Newform orbit 88.5.j.a

• Label: 88.5.j.a
• Level: $88$
• Weight: $5$
• Character orbit: 88.j
• Analytic conductor: $9.097$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 88 = 2^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	88.j (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.09655675138\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 8 q^{3} - 492 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
17.1		0		−12.5334	−	9.10608i		0		3.97815	−	12.2435i		0		−48.5546	−	66.8297i		0		49.1361	+	151.225i		0
17.2		0		−10.4394	−	7.58468i		0		−8.80609	+	27.1023i		0		7.94479	+	10.9351i		0		26.4237	+	81.3237i		0
17.3		0		−8.80073	−	6.39410i		0		10.2982	−	31.6947i		0		−1.54592	−	2.12778i		0		11.5379	+	35.5099i		0
17.4		0		−5.71659	−	4.15335i		0		−3.13149	+	9.63774i		0		17.1137	+	23.5549i		0		−9.60123	−	29.5496i		0
17.5		0		−4.51291	−	3.27882i		0		6.29746	−	19.3816i		0		33.3932	+	45.9618i		0		−15.4147	−	47.4415i		0
17.6		0		0.609130	+	0.442559i		0		−7.32841	+	22.5545i		0		7.17242	+	9.87199i		0		−24.8552	−	76.4964i		0
17.7		0		0.777393	+	0.564809i		0		−14.2996	+	44.0095i		0		−28.0017	−	38.5411i		0		−24.7450	−	76.1574i		0
17.8		0		3.79982	+	2.76073i		0		1.18231	−	3.63876i		0		−52.6193	−	72.4242i		0		−18.2134	−	56.0550i		0
17.9		0		4.67890	+	3.39942i		0		8.76633	−	26.9800i		0		34.9283	+	48.0747i		0		−14.6943	−	45.2245i		0
17.10		0		6.78289	+	4.92806i		0		10.8109	−	33.2725i		0		−40.5142	−	55.7630i		0		−3.30853	−	10.1826i		0
17.11		0		9.85015	+	7.15656i		0		−1.60048	+	4.92578i		0		21.8981	+	30.1402i		0		20.7789	+	63.9508i		0
17.12		0		13.5048	+	9.81180i		0		−6.16735	+	18.9811i		0		4.06384	+	5.59339i		0		61.0774	+	187.977i		0
41.1		0		−4.78745	−	14.7343i		0		34.8197	+	25.2980i		0		73.4782	+	23.8745i		0		−128.648	+	93.4685i		0
41.2		0		−4.46915	−	13.7546i		0		−11.7109	−	8.50850i		0		−38.1143	−	12.3841i		0		−103.686	+	75.3323i		0
41.3		0		−3.72907	−	11.4769i		0		−37.7158	−	27.4021i		0		36.5235	+	11.8672i		0		−52.2828	+	37.9857i		0
41.4		0		−2.04014	−	6.27892i		0		8.90463	+	6.46959i		0		−69.9417	−	22.7254i		0		30.2677	−	21.9908i		0
41.5		0		−1.94870	−	5.99749i		0		−2.12135	−	1.54125i		0		50.8092	+	16.5089i		0		33.3580	−	24.2360i		0
41.6		0		−1.73845	−	5.35040i		0		31.2487	+	22.7035i		0		−26.5962	−	8.64162i		0		39.9258	−	29.0078i		0
41.7		0		0.364909	+	1.12307i		0		−6.95065	−	5.04994i		0		0.103808	+	0.0337293i		0		64.4022	−	46.7910i		0
41.8		0		1.52693	+	4.69942i		0		−22.8681	−	16.6146i		0		55.5149	+	18.0379i		0		45.7774	−	33.2592i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.d	odd	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	88.5.j.a	✓	48
4.b	odd	2	1		176.5.n.d		48
11.d	odd	10	1	inner	88.5.j.a	✓	48
44.g	even	10	1		176.5.n.d		48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
88.5.j.a	✓	48	1.a	even	1	1	trivial
88.5.j.a	✓	48	11.d	odd	10	1	inner
176.5.n.d		48	4.b	odd	2	1
176.5.n.d		48	44.g	even	10	1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(88, [\chi])\).