Newform orbit 49.4.g.a

• Label: 49.4.g.a
• Level: $49$
• Weight: $4$
• Character orbit: 49.g
• Analytic conductor: $2.891$
• Analytic rank: $0$
• Dimension: $156$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	49.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.89109359028\)
Analytic rank:	\(0\)
Dimension:	\(156\)
Relative dimension:	\(13\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 156 q - 13 q^{2} - 7 q^{3} + 35 q^{4} - 7 q^{5} - 70 q^{6} - 42 q^{7} + 16 q^{8} + 198 q^{9}+O(q^{10}) \)

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	−4.61656	−	1.42402i	0.231546	−	0.589970i	12.6749	+	8.64158i	−16.6086	+	2.50334i	−1.90907	+	2.39390i	18.4370	+	1.75414i	−22.1109	−	27.7262i	19.4979	+	18.0915i	80.2392	+	12.0941i
2.2	−4.56991	−	1.40963i	2.47210	−	6.29880i	12.2871	+	8.37722i	16.6312	−	2.50675i	−20.1763	+	25.3002i	−5.68057	−	17.6276i	−20.4882	−	25.6914i	−13.7713	−	12.7779i	−79.5368	−	11.9882i
2.3	−3.74530	−	1.15527i	−3.45495	+	8.80308i	6.08268	+	4.14710i	−0.875096	+	0.131899i	23.1098	−	28.9787i	−6.85447	−	17.2051i	1.55937	+	1.95538i	−45.7650	−	42.4637i	3.42987	+	0.516970i
2.4	−3.15708	−	0.973830i	−0.0268992	+	0.0685381i	2.40889	+	1.64236i	−0.551282	+	0.0830924i	0.151667	−	0.190185i	−15.0526	+	10.7898i	10.4737	+	13.1336i	19.7884	+	18.3610i	1.82136	+	0.274526i
2.5	−1.76903	−	0.545674i	3.43379	−	8.74916i	−3.77820	−	2.57593i	−8.01721	+	1.20840i	−10.8487	+	13.6038i	−0.295568	+	18.5179i	14.5122	+	18.1977i	−44.9645	−	41.7210i	14.8421	+	2.23708i
2.6	−1.42883	−	0.440736i	−1.06746	+	2.71984i	−4.76260	−	3.24708i	14.5837	−	2.19813i	2.72395	−	3.41573i	18.2694	+	3.03821i	12.8321	+	16.0909i	13.5343	+	12.5580i	−21.8064	−	3.28678i
2.7	0.0991526	+	0.0305845i	−0.212463	+	0.541346i	−6.60101	−	4.50050i	−19.4416	+	2.93036i	−0.0376231	+	0.0471778i	0.208913	−	18.5191i	−1.03442	−	1.29712i	19.5445	+	18.1346i	−2.01731	−	0.304061i
2.8	0.724682	+	0.223535i	1.81654	−	4.62846i	−6.13471	−	4.18258i	7.40499	−	1.11612i	2.35103	−	2.94810i	−12.1170	−	14.0064i	−7.29347	−	9.14572i	1.66955	+	1.54912i	5.61575	+	0.846439i
2.9	0.871702	+	0.268884i	−2.65215	+	6.75756i	−5.92235	−	4.03779i	−5.10865	+	0.770006i	−4.12888	+	5.17746i	−7.37832	+	16.9871i	−8.62695	−	10.8178i	−18.8383	−	17.4794i	−4.66026	−	0.702422i
2.10	2.83978	+	0.875955i	2.09482	−	5.33750i	0.687126	+	0.468475i	2.48561	−	0.374645i	10.6242	−	13.3224i	17.0480	+	7.23649i	−13.2822	−	16.6554i	−4.30828	−	3.99750i	7.38675	+	1.11337i
2.11	3.77876	+	1.16559i	−0.769406	+	1.96041i	6.31048	+	4.30242i	13.4588	−	2.02859i	−5.19244	+	6.51112i	−18.2161	+	3.34267i	−0.893508	−	1.12042i	16.5412	+	15.3480i	53.2221	+	8.02194i
2.12	4.09539	+	1.26326i	−2.23891	+	5.70464i	8.56651	+	5.84055i	−5.73194	+	0.863951i	−16.3757	+	20.5344i	16.2381	−	8.90649i	6.32787	+	7.93490i	−7.73783	−	7.17966i	−24.5659	−	3.70272i
2.13	5.05101	+	1.55803i	2.29854	−	5.85659i	16.4753	+	11.2327i	−16.6099	+	2.50354i	20.7347	−	26.0005i	−18.4107	−	2.01144i	39.3507	+	49.3442i	−9.22394	−	8.55857i	−87.7972	−	13.2333i
4.1	−4.51874	−	3.08083i	0.164357	+	0.152501i	8.00480	+	20.3959i	12.0019	−	3.70210i	−0.272857	−	1.19547i	1.40336	+	18.4670i	16.9288	−	74.1698i	−2.01396	−	26.8744i	−65.6390	−	20.2470i
4.2	−3.34475	−	2.28041i	5.73197	+	5.31849i	3.06435	+	7.80784i	−15.4193	+	4.75622i	−7.04365	−	30.8603i	−18.5181	−	0.281065i	0.349197	−	1.52993i	2.55141	+	34.0463i	62.4198	+	19.2540i
4.3	−2.89337	−	1.97267i	−3.27221	−	3.03617i	1.55746	+	3.96835i	−9.83041	+	3.03228i	3.47838	+	15.2398i	18.2590	+	3.09998i	−2.91198	+	12.7582i	−0.528663	−	7.05451i	34.4248	+	10.6186i
4.4	−2.79294	−	1.90420i	1.97416	+	1.83175i	1.25183	+	3.18962i	6.58232	−	2.03038i	−2.02569	−	8.87514i	3.93477	−	18.0974i	−3.44015	+	15.0723i	−1.47573	−	19.6922i	−22.2503	−	6.86330i
4.5	−2.38820	−	1.62825i	−6.73991	−	6.25373i	0.129585	+	0.330178i	10.8842	−	3.35733i	5.91365	+	25.9094i	−18.4252	+	1.87423i	−4.91735	+	21.5443i	4.29963	+	57.3746i	−31.4602	−	9.70418i
4.6	−0.609565	−	0.415594i	5.72670	+	5.31360i	−2.72388	−	6.94033i	7.13618	−	2.20122i	−1.28249	−	5.61896i	12.7167	+	13.4643i	−2.53731	+	11.1167i	2.54302	+	33.9342i	−5.26478	−	1.62397i
4.7	−0.0954438	−	0.0650725i	−1.38664	−	1.28661i	−2.91785	−	7.43457i	−13.2250	+	4.07936i	0.0486229	+	0.213031i	−12.7050	+	13.4753i	−0.410932	+	1.80041i	−1.75032	−	23.3564i	1.52769	+	0.471231i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
49.g	even	21	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	49.4.g.a	✓	156
49.g	even	21	1	inner	49.4.g.a	✓	156
49.g	even	21	1		2401.4.a.g		78
49.h	odd	42	1		2401.4.a.f		78

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
49.4.g.a	✓	156	1.a	even	1	1	trivial
49.4.g.a	✓	156	49.g	even	21	1	inner
2401.4.a.f		78	49.h	odd	42	1
2401.4.a.g		78	49.g	even	21	1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(49, [\chi])\).