Newform orbit 61.4.k.a

• Label: 61.4.k.a
• Level: $61$
• Weight: $4$
• Character orbit: 61.k
• Analytic conductor: $3.599$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 61 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	61.k (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.59911651035\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(14\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q - 7 q^{2} - 18 q^{3} - 45 q^{4} + 47 q^{5} + 35 q^{6} - 3 q^{7} - 10 q^{8} - 228 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
4.1	−5.49981	−	0.578053i	−5.78595	−	4.20374i	22.0886	+	4.69507i	10.9067	−	12.1131i	29.3916	+	26.4644i	6.03710	−	13.5595i	−76.6934	−	24.9192i	7.46235	+	22.9667i	−66.9870	+	60.3153i
4.2	−4.79343	−	0.503810i	4.89052	+	3.55317i	14.8980	+	3.16667i	−2.91985	+	3.24283i	−21.6523	−	19.4958i	2.47787	−	5.56539i	−33.1457	−	10.7697i	2.94870	+	9.07515i	15.6299	−	14.0732i
4.3	−3.78314	−	0.397624i	0.887830	+	0.645046i	6.32889	+	1.34525i	0.290139	−	0.322232i	−3.10230	−	2.79333i	−3.39125	+	7.61688i	5.53423	+	1.79818i	−7.97130	−	24.5331i	−1.22576	+	1.10368i
4.4	−3.25787	−	0.342416i	−5.59605	−	4.06577i	2.67129	+	0.567800i	−13.3045	+	14.7762i	16.8390	+	15.1619i	10.6935	−	24.0181i	16.4156	+	5.33375i	6.44184	+	19.8259i	48.4040	−	43.5832i
4.5	−2.19088	−	0.230271i	−5.16266	−	3.75089i	−3.07825	−	0.654302i	5.02306	−	5.57868i	10.4470	+	9.40656i	−7.93585	+	17.8242i	23.3544	+	7.58832i	4.24040	+	13.0506i	−12.2895	+	11.0655i
4.6	−1.87824	−	0.197411i	7.36912	+	5.35398i	−4.33638	−	0.921727i	14.7512	−	16.3829i	−12.7840	−	11.5108i	−3.47555	+	7.80621i	22.3320	+	7.25610i	17.2954	+	53.2297i	−30.9404	+	27.8588i
4.7	−0.519615	−	0.0546138i	0.831355	+	0.604015i	−7.55816	−	1.60654i	2.25104	−	2.50003i	−0.398997	−	0.359259i	13.8673	−	31.1464i	7.81484	+	2.53920i	−8.01714	−	24.6742i	−1.30621	+	1.17612i
4.8	−0.396928	−	0.0417189i	3.94287	+	2.86467i	−7.66937	−	1.63017i	−9.53996	+	10.5952i	−1.44553	−	1.30156i	−7.35973	+	16.5302i	6.01283	+	1.95369i	−1.00352	−	3.08850i	4.22870	−	3.80754i
4.9	1.55742	+	0.163691i	−2.62329	−	1.90593i	−5.42643	−	1.15342i	9.78939	−	10.8722i	−3.77357	−	3.39774i	1.15495	−	2.59407i	−20.1772	−	6.55597i	−5.09438	−	15.6789i	17.0258	−	15.3301i
4.10	2.01915	+	0.212221i	−4.91283	−	3.56938i	−3.79325	−	0.806279i	−7.80952	+	8.67335i	−9.16224	−	8.24972i	−4.04083	+	9.07584i	−22.9353	−	7.45212i	3.05196	+	9.39296i	−17.6093	+	15.8555i
4.11	3.01786	+	0.317190i	6.26326	+	4.55052i	1.18169	+	0.251175i	−0.718490	+	0.797964i	17.4583	+	15.7195i	3.67803	−	8.26100i	−19.6012	−	6.36882i	10.1777	+	31.3237i	−2.42141	+	2.18024i
4.12	4.11322	+	0.432317i	1.12099	+	0.814445i	8.90648	+	1.89313i	8.62540	−	9.57947i	4.25877	+	3.83461i	−8.12179	+	18.2418i	4.34827	+	1.41284i	−7.75017	−	23.8526i	39.6195	−	35.6736i
4.13	4.62186	+	0.485778i	−7.73639	−	5.62081i	13.3005	+	2.82710i	5.92830	−	6.58405i	−33.0261	−	29.7368i	10.6854	−	23.9999i	24.7407	+	8.03874i	19.9147	+	61.2910i	30.5982	−	27.5507i
4.14	5.01226	+	0.526810i	0.348106	+	0.252914i	17.0200	+	3.61772i	−7.75442	+	8.61215i	1.61156	+	1.45106i	1.00402	−	2.25506i	45.0574	+	14.6400i	−8.28625	−	25.5024i	−43.4041	+	39.0812i
5.1	−2.10489	−	4.72767i	−4.50139	−	3.27045i	−12.5672	+	13.9573i	1.39106	+	0.295679i	−5.98666	+	28.1650i	−3.63545	+	0.382101i	53.0641	+	17.2416i	1.22320	+	3.76462i	−1.53016	−	7.19884i
5.2	−1.98893	−	4.46722i	6.27202	+	4.55689i	−10.6471	+	11.8248i	3.45081	+	0.733493i	7.88198	−	37.0818i	27.9973	−	2.94264i	36.7953	+	11.9555i	10.2295	+	31.4833i	−3.58676	−	16.8744i
5.3	−1.70078	−	3.82002i	1.58657	+	1.15271i	−6.34683	+	7.04887i	−6.60759	−	1.40449i	1.70497	−	8.02124i	−28.6128	+	3.00733i	5.90639	+	1.91910i	−7.15499	−	22.0208i	5.87290	+	27.6298i
5.4	−1.07568	−	2.41602i	−0.617406	−	0.448571i	0.672970	−	0.747408i	18.1532	+	3.85859i	−0.419626	+	1.97419i	5.36977	−	0.564385i	−22.6515	−	7.35991i	−8.16349	−	25.1246i	−10.2047	−	48.0093i
5.5	−1.05438	−	2.36818i	1.79208	+	1.30202i	0.856475	−	0.951211i	−20.2175	−	4.29735i	1.19389	−	5.61680i	13.6783	−	1.43765i	−22.8791	−	7.43387i	−6.82717	−	21.0119i	11.1400	+	52.4097i
5.6	−0.910769	−	2.04562i	−7.25057	−	5.26785i	1.99798	−	2.21898i	−3.25685	−	0.692265i	−4.17242	+	19.6297i	5.37991	−	0.565452i	−23.3958	−	7.60176i	16.4771	+	50.7112i	1.55013	+	7.29277i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
61.k	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	61.4.k.a	✓	112
61.k	even	30	1	inner	61.4.k.a	✓	112

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
61.4.k.a	✓	112	1.a	even	1	1	trivial
61.4.k.a	✓	112	61.k	even	30	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(61, [\chi])\).