Newform orbit 241.2.f.a

• Label: 241.2.f.a
• Level: $241$
• Weight: $2$
• Character orbit: 241.f
• Analytic conductor: $1.924$
• Analytic rank: $0$
• Dimension: $38$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	241.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.92439468871\)
Analytic rank:	\(0\)
Dimension:	\(38\)
Relative dimension:	\(19\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 38 q - 3 q^{2} - 4 q^{3} - 21 q^{4} - 4 q^{5} - 16 q^{6} - 9 q^{7} + 12 q^{8} - 13 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
16.1	−1.36918	+	2.37149i	0.0299508	+	0.0518764i	−2.74930	−	4.76193i	3.05536			−0.164032			−2.36254	−	1.36401i	9.58042			1.49821	−	2.59497i	−4.18334	+	7.24576i
16.2	−1.34788	+	2.33460i	1.49978	+	2.59769i	−2.63358	−	4.56150i	−3.42708			−8.08609			1.17291	+	0.677179i	8.80751			−2.99866	+	5.19382i	4.61931	−	8.00088i
16.3	−1.11240	+	1.92674i	−1.57519	−	2.72832i	−1.47488	−	2.55456i	1.03072			7.00899			0.265097	+	0.153054i	2.11301			−3.46247	+	5.99718i	−1.14657	+	1.98592i
16.4	−1.02210	+	1.77033i	−0.494336	−	0.856215i	−1.08938	−	1.88687i	−2.24767			2.02105			0.219409	+	0.126676i	0.365441			1.01126	−	1.75156i	2.29735	−	3.97913i
16.5	−1.00610	+	1.74262i	0.772601	+	1.33818i	−1.02448	−	1.77445i	1.74196			−3.10926			2.88263	+	1.66429i	0.0985037			0.306174	−	0.530309i	−1.75258	+	3.03557i
16.6	−0.820125	+	1.42050i	0.469732	+	0.813599i	−0.345211	−	0.597922i	−1.97904			−1.54095			−4.17023	−	2.40768i	−2.14804			1.05870	−	1.83373i	1.62306	−	2.81122i
16.7	−0.546951	+	0.947347i	−0.789117	−	1.36679i	0.401690	+	0.695747i	2.76430			1.72643			−0.609429	−	0.351854i	−3.06662			0.254589	−	0.440960i	−1.51194	+	2.61875i
16.8	−0.415659	+	0.719942i	0.269435	+	0.466674i	0.654456	+	1.13355i	1.25259			−0.447971			3.75407	+	2.16741i	−2.75076			1.35481	−	2.34660i	−0.520652	+	0.901795i
16.9	−0.268739	+	0.465470i	−1.29756	−	2.24744i	0.855559	+	1.48187i	−4.19513			1.39482			3.60587	+	2.08185i	−1.99464			−1.86734	+	3.23432i	1.12740	−	1.95271i
16.10	−0.156104	+	0.270380i	1.09341	+	1.89384i	0.951263	+	1.64764i	−2.16188			−0.682743			0.644641	+	0.372184i	−1.21840			−0.891086	+	1.54341i	0.337478	−	0.584530i
16.11	0.0463729	−	0.0803203i	0.665396	+	1.15250i	0.995699	+	1.72460i	4.04199			0.123425			−2.87243	−	1.65840i	0.370186			0.614496	−	1.06434i	0.187439	−	0.324654i
16.12	0.115311	−	0.199724i	−1.29659	−	2.24576i	0.973407	+	1.68599i	−1.11088			−0.598043			−4.30512	−	2.48556i	0.910220			−1.86229	+	3.22558i	−0.128096	+	0.221869i
16.13	0.373224	−	0.646443i	−0.201285	−	0.348635i	0.721407	+	1.24951i	−0.792043			−0.300497			0.107560	+	0.0620996i	2.56988			1.41897	−	2.45773i	−0.295610	+	0.512011i
16.14	0.644290	−	1.11594i	−0.545999	−	0.945698i	0.169781	+	0.294069i	1.23465			−1.40713			1.78210	+	1.02890i	3.01471			0.903770	−	1.56538i	0.795473	−	1.37780i
16.15	0.761723	−	1.31934i	1.36854	+	2.37038i	−0.160443	−	0.277896i	−0.0208296			4.16979			−1.63410	−	0.943449i	2.55804			−2.24580	+	3.88983i	−0.0158664	+	0.0274814i
16.16	1.06785	−	1.84957i	−1.38303	−	2.39547i	−1.28062	−	2.21809i	3.60391			−5.90748			−2.44230	−	1.41007i	−1.19863			−2.32553	+	4.02793i	3.84844	−	6.66569i
16.17	1.12538	−	1.94921i	−1.19791	−	2.07484i	−1.53296	−	2.65516i	−2.00767			−5.39242			2.14765	+	1.23995i	−2.39911			−1.36999	+	2.37289i	−2.25939	+	3.91338i
16.18	1.14104	−	1.97633i	−0.0555218	−	0.0961666i	−1.60393	−	2.77808i	−3.81601			−0.253410			−3.24881	−	1.87570i	−2.75640			1.49383	−	2.58740i	−4.35420	+	7.54170i
16.19	1.29006	−	2.23444i	0.667706	+	1.15650i	−2.32849	−	4.03306i	1.03275			3.44552			0.563034	+	0.325068i	−6.85531			0.608336	−	1.05367i	1.33230	−	2.30761i
226.1	−1.36918	−	2.37149i	0.0299508	−	0.0518764i	−2.74930	+	4.76193i	3.05536			−0.164032			−2.36254	+	1.36401i	9.58042			1.49821	+	2.59497i	−4.18334	−	7.24576i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
241.f	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	241.2.f.a	✓	38
241.f	even	6	1	inner	241.2.f.a	✓	38

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
241.2.f.a	✓	38	1.a	even	1	1	trivial
241.2.f.a	✓	38	241.f	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(241, [\chi])\).