Newform orbit 47.4.c.a

• Label: 47.4.c.a
• Level: $47$
• Weight: $4$
• Character orbit: 47.c
• Analytic conductor: $2.773$
• Analytic rank: $0$
• Dimension: $242$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 47 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	47.c (of order \(23\), degree \(22\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.77308977027\)
Analytic rank:	\(0\)
Dimension:	\(242\)
Relative dimension:	\(11\) over \(\Q(\zeta_{23})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{23}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 242 q - 21 q^{2} - 25 q^{3} - 73 q^{4} - 31 q^{5} - 31 q^{6} - 17 q^{7} - 41 q^{8} - 74 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−5.15418	−	0.708426i	0.181025	−	0.110084i	18.3604	+	5.14433i	−5.71869	−	16.0909i	−1.01102	+	0.439148i	−1.05976	+	15.4931i	−52.8129	−	22.9399i	−12.4011	+	23.9330i	18.0760	+	86.9864i
2.2	−4.04801	−	0.556386i	6.68482	−	4.06513i	8.37345	+	2.34613i	2.74889	+	7.73463i	−29.3220	+	12.7363i	1.40277	−	20.5078i	−2.60816	−	1.13288i	15.7398	−	30.3764i	−6.82408	−	32.8393i
2.3	−3.56582	−	0.490110i	−7.15334	+	4.35005i	4.77151	+	1.33691i	0.237965	+	0.669570i	27.6395	−	12.0055i	0.815463	−	11.9216i	10.0518	+	4.36610i	19.8257	−	38.2618i	−0.520377	−	2.50419i
2.4	−2.73518	−	0.375942i	0.530623	−	0.322679i	−0.363459	−	0.101836i	1.89686	+	5.33724i	−1.57266	+	0.683102i	−0.928647	+	13.5763i	21.2144	+	9.21473i	−12.2443	+	23.6305i	−3.18175	−	15.3114i
2.5	−0.478837	−	0.0658147i	6.24061	−	3.79500i	−7.47838	−	2.09535i	−6.41556	−	18.0516i	−3.23800	+	1.40646i	−1.36094	+	19.8962i	6.98962	+	3.03602i	12.1214	−	23.3933i	1.88395	+	9.06604i
2.6	−0.0676048	−	0.00929206i	−1.40242	+	0.852830i	−7.69885	−	2.15712i	−3.09419	−	8.70621i	0.102735	−	0.0446240i	2.20846	−	32.2865i	1.00116	+	0.434866i	−11.1823	+	21.5808i	0.128283	+	0.617333i
2.7	0.676553	+	0.0929900i	−2.77999	+	1.69055i	−7.25426	−	2.03255i	5.82278	+	16.3837i	−2.03801	+	0.885234i	−1.13667	+	16.6176i	−9.72989	−	4.22629i	−7.55137	+	14.5735i	2.41589	+	11.6259i
2.8	1.97477	+	0.271426i	−7.66193	+	4.65932i	−3.87728	−	1.08636i	−4.31445	−	12.1397i	−16.3952	+	7.12145i	−1.77648	+	25.9712i	−21.9884	−	9.55091i	24.5741	−	47.4258i	−5.22503	−	25.1442i
2.9	2.35651	+	0.323895i	6.31823	−	3.84220i	−2.25510	−	0.631849i	3.52793	+	9.92665i	16.1335	−	7.00775i	0.548900	−	8.02463i	−22.5634	−	9.80068i	12.7358	−	24.5790i	5.09842	+	24.5350i
2.10	4.32143	+	0.593966i	1.23824	−	0.752993i	10.6186	+	2.97519i	−2.25235	−	6.33750i	5.79823	−	2.51853i	−0.499337	+	7.30004i	12.1129	+	5.26136i	−11.4555	+	22.1081i	−5.96910	−	28.7249i
2.11	4.75744	+	0.653896i	−6.79067	+	4.12950i	14.5024	+	4.06337i	5.04422	+	14.1931i	−35.0065	+	15.2055i	1.96324	−	28.7016i	31.1003	+	13.5088i	16.6387	−	32.1112i	14.7168	+	70.8211i
3.1	−4.16509	+	2.53285i	0.249937	−	0.354081i	7.25214	−	13.9960i	−1.38842	−	6.68146i	−0.144180	+	2.10783i	14.6643	−	4.10874i	2.58258	+	37.7560i	8.97885	+	25.2640i	22.7060	+	24.3122i
3.2	−3.41936	+	2.07936i	−5.32799	+	7.54804i	3.68775	−	7.11704i	3.01417	+	14.5050i	2.52322	−	36.8882i	−5.29584	+	1.48382i	0.00429289	+	0.0627599i	−19.5437	−	54.9907i	−40.4676	−	43.3302i
3.3	−2.95041	+	1.79419i	1.26856	−	1.79714i	1.80532	−	3.48411i	1.62096	+	7.80048i	−0.518372	+	7.57833i	−25.4080	+	7.11897i	−0.960495	−	14.0419i	7.42129	+	20.8815i	−18.7780	−	20.1063i
3.4	−1.69708	+	1.03202i	5.10737	−	7.23550i	−1.86550	+	3.60025i	−2.89852	−	13.9485i	−1.20046	+	17.5501i	−9.59591	+	2.68865i	−1.63398	−	23.8880i	−17.2254	−	48.4677i	19.3141	+	20.6803i
3.5	−1.49744	+	0.910611i	−3.69312	+	5.23196i	−2.26742	+	4.37592i	−4.09988	−	19.7297i	0.765931	−	11.1975i	6.80260	−	1.90600i	−1.54625	−	22.6053i	−4.69252	−	13.2035i	24.1054	+	25.8105i
3.6	−0.801524	+	0.487418i	2.38752	−	3.38235i	−3.27565	+	6.32173i	2.73456	+	13.1594i	−0.265040	+	3.87475i	31.1035	−	8.71480i	−0.967946	−	14.1509i	3.30174	+	9.29020i	−8.60594	−	9.21472i
3.7	0.618674	−	0.376224i	−1.88025	+	2.66370i	−3.43931	+	6.63756i	0.575289	+	2.76844i	−0.161111	+	2.35536i	−16.3472	+	4.58028i	0.764708	+	11.1796i	5.48176	+	15.4242i	1.39747	+	1.49633i
3.8	2.54587	−	1.54818i	2.03823	−	2.88752i	0.404069	−	0.779817i	−2.61060	−	12.5629i	0.718683	−	10.5068i	12.2925	−	3.44421i	1.44812	+	21.1707i	4.85839	+	13.6702i	−26.0959	−	27.9418i
3.9	2.71721	−	1.65237i	−5.07813	+	7.19407i	0.972388	−	1.87662i	1.83338	+	8.82273i	−1.91107	+	27.9388i	26.4185	−	7.40213i	1.27749	+	18.6762i	−16.9255	−	47.6238i	19.5602	+	20.9438i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
47.c	even	23	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	47.4.c.a	✓	242
47.c	even	23	1	inner	47.4.c.a	✓	242
47.c	even	23	1		2209.4.a.o		121
47.d	odd	46	1		2209.4.a.n		121

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
47.4.c.a	✓	242	1.a	even	1	1	trivial
47.4.c.a	✓	242	47.c	even	23	1	inner
2209.4.a.n		121	47.d	odd	46	1
2209.4.a.o		121	47.c	even	23	1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(47, [\chi])\).