Newform orbit 538.4.b.a

• Label: 538.4.b.a
• Level: $538$
• Weight: $4$
• Character orbit: 538.b
• Analytic conductor: $31.743$
• Analytic rank: $0$
• Dimension: $68$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 538 = 2 \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	538.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(31.7430275831\)
Analytic rank:	\(0\)
Dimension:	\(68\)
Twist minimal:	yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 272 q^{4} + 38 q^{5} - 4 q^{6} - 594 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
537.1		−	2.00000i		−	9.88791i	−4.00000			9.87298			−19.7758				−	5.28763i			8.00000i	−70.7708				−	19.7460i
537.2		−	2.00000i		−	9.56359i	−4.00000			−14.2576			−19.1272				−	26.9416i			8.00000i	−64.4623					28.5153i
537.3		−	2.00000i		−	9.28480i	−4.00000			8.94524			−18.5696					16.2739i			8.00000i	−59.2075				−	17.8905i
537.4		−	2.00000i		−	8.17841i	−4.00000			10.4648			−16.3568					14.5901i			8.00000i	−39.8864				−	20.9295i
537.5		−	2.00000i		−	7.77553i	−4.00000			−14.6725			−15.5511					24.2023i			8.00000i	−33.4588					29.3451i
537.6		−	2.00000i		−	7.25003i	−4.00000			−10.7929			−14.5001					6.91932i			8.00000i	−25.5630					21.5858i
537.7		−	2.00000i		−	5.56172i	−4.00000			10.6999			−11.1234				−	28.6762i			8.00000i	−3.93276				−	21.3998i
537.8		−	2.00000i		−	5.50515i	−4.00000			−2.58904			−11.0103					31.4556i			8.00000i	−3.30666					5.17808i
537.9		−	2.00000i		−	5.25034i	−4.00000			11.3357			−10.5007				−	15.7937i			8.00000i	−0.566075				−	22.6713i
537.10		−	2.00000i		−	4.97105i	−4.00000			0.491173			−9.94210				−	17.1942i			8.00000i	2.28866				−	0.982346i
537.11		−	2.00000i		−	4.55526i	−4.00000			−7.10991			−9.11052				−	12.2116i			8.00000i	6.24961					14.2198i
537.12		−	2.00000i		−	3.80249i	−4.00000			21.0758			−7.60499					19.4523i			8.00000i	12.5410				−	42.1517i
537.13		−	2.00000i		−	2.90912i	−4.00000			−14.2265			−5.81824				−	12.8404i			8.00000i	18.5370					28.4531i
537.14		−	2.00000i		−	2.52250i	−4.00000			−0.152611			−5.04499					7.33691i			8.00000i	20.6370					0.305221i
537.15		−	2.00000i		−	1.31750i	−4.00000			−18.2850			−2.63499					19.3161i			8.00000i	25.2642					36.5699i
537.16		−	2.00000i		−	1.13081i	−4.00000			−16.2280			−2.26163				−	31.7390i			8.00000i	25.7213					32.4561i
537.17		−	2.00000i		−	0.122336i	−4.00000			7.26358			−0.244671					31.0626i			8.00000i	26.9850				−	14.5272i
537.18		−	2.00000i			0.374041i	−4.00000			15.5232			0.748083				−	10.9338i			8.00000i	26.8601				−	31.0464i
537.19		−	2.00000i			0.568496i	−4.00000			5.35960			1.13699				−	6.64699i			8.00000i	26.6768				−	10.7192i
537.20		−	2.00000i			2.32411i	−4.00000			−7.44884			4.64822					10.6572i			8.00000i	21.5985					14.8977i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
269.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	538.4.b.a	✓	68
269.b	even	2	1	inner	538.4.b.a	✓	68

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
538.4.b.a	✓	68	1.a	even	1	1	trivial
538.4.b.a	✓	68	269.b	even	2	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(538, [\chi])\).