Newform orbit 1216.3.g.e

• Label: 1216.3.g.e
• Level: $1216$
• Weight: $3$
• Character orbit: 1216.g
• Analytic conductor: $33.134$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1216.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(33.1336001462\)
Analytic rank:	\(0\)
Dimension:	\(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 264 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
417.1		0		−5.07504				0				5.51741i		0		−8.74188				0		16.7560				0
417.2		0		−5.07504				0			−	5.51741i		0		8.74188				0		16.7560				0
417.3		0		−5.07504				0				5.51741i		0		8.74188				0		16.7560				0
417.4		0		−5.07504				0			−	5.51741i		0		−8.74188				0		16.7560				0
417.5		0		−4.81671				0				9.51528i		0		−8.13398				0		14.2007				0
417.6		0		−4.81671				0			−	9.51528i		0		8.13398				0		14.2007				0
417.7		0		−4.81671				0				9.51528i		0		8.13398				0		14.2007				0
417.8		0		−4.81671				0			−	9.51528i		0		−8.13398				0		14.2007				0
417.9		0		−4.58612				0			−	0.323607i		0		−8.09640				0		12.0325				0
417.10		0		−4.58612				0			−	0.323607i		0		8.09640				0		12.0325				0
417.11		0		−4.58612				0				0.323607i		0		8.09640				0		12.0325				0
417.12		0		−4.58612				0				0.323607i		0		−8.09640				0		12.0325				0
417.13		0		−2.88498				0			−	4.10261i		0		−9.53824				0		−0.676870				0
417.14		0		−2.88498				0				4.10261i		0		9.53824				0		−0.676870				0
417.15		0		−2.88498				0			−	4.10261i		0		9.53824				0		−0.676870				0
417.16		0		−2.88498				0				4.10261i		0		−9.53824				0		−0.676870				0
417.17		0		−2.73941				0				5.26292i		0		−4.69464				0		−1.49561				0
417.18		0		−2.73941				0			−	5.26292i		0		4.69464				0		−1.49561				0
417.19		0		−2.73941				0				5.26292i		0		4.69464				0		−1.49561				0
417.20		0		−2.73941				0			−	5.26292i		0		−4.69464				0		−1.49561				0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
8.b	even	2	1	inner
8.d	odd	2	1	inner
19.b	odd	2	1	inner
76.d	even	2	1	inner
152.b	even	2	1	inner
152.g	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1216.3.g.e	✓	48
4.b	odd	2	1	inner	1216.3.g.e	✓	48
8.b	even	2	1	inner	1216.3.g.e	✓	48
8.d	odd	2	1	inner	1216.3.g.e	✓	48
19.b	odd	2	1	inner	1216.3.g.e	✓	48
76.d	even	2	1	inner	1216.3.g.e	✓	48
152.b	even	2	1	inner	1216.3.g.e	✓	48
152.g	odd	2	1	inner	1216.3.g.e	✓	48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1216.3.g.e	✓	48	1.a	even	1	1	trivial
1216.3.g.e	✓	48	4.b	odd	2	1	inner
1216.3.g.e	✓	48	8.b	even	2	1	inner
1216.3.g.e	✓	48	8.d	odd	2	1	inner
1216.3.g.e	✓	48	19.b	odd	2	1	inner
1216.3.g.e	✓	48	76.d	even	2	1	inner
1216.3.g.e	✓	48	152.b	even	2	1	inner
1216.3.g.e	✓	48	152.g	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1216, [\chi])\):

\( T_{3}^{12} - 87T_{3}^{10} + 2899T_{3}^{8} - 46005T_{3}^{6} + 351080T_{3}^{4} - 1140656T_{3}^{2} + 928896 \)

\( T_{5}^{12} + 170T_{5}^{10} + 9353T_{5}^{8} + 217852T_{5}^{6} + 2092864T_{5}^{4} + 5849088T_{5}^{2} + 589824 \)