Newform orbit 3520.2.p.f

• Label: 3520.2.p.f
• Level: $3520$
• Weight: $2$
• Character orbit: 3520.p
• Analytic conductor: $28.107$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3520 = 2^{6} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3520.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(28.1073415115\)
Analytic rank:	\(0\)
Dimension:	\(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{3} + 40 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
351.1		0		−2.75916				0				1.00000i		0		1.24484				0		4.61295				0
351.2		0		−2.75916				0				1.00000i		0		−1.24484				0		4.61295				0
351.3		0		−2.75916				0			−	1.00000i		0		−1.24484				0		4.61295				0
351.4		0		−2.75916				0			−	1.00000i		0		1.24484				0		4.61295				0
351.5		0		−1.96322				0			−	1.00000i		0		−3.19572				0		0.854245				0
351.6		0		−1.96322				0			−	1.00000i		0		3.19572				0		0.854245				0
351.7		0		−1.96322				0				1.00000i		0		3.19572				0		0.854245				0
351.8		0		−1.96322				0				1.00000i		0		−3.19572				0		0.854245				0
351.9		0		−1.34165				0			−	1.00000i		0		−3.41100				0		−1.19997				0
351.10		0		−1.34165				0			−	1.00000i		0		3.41100				0		−1.19997				0
351.11		0		−1.34165				0				1.00000i		0		3.41100				0		−1.19997				0
351.12		0		−1.34165				0				1.00000i		0		−3.41100				0		−1.19997				0
351.13		0		−0.266719				0			−	1.00000i		0		−2.75342				0		−2.92886				0
351.14		0		−0.266719				0			−	1.00000i		0		2.75342				0		−2.92886				0
351.15		0		−0.266719				0				1.00000i		0		2.75342				0		−2.92886				0
351.16		0		−0.266719				0				1.00000i		0		−2.75342				0		−2.92886				0
351.17		0		1.13269				0			−	1.00000i		0		−0.810605				0		−1.71701				0
351.18		0		1.13269				0			−	1.00000i		0		0.810605				0		−1.71701				0
351.19		0		1.13269				0				1.00000i		0		0.810605				0		−1.71701				0
351.20		0		1.13269				0				1.00000i		0		−0.810605				0		−1.71701				0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	inner
11.b	odd	2	1	inner
88.g	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3520.2.p.f	yes	32
4.b	odd	2	1		3520.2.p.e	✓	32
8.b	even	2	1		3520.2.p.e	✓	32
8.d	odd	2	1	inner	3520.2.p.f	yes	32
11.b	odd	2	1	inner	3520.2.p.f	yes	32
44.c	even	2	1		3520.2.p.e	✓	32
88.b	odd	2	1		3520.2.p.e	✓	32
88.g	even	2	1	inner	3520.2.p.f	yes	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3520.2.p.e	✓	32	4.b	odd	2	1
3520.2.p.e	✓	32	8.b	even	2	1
3520.2.p.e	✓	32	44.c	even	2	1
3520.2.p.e	✓	32	88.b	odd	2	1
3520.2.p.f	yes	32	1.a	even	1	1	trivial
3520.2.p.f	yes	32	8.d	odd	2	1	inner
3520.2.p.f	yes	32	11.b	odd	2	1	inner
3520.2.p.f	yes	32	88.g	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3520, [\chi])\):

\( T_{3}^{8} - 2T_{3}^{7} - 15T_{3}^{6} + 24T_{3}^{5} + 68T_{3}^{4} - 76T_{3}^{3} - 92T_{3}^{2} + 72T_{3} + 24 \)

\( T_{59}^{8} + 20T_{59}^{7} + 16T_{59}^{6} - 1232T_{59}^{5} - 2480T_{59}^{4} + 15552T_{59}^{3} + 35776T_{59}^{2} - 3072T_{59} - 25344 \)