Newform orbit 61.4.e.a

• Label: 61.4.e.a
• Level: $61$
• Weight: $4$
• Character orbit: 61.e
• Analytic conductor: $3.599$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 61 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	61.e (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.59911651035\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(15\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q - q^{2} - 3 q^{3} - 75 q^{4} - 33 q^{5} - 13 q^{6} - 29 q^{7} + 37 q^{8} - 180 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
9.1	−4.17097	−	3.03039i	−4.37650	−	3.17971i	5.74160	+	17.6708i	1.95438	+	6.01496i	8.61848	+	26.5250i	−12.1707	+	8.84249i	16.8561	−	51.8776i	0.699712	+	2.15349i	10.0760	−	31.0108i
9.2	−3.85735	−	2.80253i	2.47493	+	1.79815i	4.55283	+	14.0122i	−2.57166	−	7.91476i	−4.50733	−	13.8721i	17.1119	−	12.4325i	9.92064	−	30.5326i	−5.45148	−	16.7779i	−12.2615	+	37.7371i
9.3	−3.14619	−	2.28584i	6.28560	+	4.56676i	2.20132	+	6.77497i	2.08047	+	6.40302i	−9.33683	−	28.7358i	−25.8397	+	18.7736i	−1.05318	+	3.24135i	10.3100	+	31.7311i	8.09075	−	24.9008i
9.4	−2.21813	−	1.61157i	−6.46050	−	4.69383i	−0.149177	−	0.459121i	−2.44308	−	7.51901i	6.76582	+	20.8230i	6.41738	−	4.66250i	−7.18701	+	22.1194i	11.3626	+	34.9704i	−6.69833	+	20.6153i
9.5	−1.98576	−	1.44274i	0.715865	+	0.520107i	−0.610382	−	1.87856i	6.02539	+	18.5442i	−0.671160	−	2.06562i	21.4251	−	15.5663i	−7.56616	+	23.2862i	−8.10151	−	24.9339i	14.7896	−	45.5176i
9.6	−1.55944	−	1.13300i	−0.272682	−	0.198115i	−1.32396	−	4.07474i	−0.463403	−	1.42621i	0.200768	+	0.617900i	−12.7404	+	9.25643i	−7.31728	+	22.5203i	−8.30835	−	25.5705i	−0.893247	+	2.74913i
9.7	−1.10401	−	0.802111i	7.37102	+	5.35536i	−1.89668	−	5.83737i	−5.12662	−	15.7781i	−3.84209	−	11.8248i	18.0847	−	13.1393i	−5.96183	+	18.3486i	17.3086	+	53.2703i	−6.99596	+	21.5313i
9.8	0.540539	+	0.392725i	0.203596	+	0.147921i	−2.33419	−	7.18389i	−3.57937	−	11.0162i	0.0519594	+	0.159915i	−8.91276	+	6.47550i	3.21131	−	9.88340i	−8.32389	−	25.6183i	2.39153	−	7.36038i
9.9	0.607304	+	0.441232i	−6.50109	−	4.72332i	−2.29800	−	7.07253i	6.67776	+	20.5520i	−1.86406	−	5.73698i	−25.6752	+	18.6541i	3.58080	−	11.0206i	11.6110	+	35.7348i	−5.01279	+	15.4278i
9.10	1.34866	+	0.979858i	6.08999	+	4.42464i	−1.61338	−	4.96546i	3.29047	+	10.1270i	3.87780	+	11.9347i	1.35909	−	0.987437i	6.81069	−	20.9611i	9.16710	+	28.2134i	−5.48532	+	16.8821i
9.11	1.66802	+	1.21189i	−3.68961	−	2.68066i	−1.15851	−	3.56554i	0.331043	+	1.01885i	−2.90569	−	8.94278i	23.8197	−	17.3060i	7.48563	−	23.0384i	−1.91618	−	5.89740i	−0.682541	+	2.10065i
9.12	3.11983	+	2.26669i	2.49714	+	1.81428i	2.12334	+	6.53496i	1.76623	+	5.43588i	3.67826	+	11.3205i	−10.4534	+	7.59487i	1.34508	−	4.13973i	−5.39935	−	16.6175i	−6.81114	+	20.9625i
9.13	3.12537	+	2.27072i	−7.35424	−	5.34317i	2.13966	+	6.58520i	−5.87041	−	18.0673i	−10.8519	−	33.3988i	−13.4987	+	9.80739i	1.28440	−	3.95299i	17.1919	+	52.9113i	22.6784	−	69.7969i
9.14	3.78376	+	2.74906i	4.79945	+	3.48700i	4.28735	+	13.1951i	−5.24672	−	16.1477i	8.57396	+	26.3879i	2.15166	−	1.56327i	−8.48974	+	26.1287i	2.53204	+	7.79282i	24.5388	−	75.5227i
9.15	4.15739	+	3.02052i	−3.09200	−	2.24647i	5.68821	+	17.5065i	3.31078	+	10.1895i	−6.06914	−	18.6789i	7.75817	−	5.63664i	−16.5268	+	50.8643i	−3.82962	−	11.7864i	−17.0135	+	52.3621i
20.1	−1.72824	−	5.31898i	1.72601	+	5.31210i	−18.8326	+	13.6827i	2.90032	−	2.10721i	25.2720	−	18.3612i	−10.5273	+	32.3996i	69.1282	+	50.2246i	−3.39586	+	2.46724i	−16.2206	−	11.7850i
20.2	−1.60784	−	4.94844i	−2.62270	−	8.07183i	−15.4297	+	11.2104i	6.03391	−	4.38389i	−35.7260	+	25.9565i	8.09382	−	24.9102i	46.6072	+	33.8621i	−36.4324	+	26.4697i	−31.3950	−	22.8098i
20.3	−1.19167	−	3.66759i	−0.892419	−	2.74658i	−5.55900	+	4.03885i	−14.3907	+	10.4555i	−9.00987	+	6.54606i	−2.38782	+	7.34897i	−3.52136	−	2.55842i	15.0961	−	10.9680i	55.4953	+	40.3197i
20.4	−1.12586	−	3.46504i	1.20243	+	3.70069i	−4.26680	+	3.10001i	4.45515	−	3.23686i	11.4693	−	8.33291i	7.10204	−	21.8578i	−8.03484	−	5.83765i	9.59420	−	6.97059i	−16.2317	−	11.7930i
20.5	−0.776790	−	2.39071i	−1.36398	−	4.19789i	1.36003	−	0.988118i	14.5498	−	10.5711i	−8.97643	+	6.52176i	−6.69639	+	20.6094i	−19.6881	−	14.3042i	6.08161	−	4.41854i	−36.5745	−	26.5729i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
61.e	even	5	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	61.4.e.a	✓	60
61.e	even	5	1	inner	61.4.e.a	✓	60

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
61.4.e.a	✓	60	1.a	even	1	1	trivial
61.4.e.a	✓	60	61.e	even	5	1	inner

Hecke kernels

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