Newform orbit 229.4.e.a

• Label: 229.4.e.a
• Level: $229$
• Weight: $4$
• Character orbit: 229.e
• Analytic conductor: $13.511$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 229 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	229.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.5114373913\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(56\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q + 3 q^{3} - 412 q^{4} + 12 q^{5} + 21 q^{6} - 57 q^{7} - 513 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
95.1		−	5.59890i	2.13688	+	3.70118i	−23.3477			0.397054	−	0.687718i	20.7225	−	11.9642i	−23.3838	−	13.5007i			85.9303i	4.36752	−	7.56476i	−3.85047	−	2.22307i
95.2		−	5.24809i	1.35071	+	2.33949i	−19.5425			3.56085	−	6.16757i	12.2779	−	7.08863i	26.9437	+	15.5559i			60.5760i	9.85119	−	17.0628i	−32.3680	−	18.6877i
95.3		−	5.16927i	−3.57187	−	6.18665i	−18.7214			−2.27892	+	3.94721i	−31.9805	+	18.4640i	−12.1333	−	7.00516i			55.4219i	−12.0164	+	20.8131i	20.4042	+	11.7804i
95.4		−	4.89891i	−2.60564	−	4.51310i	−15.9994			9.09963	−	15.7610i	−22.1093	+	12.7648i	7.54611	+	4.35675i			39.1882i	−0.0787278	+	0.136360i	−77.2119	−	44.5783i
95.5		−	4.88435i	−0.341672	−	0.591793i	−15.8569			−9.80902	+	16.9897i	−2.89053	+	1.66885i	9.50542	+	5.48796i			38.3757i	13.2665	−	22.9783i	82.9838	+	47.9107i
95.6		−	4.57710i	4.53130	+	7.84844i	−12.9499			9.29894	−	16.1062i	35.9231	−	20.7402i	−9.47962	−	5.47306i			22.6560i	−27.5653	+	47.7446i	−73.7199	−	42.5622i
95.7		−	4.49258i	3.29550	+	5.70797i	−12.1833			−4.76266	+	8.24917i	25.6435	−	14.8053i	2.69506	+	1.55599i			18.7936i	−8.22062	+	14.2385i	37.0601	+	21.3966i
95.8		−	4.45453i	−4.21061	−	7.29299i	−11.8429			−3.05880	+	5.29800i	−32.4869	+	18.7563i	27.0271	+	15.6041i			17.1182i	−21.9585	+	38.0332i	23.6001	+	13.6255i
95.9		−	4.25392i	−0.572421	−	0.991463i	−10.0958			2.83491	−	4.91021i	−4.21760	+	2.43503i	−9.49060	−	5.47940i			8.91539i	12.8447	−	22.2476i	−20.8876	−	12.0595i
95.10		−	3.93710i	4.04600	+	7.00788i	−7.50078			−0.327278	+	0.566862i	27.5907	−	15.9295i	8.49678	+	4.90562i		−	1.96549i	−19.2403	+	33.3251i	2.23179	+	1.28853i
95.11		−	3.88906i	−0.370198	−	0.641201i	−7.12477			8.76886	−	15.1881i	−2.49367	+	1.43972i	−13.2364	−	7.64205i		−	3.40383i	13.2259	−	22.9079i	−59.0674	−	34.1026i
95.12		−	3.71526i	2.77922	+	4.81375i	−5.80315			−8.74246	+	15.1424i	17.8843	−	10.3255i	−30.0974	−	17.3767i		−	8.16188i	−1.94812	+	3.37424i	56.2579	+	32.4805i
95.13		−	3.12602i	−1.86331	−	3.22736i	−1.77198			−0.838868	+	1.45296i	−10.0888	+	5.82475i	15.9054	+	9.18298i		−	19.4689i	6.55612	−	11.3555i	4.54199	+	2.62232i
95.14		−	3.11916i	−5.14483	−	8.91110i	−1.72913			7.94771	−	13.7658i	−27.7951	+	16.0475i	−8.25062	−	4.76350i		−	19.5598i	−39.4385	+	68.3095i	−42.9378	−	24.7901i
95.15		−	3.03910i	−3.43070	−	5.94215i	−1.23613			−4.75713	+	8.23959i	−18.0588	+	10.4262i	−24.7427	−	14.2852i		−	20.5561i	−10.0394	+	17.3888i	25.0410	+	14.4574i
95.16		−	2.71014i	0.766356	+	1.32737i	0.655139			−2.06905	+	3.58371i	3.59735	−	2.07693i	11.5089	+	6.64469i		−	23.4566i	12.3254	−	21.3482i	9.71235	+	5.60743i
95.17		−	2.67545i	−0.911074	−	1.57803i	0.841950			−7.26188	+	12.5779i	−4.22194	+	2.43754i	0.0589830	+	0.0340538i		−	23.6562i	11.8399	−	20.5073i	33.6517	+	19.4288i
95.18		−	2.65057i	−3.09449	−	5.35982i	0.974499			3.56708	−	6.17836i	−14.2066	+	8.20216i	21.8927	+	12.6398i		−	23.7875i	−5.65180	+	9.78920i	−16.3761	−	9.45477i
95.19		−	2.34635i	2.35926	+	4.08636i	2.49466			9.80565	−	16.9839i	9.58802	−	5.53565i	22.4546	+	12.9642i		−	24.6241i	2.36776	−	4.10108i	−39.8501	−	23.0075i
95.20		−	2.09994i	1.89908	+	3.28931i	3.59026			5.51530	−	9.55279i	6.90734	−	3.98796i	−26.8118	−	15.4798i		−	24.3388i	6.28697	−	10.8894i	−20.0603	−	11.5818i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
229.e	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	229.4.e.a	✓	112
229.e	even	6	1	inner	229.4.e.a	✓	112

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
229.4.e.a	✓	112	1.a	even	1	1	trivial
229.4.e.a	✓	112	229.e	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(229, [\chi])\).