Newform orbit 360.2.bk.a

• Label: 360.2.bk.a
• Level: $360$
• Weight: $2$
• Character orbit: 360.bk
• Analytic conductor: $2.875$
• Analytic rank: $0$
• Dimension: $136$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	360.bk (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.87461447277\)
Analytic rank:	\(0\)
Dimension:	\(136\)
Relative dimension:	\(68\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 136 q - 2 q^{4} - 8 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
229.1	−1.40884	+	0.123126i	0.777229	+	1.54787i	1.96968	−	0.346930i	1.94405	+	1.10484i	−1.28558	−	2.08502i	−2.92425	+	1.68832i	−2.73226	+	0.731288i	−1.79183	+	2.40611i	−2.87489	−	1.31719i
229.2	−1.40262	−	0.180707i	−1.01615	−	1.40266i	1.93469	+	0.506927i	−1.43541	+	1.71453i	1.17180	+	2.15102i	2.42894	−	1.40235i	−2.62203	−	1.06064i	−0.934882	+	2.85061i	2.32316	−	2.14545i
229.3	−1.39716	+	0.218938i	−1.67702	+	0.433134i	1.90413	−	0.611784i	−1.45294	−	1.69970i	2.24824	−	0.972322i	−0.435632	+	0.251512i	−2.52644	+	1.27165i	2.62479	−	1.45275i	2.40212	+	2.05665i
229.4	−1.39375	+	0.239699i	1.69556	+	0.353674i	1.88509	−	0.668161i	0.971012	−	2.01423i	−2.44796	−	0.0865103i	−1.11197	+	0.641993i	−2.46719	+	1.38310i	2.74983	+	1.19935i	−0.870541	+	3.04009i
229.5	−1.38109	+	0.304279i	1.07030	−	1.36178i	1.81483	−	0.840474i	−1.96832	+	1.06099i	−1.06383	+	2.20642i	−1.60442	+	0.926310i	−2.25071	+	1.71299i	−0.708903	−	2.91504i	2.39560	−	2.06425i
229.6	−1.37959	−	0.311008i	1.28537	−	1.16096i	1.80655	+	0.858129i	2.15709	+	0.589033i	−2.13435	+	1.20190i	2.73484	−	1.57896i	−2.22541	−	1.74572i	0.304332	−	2.98452i	−2.79271	−	1.48350i
229.7	−1.35366	−	0.409391i	−0.0938520	−	1.72951i	1.66480	+	1.10835i	0.465822	−	2.18701i	−0.581001	+	2.37959i	−3.18214	+	1.83721i	−1.79982	−	2.18189i	−2.98238	+	0.324635i	−1.52591	+	2.76977i
229.8	−1.34429	−	0.439174i	−0.277804	+	1.70963i	1.61425	+	1.18076i	−2.16653	+	0.553294i	1.12427	−	2.17624i	−0.442721	+	0.255605i	−1.65147	−	2.29622i	−2.84565	−	0.949882i	3.15545	+	0.207696i
229.9	−1.30878	+	0.535802i	−0.883424	+	1.48982i	1.42583	−	1.40250i	1.08655	+	1.95433i	0.357964	−	2.42319i	4.23238	−	2.44357i	−1.11465	+	2.59953i	−1.43912	−	2.63229i	−2.46920	−	1.97562i
229.10	−1.25986	−	0.642465i	1.66988	+	0.459910i	1.17448	+	1.61883i	−1.65905	−	1.49919i	−1.80833	−	1.65226i	1.92958	−	1.11405i	−0.439633	−	2.79405i	2.57697	+	1.53598i	1.12698	+	2.95464i
229.11	−1.25228	+	0.657113i	−1.28051	−	1.16631i	1.13640	−	1.64578i	2.01086	−	0.977977i	2.36996	+	0.619105i	1.13221	−	0.653679i	−0.341633	+	2.80772i	0.279433	+	2.98696i	−1.87552	+	2.54606i
