Newform orbit 363.5.c.e

• Label: 363.5.c.e
• Level: $363$
• Weight: $5$
• Character orbit: 363.c
• Analytic conductor: $37.523$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	363.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(37.5232965994\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Twist minimal:	no (minimal twist has level 33)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 244 q^{4} + 36 q^{5} + 864 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
241.1		−	7.67147i	−5.19615			−42.8515			7.05857					39.8622i			48.7800i			205.991i	27.0000				−	54.1497i
241.2		−	7.43699i	−5.19615			−39.3088			30.7742					38.6437i		−	42.7028i			173.347i	27.0000				−	228.867i
241.3		−	7.15959i	5.19615			−35.2597			−41.1121				−	37.2023i			33.8668i			137.891i	27.0000					294.346i
241.4		−	6.81736i	5.19615			−30.4764			6.82126				−	35.4240i		−	21.1733i			98.6907i	27.0000				−	46.5030i
241.5		−	6.74807i	−5.19615			−29.5364			−31.7452					35.0640i			78.9385i			91.3448i	27.0000					214.219i
241.6		−	5.76597i	5.19615			−17.2464			−5.95469				−	29.9609i		−	14.6130i			7.18697i	27.0000					34.3346i
241.7		−	4.72406i	5.19615			−6.31678			17.3409				−	24.5470i			82.0604i		−	45.7442i	27.0000				−	81.9194i
241.8		−	4.63229i	−5.19615			−5.45813			41.2782					24.0701i		−	64.2682i		−	48.8330i	27.0000				−	191.213i
241.9		−	4.16193i	5.19615			−1.32170			32.1440				−	21.6260i			16.6332i		−	61.0901i	27.0000				−	133.781i
241.10		−	3.32762i	−5.19615			4.92693			−8.19957					17.2908i			74.8598i		−	69.6369i	27.0000					27.2851i
241.11		−	3.32484i	−5.19615			4.94542			−43.8015					17.2764i			8.46019i		−	69.6402i	27.0000					145.633i
241.12		−	1.73343i	−5.19615			12.9952			39.5641					9.00717i			28.7171i		−	50.2612i	27.0000				−	68.5817i
241.13		−	0.781598i	5.19615			15.3891			−31.4452				−	4.06130i		−	28.7139i		−	24.5337i	27.0000					24.5775i
241.14		−	0.651818i	5.19615			15.5751			40.5487				−	3.38694i			76.2719i		−	20.5812i	27.0000				−	26.4303i
241.15		−	0.182175i	−5.19615			15.9668			−20.7326					0.946610i			21.5660i		−	5.82356i	27.0000					3.77697i
241.16		−	0.150791i	5.19615			15.9773			−14.5389				−	0.783534i		−	70.1333i		−	4.82189i	27.0000					2.19234i
241.17			0.150791i	5.19615			15.9773			−14.5389					0.783534i			70.1333i			4.82189i	27.0000				−	2.19234i
241.18			0.182175i	−5.19615			15.9668			−20.7326				−	0.946610i		−	21.5660i			5.82356i	27.0000				−	3.77697i
241.19			0.651818i	5.19615			15.5751			40.5487					3.38694i		−	76.2719i			20.5812i	27.0000					26.4303i
241.20			0.781598i	5.19615			15.3891			−31.4452					4.06130i			28.7139i			24.5337i	27.0000				−	24.5775i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	363.5.c.e		32
11.b	odd	2	1	inner	363.5.c.e		32
11.c	even	5	1		33.5.g.a	✓	32
11.d	odd	10	1		33.5.g.a	✓	32
33.f	even	10	1		99.5.k.c		32
33.h	odd	10	1		99.5.k.c		32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
33.5.g.a	✓	32	11.c	even	5	1
33.5.g.a	✓	32	11.d	odd	10	1
99.5.k.c		32	33.f	even	10	1
99.5.k.c		32	33.h	odd	10	1
363.5.c.e		32	1.a	even	1	1	trivial
363.5.c.e		32	11.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^32 + 378*T2^30 + 63441*T2^28 + 6233382*T2^26 + 398097868*T2^24 + 17352244146*T2^22 + 527453950401*T2^20 + 11224733196966*T2^18 + 165378112584441*T2^16 + 1640119944339024*T2^14 + 10392208440043536*T2^12 + 38337264046611840*T2^10 + 69545550675007840*T2^8 + 48250438321827072*T2^6 + 11833369345631232*T2^4 + 553459310449152*T2^2 + 7015378606336