Newform orbit 196.4.p.a

• Label: 196.4.p.a
• Level: $196$
• Weight: $4$
• Character orbit: 196.p
• Analytic conductor: $11.564$
• Analytic rank: $0$
• Dimension: $984$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	196.p (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.5643743611\)
Analytic rank:	\(0\)
Dimension:	\(984\)
Relative dimension:	\(82\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 984 q - 13 q^{2} - 13 q^{4} - 22 q^{5} - 14 q^{6} - 52 q^{8} + 676 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3.1	−2.82651	−	0.104251i	−0.724468	+	9.66735i	7.97826	+	0.589329i	−6.94687	+	10.1892i	3.05554	−	27.2493i	15.4614	−	10.1954i	−22.4892	−	2.49748i	−66.2344	−	9.98322i	20.6976	−	28.0756i
3.2	−2.82631	+	0.109333i	0.329697	−	4.39950i	7.97609	−	0.618017i	−3.40513	+	4.99441i	−0.450818	+	12.4704i	11.0694	−	14.8482i	−22.4754	+	2.61876i	7.45151	+	1.12313i	9.07791	−	14.4880i
3.3	−2.82047	−	0.212021i	0.472707	−	6.30783i	7.91009	+	1.19600i	10.1978	−	14.9574i	−2.67065	+	17.6908i	−18.5165	+	0.371271i	−22.0566	−	5.05039i	−12.8669	−	1.93937i	−31.9337	+	40.0246i
3.4	−2.81992	−	0.219246i	0.425493	−	5.67781i	7.90386	+	1.23651i	−12.0529	+	17.6783i	−2.44469	+	15.9177i	−13.0992	+	13.0924i	−22.0171	−	5.21976i	−5.35803	−	0.807594i	37.8640	−	47.2088i
3.5	−2.81552	−	0.269932i	−0.111573	+	1.48884i	7.85427	+	1.52000i	0.866870	−	1.27146i	0.716023	−	4.16175i	−17.2840	−	6.65301i	−21.7035	−	6.39970i	24.4942	+	3.69191i	−2.78390	+	3.34583i
3.6	−2.80822	+	0.337458i	−0.356863	+	4.76201i	7.77224	−	1.89532i	3.75839	−	5.51255i	−0.604828	−	13.4932i	8.23084	+	16.5908i	−21.1866	+	7.94528i	4.14905	+	0.625368i	−8.69415	+	16.7488i
3.7	−2.69550	+	0.856917i	−0.0333098	+	0.444488i	6.53139	−	4.61963i	7.11917	−	10.4419i	−0.291103	−	1.22666i	15.0715	−	10.7633i	−13.6467	+	18.0490i	26.5020	+	3.99453i	−10.2418	+	34.2466i
3.8	−2.68159	+	0.899500i	−0.483912	+	6.45735i	6.38180	−	4.82417i	−7.21425	+	10.5814i	−4.51074	−	17.7512i	−15.9336	+	9.44044i	−12.7740	+	18.6769i	−14.7648	−	2.22543i	9.82770	−	34.8641i
3.9	−2.64689	+	0.996971i	0.570652	−	7.61482i	6.01210	−	5.27775i	0.572835	−	0.840195i	6.08129	+	20.7245i	3.91364	+	18.1020i	−10.6516	+	19.9635i	−30.9614	−	4.66668i	−0.678585	+	2.79501i
3.10	−2.58624	−	1.14515i	0.697248	−	9.30412i	5.37725	+	5.92327i	2.80647	−	4.11634i	−12.4579	+	23.2642i	18.1067	−	3.89210i	−7.12381	−	21.4768i	−59.3821	−	8.95041i	−11.9720	+	7.43199i
3.11	−2.56331	−	1.19560i	−0.0591988	+	0.789953i	5.14110	+	6.12937i	−6.48792	+	9.51603i	1.09621	−	1.95412i	17.5229	+	5.99566i	−5.84996	−	21.8581i	26.0779	+	3.93061i	28.0079	−	16.6356i
3.12	−2.53117	−	1.26221i	−0.724461	+	9.66726i	4.81366	+	6.38973i	7.95361	−	11.6658i	14.0358	−	23.5551i	−10.0818	+	15.5357i	−4.11903	−	22.2494i	−66.2326	−	9.98296i	−34.8566	+	19.4890i
3.13	−2.52899	−	1.26658i	0.175251	−	2.33856i	4.79155	+	6.40633i	7.69218	−	11.2824i	−3.40518	+	5.69222i	4.79914	+	17.8877i	−4.00363	−	22.2704i	21.2603	+	3.20447i	−33.7434	+	18.7902i
3.14	−2.52541	−	1.27370i	−0.417652	+	5.57317i	4.75538	+	6.43322i	−3.63461	+	5.33100i	8.15329	−	13.5426i	−7.33529	−	17.0057i	−3.81530	−	22.3034i	−4.18739	−	0.631148i	15.9690	−	8.83355i
3.15	−2.49159	+	1.33865i	−0.573352	+	7.65085i	4.41606	−	6.67071i	8.08712	−	11.8616i	−8.81322	−	19.8303i	−10.6785	−	15.1318i	−2.07330	+	22.5322i	−31.5084	−	4.74912i	−4.27129	+	40.3801i
3.16	−2.33199	+	1.60057i	0.140561	−	1.87565i	2.87637	−	7.46502i	−7.16265	+	10.5057i	2.67432	+	4.59899i	−4.09573	−	18.0617i	5.24060	+	22.0122i	23.2001	+	3.49685i	−0.111807	−	35.9635i
3.17	−2.32179	−	1.61533i	−0.214899	+	2.86763i	2.78144	+	7.50091i	10.8964	−	15.9820i	5.13111	−	6.31091i	4.29755	−	18.0147i	5.65850	−	21.9085i	18.5213	+	2.79164i	−51.1152	+	19.5057i
3.18	−2.26067	+	1.69981i	−0.192224	+	2.56506i	2.22129	−	7.68543i	−9.76041	+	14.3159i	−3.92556	−	6.12550i	14.7343	+	11.2205i	8.04218	+	21.1500i	20.1559	+	3.03801i	−2.26921	−	48.9544i
3.19	−2.20019	+	1.77741i	0.760058	−	10.1423i	1.68165	−	7.82126i	−2.46545	+	3.61615i	16.3547	+	23.6658i	−10.5458	−	15.2245i	10.2016	+	20.1972i	−75.5896	−	11.3933i	−1.00292	−	12.3383i
3.20	−2.14647	+	1.84192i	0.113405	−	1.51329i	1.21467	−	7.90725i	3.68683	−	5.40758i	2.54393	+	3.45711i	−17.2959	+	6.62196i	11.9572	+	19.2100i	24.4213	+	3.68091i	2.04666	+	18.3981i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
49.h	odd	42	1	inner
196.p	even	42	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	196.4.p.a	✓	984
4.b	odd	2	1	inner	196.4.p.a	✓	984
49.h	odd	42	1	inner	196.4.p.a	✓	984
196.p	even	42	1	inner	196.4.p.a	✓	984

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
196.4.p.a	✓	984	1.a	even	1	1	trivial
196.4.p.a	✓	984	4.b	odd	2	1	inner
196.4.p.a	✓	984	49.h	odd	42	1	inner
196.4.p.a	✓	984	196.p	even	42	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(196, [\chi])\).