Newform orbit 61.4.i.a

• Label: 61.4.i.a
• Level: $61$
• Weight: $4$
• Character orbit: 61.i
• Analytic conductor: $3.599$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 61 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	61.i (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.59911651035\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(15\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 11 q^{2} - 18 q^{3} + 63 q^{4} + 21 q^{5} + 13 q^{6} - 73 q^{7} + 26 q^{8} - 234 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
12.1	−4.88163	−	2.17344i	3.96513	−	2.88084i	13.7534	+	15.2747i	−7.43084	+	1.57947i	−25.6176	+	5.44520i	−3.20985	+	30.5397i	−20.7303	−	63.8014i	−0.920419	+	2.83276i	39.7075	+	8.44010i
12.2	−4.27961	−	1.90541i	−1.26945	+	0.922307i	9.33148	+	10.3637i	7.47081	−	1.58797i	7.19011	−	1.52831i	2.75394	−	26.2020i	−8.60712	−	26.4900i	−7.58261	+	23.3369i	−34.9979	−	7.43904i
12.3	−3.96443	−	1.76508i	−8.12469	+	5.90294i	7.24818	+	8.04992i	−17.2583	+	3.66836i	42.6290	−	9.06106i	−0.834850	+	7.94307i	−3.79807	−	11.6893i	22.8225	−	70.2405i	74.8942	+	15.9193i
12.4	−2.82114	−	1.25605i	−1.61991	+	1.17694i	1.02812	+	1.14184i	−1.38165	+	0.293680i	6.04829	−	1.28560i	0.268729	−	2.55679i	6.16800	+	18.9832i	−7.10452	+	21.8655i	4.26672	+	0.906919i
12.5	−2.76697	−	1.23193i	6.90547	−	5.01711i	0.785402	+	0.872277i	15.6478	−	3.32603i	−25.2879	+	5.37512i	−0.541506	+	5.15209i	6.38907	+	19.6635i	14.1706	−	43.6125i	−47.3943	−	10.0740i
12.6	−1.70447	−	0.758878i	4.55013	−	3.30586i	−3.02373	−	3.35819i	−19.6911	+	4.18547i	−10.2643	+	2.18174i	0.566991	−	5.39456i	7.21784	+	22.2142i	1.43148	−	4.40565i	36.7391	+	7.80913i
12.7	−1.42999	−	0.636672i	−2.01035	+	1.46060i	−3.71353	−	4.12429i	5.62581	−	1.19580i	3.80470	−	0.808715i	−2.32760	+	22.1456i	6.55416	+	20.1716i	−6.43532	+	19.8059i	−8.80617	−	1.87181i
12.8	−0.424293	−	0.188907i	−7.72570	+	5.61305i	−5.20871	−	5.78485i	14.8472	−	3.15587i	4.33831	−	0.922136i	0.743920	−	7.07793i	2.26539	+	6.97216i	19.8367	−	61.0510i	−6.89572	−	1.46573i
12.9	0.887392	+	0.395092i	4.07001	−	2.95704i	−4.72168	−	5.24395i	4.92514	−	1.04687i	4.78000	−	1.01602i	1.46368	−	13.9259i	−4.51950	−	13.9096i	−0.522523	+	1.60816i	4.78415	+	1.01690i
12.10	1.21965	+	0.543021i	−4.02604	+	2.92509i	−4.16038	−	4.62057i	−7.94073	+	1.68786i	−6.49872	+	1.38135i	3.19830	−	30.4298i	−5.86559	−	18.0524i	−0.690615	+	2.12550i	−10.6014	−	2.25340i
12.11	1.82912	+	0.814375i	−3.24927	+	2.36073i	−2.67058	−	2.96598i	−15.0436	+	3.19761i	−7.86582	+	1.67193i	−2.87023	+	27.3085i	−7.41914	−	22.8338i	−3.35877	+	10.3372i	−30.1205	−	6.40231i
12.12	3.04497	+	1.35571i	7.67621	−	5.57709i	2.08083	+	2.31100i	−7.33560	+	1.55923i	30.9347	−	6.57538i	−2.93425	+	27.9175i	−5.03692	−	15.5020i	19.4768	−	59.9434i	−24.4505	−	5.19711i
12.13	3.06474	+	1.36451i	−0.669740	+	0.486595i	2.17770	+	2.41858i	19.4302	−	4.13003i	−2.71654	+	0.577419i	−1.32550	+	12.6113i	−4.91955	−	15.1408i	−8.13168	+	25.0267i	65.1841	+	13.8553i
12.14	4.36442	+	1.94317i	−6.75730	+	4.90947i	9.91924	+	11.0164i	−1.46328	+	0.311029i	−39.0316	+	8.29643i	−0.247302	+	2.35292i	10.0745	+	31.0060i	13.2148	−	40.6710i	−6.99073	−	1.48593i
12.15	4.36603	+	1.94388i	2.12239	−	1.54201i	9.93052	+	11.0290i	−4.00953	+	0.852252i	12.2639	−	2.60677i	1.37608	−	13.0926i	10.1031	+	31.0941i	−6.21670	+	19.1330i	−19.1624	−	4.07310i
15.1	−0.571761	−	5.43994i	−4.32136	+	3.13965i	−21.4408	+	4.55739i	0.634470	+	0.704650i	19.5503	+	21.7128i	13.6052	−	6.05745i	23.5286	+	72.4136i	0.473277	−	1.45660i	3.47049	−	3.85437i
15.2	−0.469431	−	4.46633i	5.46760	−	3.97244i	−11.9026	+	2.52998i	−2.55448	−	2.83704i	−20.3089	−	22.5553i	1.30620	−	0.581560i	5.78496	+	17.8043i	5.77088	−	17.7609i	−11.4720	+	12.7410i
15.3	−0.366610	−	3.48806i	−2.44817	+	1.77870i	−4.20701	+	0.894228i	−7.30653	−	8.11473i	7.10174	+	7.88729i	−20.8609	+	9.28786i	−4.00901	−	12.3385i	−5.51369	+	16.9694i	−25.6261	+	28.4606i
15.4	−0.324338	−	3.08587i	−3.77561	+	2.74314i	−1.59224	+	0.338440i	14.3363	+	15.9221i	9.68957	+	10.7614i	−2.46286	+	1.09654i	−6.10991	−	18.8044i	−1.61304	+	4.96443i	44.4838	−	49.4042i
15.5	−0.262789	−	2.50027i	2.74143	−	1.99177i	1.64287	−	0.349202i	4.36835	+	4.85154i	−5.70038	−	6.33091i	22.3858	−	9.96681i	−7.51989	−	23.1439i	−4.79515	+	14.7580i	10.9822	−	12.1970i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
61.i	even	15	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	61.4.i.a	✓	120
61.i	even	15	1	inner	61.4.i.a	✓	120

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
61.4.i.a	✓	120	1.a	even	1	1	trivial
61.4.i.a	✓	120	61.i	even	15	1	inner

Hecke kernels

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