1009.83	+	583.025i		0		−1699.40	−	1696.12i		0			0			0
53.3		0			0			0		−724.791	+	418.458i		0		613.419	+	2321.32i		0			0			0
53.4		0			0			0		−700.103	+	404.204i		0		−641.390	+	2313.75i		0			0			0
53.5		0			0			0		−674.478	+	389.410i		0		−1634.98	−	1758.31i		0			0			0
53.6		0			0			0		−517.869	+	298.992i		0		2338.25	−	545.349i		0			0			0
53.7		0			0			0		−349.907	+	202.019i		0		−1821.32	+	1564.48i		0			0			0
53.8		0			0			0		−318.126	+	183.670i		0		1593.54	+	1795.95i		0			0			0
53.9		0			0			0		−173.159	+	99.9732i		0		1613.03	−	1778.47i		0			0			0
53.10		0			0			0		−36.7886	+	21.2399i		0		−115.865	−	2398.20i		0			0			0
53.11		0			0			0		−23.4272	+	13.5257i		0		−2307.52	+	663.449i		0			0			0
53.12		0			0			0		23.4272	−	13.5257i		0		−2307.52	+	663.449i		0			0			0
53.13		0			0			0		36.7886	−	21.2399i		0		−115.865	−	2398.20i		0			0			0
53.14		0			0			0		173.159	−	99.9732i		0		1613.03	−	1778.47i		0			0			0
53.15		0			0			0		318.126	−	183.670i		0		1593.54	+	1795.95i		0			0			0
53.16		0			0			0		349.907	−	202.019i		0		−1821.32	+	1564.48i		0			0			0
53.17		0			0			0		517.869	−	298.992i		0		2338.25	−	545.349i		0			0			0
53.18		0			0			0		674.478	−	389.410i		0		−1634.98	−	1758.31i		0			0			0
53.19		0			0			0		700.103	−	404.204i		0		−641.390	+	2313.75i		0			0			0
53.20		0			0			0		724.791	−	418.458i		0		613.419	+	2321.32i		0			0			0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.c	even	3	1	inner
21.h	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	252.9.bk.a	✓	44
3.b	odd	2	1	inner	252.9.bk.a	✓	44
7.c	even	3	1	inner	252.9.bk.a	✓	44
21.h	odd	6	1	inner	252.9.bk.a	✓	44

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
252.9.bk.a	✓	44	1.a	even	1	1	trivial
252.9.bk.a	✓	44	3.b	odd	2	1	inner
252.9.bk.a	✓	44	7.c	even	3	1	inner
252.9.bk.a	✓	44	21.h	odd	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(252, [\chi])\).