Newform orbit 241.4.n.a

• Label: 241.4.n.a
• Level: $241$
• Weight: $4$
• Character orbit: 241.n
• Analytic conductor: $14.219$
• Analytic rank: $0$
• Dimension: $480$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 241 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	241.n (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.2194603114\)
Analytic rank:	\(0\)
Dimension:	\(480\)
Relative dimension:	\(60\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 480 q - 4 q^{2} - 13 q^{3} - 970 q^{4} + 22 q^{5} + 8 q^{6} - 85 q^{7} + 48 q^{8} + 575 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
10.1	−2.81425	+	4.87443i	6.27119	−	6.96487i	−11.8400	−	20.5075i	−5.28351	+	16.2610i	16.3010	+	50.1694i	−4.67237	−	10.4943i	88.2554			−6.35922	−	60.5040i	−64.3938	−	71.5166i
10.2	−2.71572	+	4.70376i	−0.216559	+	0.240513i	−10.7503	−	18.6200i	5.32947	−	16.4024i	−0.543204	−	1.67181i	−9.55652	−	21.4643i	73.3272			2.81132	+	26.7479i	62.6798	+	69.6129i
10.3	−2.60584	+	4.51345i	−1.31859	+	1.46444i	−9.58081	−	16.5944i	−1.39752	+	4.30112i	−3.17365	−	9.76749i	4.29782	+	9.65305i	58.1708			2.41636	+	22.9901i	−15.7712	−	17.5157i
10.4	−2.53670	+	4.39369i	−2.92047	+	3.24351i	−8.86970	−	15.3628i	−5.82466	+	17.9264i	−6.84264	−	21.0595i	8.28338	+	18.6048i	49.4119			0.831052	+	7.90693i	−63.9879	−	71.0658i
10.5	−2.51391	+	4.35422i	−6.69699	+	7.43776i	−8.63950	−	14.9641i	3.11889	−	9.59895i	−15.5500	−	47.8580i	5.66447	+	12.7226i	46.6532			−7.64833	−	72.7690i	33.9554	+	37.7112i
10.6	−2.49852	+	4.32757i	3.41641	−	3.79431i	−8.48522	−	14.6968i	2.09757	−	6.45564i	7.88415	+	24.2649i	8.12581	+	18.2509i	44.8257			0.0973523	+	0.926245i	22.6964	+	25.2069i
10.7	−2.48908	+	4.31121i	−4.51240	+	5.01153i	−8.39101	−	14.5337i	−1.60679	+	4.94518i	−10.3740	−	31.9280i	−7.68684	−	17.2649i	43.7183			−1.93139	−	18.3759i	−17.3203	−	19.2361i
10.8	−2.21812	+	3.84189i	3.06458	−	3.40357i	−5.84007	−	10.1153i	−0.612849	+	1.88616i	6.27852	+	19.3233i	−5.37038	−	12.0621i	16.3259			0.629687	+	5.99107i	−5.88703	−	6.53821i
10.9	−2.15579	+	3.73393i	6.21367	−	6.90098i	−5.29485	−	9.17094i	5.85135	−	18.0086i	12.3724	+	38.0785i	−4.16554	−	9.35596i	11.1656			−6.19155	−	58.9087i	54.6287	+	60.6713i
10.10	−2.04306	+	3.53868i	0.591934	−	0.657409i	−4.34817	−	7.53125i	−3.60822	+	11.1050i	1.11700	+	3.43779i	−9.39000	−	21.0903i	2.84530			2.74047	+	26.0738i	−31.9251	−	35.4564i
10.11	−2.02898	+	3.51430i	−1.92591	+	2.13895i	−4.23353	−	7.33268i	4.87800	−	15.0130i	−3.60925	−	11.1081i	9.59090	+	21.5415i	1.89528			1.95633	+	18.6132i	42.8626	+	47.6038i
10.12	−1.86962	+	3.23827i	4.85968	−	5.39722i	−2.99094	−	5.18046i	−0.153089	+	0.471158i	8.39193	+	25.8277i	10.2820	+	23.0938i	−7.54623			−2.69124	−	25.6055i	−1.23952	−	1.37663i
10.13	−1.80684	+	3.12955i	−4.03947	+	4.48628i	−2.52938	−	4.38101i	−0.0589808	+	0.181524i	−6.74134	−	20.7477i	3.00764	+	6.75528i	−10.6288			−0.987171	−	9.39230i	−0.461520	−	0.512569i
10.14	−1.72334	+	2.98491i	−6.33652	+	7.03742i	−1.93978	−	3.35979i	−6.67314	+	20.5378i	−10.0861	−	31.0417i	−6.14425	−	13.8002i	−14.2018			−6.55150	−	62.3333i	−49.8034	−	55.3123i
10.15	−1.72314	+	2.98456i	3.54834	−	3.94084i	−1.93841	−	3.35743i	−6.38215	+	19.6422i	5.64739	+	17.3809i	8.14366	+	18.2910i	−14.2096			−0.117170	−	1.11480i	−47.6261	−	52.8942i
10.16	−1.70445	+	2.95219i	−4.09994	+	4.55344i	−1.81027	−	3.13547i	3.36213	−	10.3476i	−6.45449	−	19.8649i	−9.86917	−	22.1665i	−14.9291			−1.10208	−	10.4856i	24.8174	+	27.5625i
10.17	−1.39465	+	2.41561i	−1.87570	+	2.08318i	0.109896	+	0.190346i	−2.49906	+	7.69132i	−2.41619	−	7.43627i	7.14617	+	16.0506i	−22.9275			2.00089	+	19.0372i	−15.0939	−	16.7635i
10.18	−1.28188	+	2.22029i	3.29366	−	3.65798i	0.713553	+	1.23591i	4.85017	−	14.9273i	3.89968	+	12.0020i	−7.57092	−	17.0046i	−24.1689			0.289647	+	2.75580i	26.9255	+	29.9038i
10.19	−1.20907	+	2.09417i	0.552773	−	0.613916i	1.07631	+	1.86422i	4.52166	−	13.9162i	0.617303	+	1.89986i	−2.32606	−	5.22443i	−24.5504			2.75093	+	26.1734i	23.6759	+	26.2948i
10.20	−1.19307	+	2.06646i	6.85866	−	7.61732i	1.15316	+	1.99733i	−1.11633	+	3.43570i	7.55801	+	23.2612i	−1.97466	−	4.43515i	−24.5924			−8.15999	−	77.6372i	−5.76787	−	6.40587i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
241.n	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	241.4.n.a	✓	480
241.n	even	30	1	inner	241.4.n.a	✓	480

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
241.4.n.a	✓	480	1.a	even	1	1	trivial
241.4.n.a	✓	480	241.n	even	30	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(241, [\chi])\).