229.12	−1.19676	−	0.753499i	−0.761001	+	1.55592i	0.864479	+	1.80352i	1.70599	−	1.44555i	2.08312	−	1.28865i	−1.65398	+	0.954927i	0.324373	−	2.80977i	−1.84175	−	2.36811i	−3.13088	+	0.444516i
229.13	−1.19522	+	0.755949i	1.28051	+	1.16631i	0.857083	−	1.80704i	−2.01086	+	0.977977i	−2.41216	−	0.425992i	1.13221	−	0.653679i	0.341633	+	2.80772i	0.279433	+	2.98696i	1.66411	−	2.68900i
229.14	−1.11841	+	0.865540i	0.883424	−	1.48982i	0.501683	−	1.93606i	−1.08655	−	1.95433i	0.301467	+	2.43087i	4.23238	−	2.44357i	1.11465	+	2.59953i	−1.43912	−	2.63229i	2.90676	+	1.24528i
229.15	−1.07641	−	0.917248i	1.70505	−	0.304648i	0.317314	+	1.97467i	−0.596890	+	2.15493i	−2.11477	−	1.23603i	−4.00243	+	2.31080i	1.46970	−	2.41661i	2.81438	−	1.03888i	2.61910	−	1.77209i
229.16	−1.04987	−	0.947511i	−1.58768	+	0.692297i	0.204446	+	1.98952i	0.371573	+	2.20498i	2.32281	+	0.777522i	1.08980	−	0.629197i	1.67045	−	2.28245i	2.04145	−	2.19829i	1.69914	−	2.66701i
229.17	−1.00550	−	0.994471i	−1.61972	−	0.613615i	0.0220563	+	1.99988i	−0.521343	−	2.17444i	1.01840	+	2.22775i	2.84021	−	1.63979i	1.96664	−	2.03281i	2.24695	+	1.98776i	−1.63821	+	2.70486i
229.18	−0.954059	+	1.04392i	−1.07030	+	1.36178i	−0.179543	−	1.99192i	1.96832	−	1.06099i	−0.400461	−	2.41653i	−1.60442	+	0.926310i	2.25071	+	1.71299i	−0.708903	−	2.91504i	−0.770302	+	3.06702i
229.19	−0.904461	+	1.08718i	−1.69556	−	0.353674i	−0.363900	−	1.96662i	−0.971012	+	2.01423i	1.91807	−	1.52348i	−1.11197	+	0.641993i	2.46719	+	1.38310i	2.74983	+	1.19935i	−1.31158	−	2.87746i
229.20	−0.888188	+	1.10051i	1.67702	−	0.433134i	−0.422246	−	1.95492i	1.45294	+	1.69970i	−1.01284	+	2.23028i	−0.435632	+	0.251512i	2.52644	+	1.27165i	2.62479	−	1.45275i	−3.16102	+	0.0893258i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
8.b	even	2	1	inner
9.c	even	3	1	inner
40.f	even	2	1	inner
45.j	even	6	1	inner
72.n	even	6	1	inner
360.bk	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	360.2.bk.a	✓	136
5.b	even	2	1	inner	360.2.bk.a	✓	136
8.b	even	2	1	inner	360.2.bk.a	✓	136
9.c	even	3	1	inner	360.2.bk.a	✓	136
40.f	even	2	1	inner	360.2.bk.a	✓	136
45.j	even	6	1	inner	360.2.bk.a	✓	136
72.n	even	6	1	inner	360.2.bk.a	✓	136
360.bk	even	6	1	inner	360.2.bk.a	✓	136

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
360.2.bk.a	✓	136	1.a	even	1	1	trivial
360.2.bk.a	✓	136	5.b	even	2	1	inner
360.2.bk.a	✓	136	8.b	even	2	1	inner
360.2.bk.a	✓	136	9.c	even	3	1	inner
360.2.bk.a	✓	136	40.f	even	2	1	inner
360.2.bk.a	✓	136	45.j	even	6	1	inner
360.2.bk.a	✓	136	72.n	even	6	1	inner
360.2.bk.a	✓	136	360.bk	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(360, [\chi])\).