Newform orbit 704.4.e.g

• Label: 704.4.e.g
• Level: $704$
• Weight: $4$
• Character orbit: 704.e
• Analytic conductor: $41.537$
• Analytic rank: $0$
• Dimension: $36$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 704 = 2^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	704.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(41.5373446440\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Twist minimal:	no (minimal twist has level 352)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 364 q^{9} + 1236 q^{25} + 592 q^{33} - 984 q^{45} + 2372 q^{49} - 376 q^{53} + 2824 q^{69} + 2592 q^{77} + 6756 q^{81} + 512 q^{89} + 872 q^{93} + 3904 q^{97}+O(q^{100}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
703.1		0			−	9.41697i		0		17.7991				0		30.3804				0		−61.6793				0
703.2		0				9.41697i		0		17.7991				0		30.3804				0		−61.6793				0
703.3		0			−	3.08315i		0		19.1950				0		20.3180				0		17.4942				0
703.4		0				3.08315i		0		19.1950				0		20.3180				0		17.4942				0
703.5		0			−	2.43252i		0		−17.0450				0		−13.3052				0		21.0829				0
703.6		0				2.43252i		0		−17.0450				0		−13.3052				0		21.0829				0
703.7		0			−	7.83879i		0		−16.0908				0		−7.53038				0		−34.4466				0
703.8		0				7.83879i		0		−16.0908				0		−7.53038				0		−34.4466				0
703.9		0			−	6.71366i		0		6.05508				0		15.0900				0		−18.0733				0
703.10		0				6.71366i		0		6.05508				0		15.0900				0		−18.0733				0
703.11		0			−	9.41697i		0		17.7991				0		−30.3804				0		−61.6793				0
703.12		0				9.41697i		0		17.7991				0		−30.3804				0		−61.6793				0
703.13		0			−	5.16105i		0		−0.643894				0		−13.2477				0		0.363555				0
703.14		0				5.16105i		0		−0.643894				0		−13.2477				0		0.363555				0
703.15		0			−	6.71366i		0		6.05508				0		−15.0900				0		−18.0733				0
703.16		0				6.71366i		0		6.05508				0		−15.0900				0		−18.0733				0
703.17		0			−	2.43252i		0		−17.0450				0		13.3052				0		21.0829				0
703.18		0				2.43252i		0		−17.0450				0		13.3052				0		21.0829				0
703.19		0			−	3.08315i		0		19.1950				0		−20.3180				0		17.4942				0
703.20		0				3.08315i		0		19.1950				0		−20.3180				0		17.4942				0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
11.b	odd	2	1	inner
44.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	704.4.e.g		36
4.b	odd	2	1	inner	704.4.e.g		36
8.b	even	2	1		352.4.e.a	✓	36
8.d	odd	2	1		352.4.e.a	✓	36
11.b	odd	2	1	inner	704.4.e.g		36
44.c	even	2	1	inner	704.4.e.g		36
88.b	odd	2	1		352.4.e.a	✓	36
88.g	even	2	1		352.4.e.a	✓	36

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
352.4.e.a	✓	36	8.b	even	2	1
352.4.e.a	✓	36	8.d	odd	2	1
352.4.e.a	✓	36	88.b	odd	2	1
352.4.e.a	✓	36	88.g	even	2	1
704.4.e.g		36	1.a	even	1	1	trivial
704.4.e.g		36	4.b	odd	2	1	inner
704.4.e.g		36	11.b	odd	2	1	inner
704.4.e.g		36	44.c	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 \(S_{4}^{\mathrm{new}}(704, [\chi])\):

T3^18 + 334*T3^16 + 44687*T3^14 + 3074352*T3^12 + 116680607*T3^10 + 2459687638*T3^8 + 28201760193*T3^6 + 171012433228*T3^4 + 500918319920*T3^2 + 529087483456

T7^18 - 3680*T7^16 + 5338416*T7^14 - 3995877888*T7^12 + 1707146380032*T7^10 - 432197262319616*T7^8 + 64656429569478656*T7^6 - 5463278965577089024*T7^4 + 233375088041493790720*T7^2 - 3864704007745315012608