Newform orbit 336.2.bu.a

• Label: 336.2.bu.a
• Level: $336$
• Weight: $2$
• Character orbit: 336.bu
• Analytic conductor: $2.683$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	336.bu (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.68297350792\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(60\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 2 q^{3} - 4 q^{4} - 8 q^{6} - 16 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−1.41143	−	0.0886154i	−1.41967	+	0.992245i	1.98429	+	0.250150i	2.16856	+	0.581063i	2.09169	−	1.27469i	2.64341	−	0.111231i	−2.77853	−	0.528909i	1.03090	−	2.81731i	−3.00929	−	1.01230i
11.2	−1.38808	+	0.270599i	0.495329	+	1.65971i	1.85355	−	0.751228i	−1.75704	−	0.470799i	−1.13667	−	2.16979i	2.40884	+	1.09429i	−2.36960	+	1.54434i	−2.50930	+	1.64421i	2.56632	+	0.178054i
11.3	−1.37508	−	0.330401i	1.72844	−	0.111804i	1.78167	+	0.908653i	−0.754514	−	0.202171i	−2.41368	−	0.417338i	−2.17010	+	1.51350i	−2.14971	−	1.83813i	2.97500	−	0.386494i	0.970717	+	0.527293i
11.4	−1.37478	+	0.331648i	−1.48697	−	0.888215i	1.78002	−	0.911883i	−1.88832	−	0.505975i	2.33882	+	0.727948i	0.358137	+	2.62140i	−2.14470	+	1.84397i	1.42215	+	2.64150i	2.76383	+	0.0693439i
11.5	−1.37118	+	0.346233i	1.54917	−	0.774639i	1.76025	−	0.949493i	2.52702	+	0.677113i	−1.85598	+	1.59854i	0.885088	−	2.49331i	−2.08486	+	1.91138i	1.79987	−	2.40010i	−3.69943	−	0.0535028i
11.6	−1.35622	+	0.400842i	0.102444	−	1.72902i	1.67865	−	1.08726i	−0.613008	−	0.164255i	0.554126	+	2.38599i	−2.64440	−	0.0846229i	−1.84080	+	2.14743i	−2.97901	−	0.354255i	0.897213	−	0.0229536i
11.7	−1.33024	−	0.480058i	−0.332434	−	1.69985i	1.53909	+	1.27719i	0.419196	+	0.112323i	−0.373809	+	2.42080i	1.95186	−	1.78613i	−1.43423	−	2.43782i	−2.77897	+	1.13018i	−0.503710	−	0.350655i
11.8	−1.32564	−	0.492619i	−1.70566	+	0.301217i	1.51465	+	1.30607i	−3.93617	−	1.05469i	2.40948	+	0.440934i	−1.10769	−	2.40271i	−1.36449	−	2.47753i	2.81854	−	1.02755i	4.69840	+	3.33718i
11.9	−1.30635	+	0.541709i	1.11887	+	1.32217i	1.41310	−	1.41532i	3.95276	+	1.05914i	−2.17787	−	1.12111i	−1.79142	+	1.94700i	−1.07931	+	2.61440i	−0.496255	+	2.95867i	−5.73744	+	0.757640i
11.10	−1.25435	−	0.653142i	−1.45082	−	0.946102i	1.14681	+	1.63854i	3.01819	+	0.808721i	1.20191	+	2.13434i	−2.25133	+	1.38979i	−0.368304	−	2.80435i	1.20978	+	2.74525i	−3.25767	−	2.98573i
11.11	−1.14092	−	0.835638i	−0.269460	+	1.71096i	0.603419	+	1.90680i	−1.67179	−	0.447955i	1.73718	−	1.72691i	−0.837482	+	2.50971i	0.904939	−	2.67975i	−2.85478	−	0.922073i	1.53306	+	1.90810i
11.12	−1.12647	+	0.855017i	−1.69811	−	0.341188i	0.537892	−	1.92631i	0.483584	+	0.129576i	2.20460	−	1.06758i	0.378986	−	2.61847i	1.04111	+	2.62985i	2.76718	+	1.15875i	−0.655534	+	0.267508i
11.13	−1.11569	+	0.869039i	−0.392190	+	1.68706i	0.489541	−	1.93916i	−1.41326	−	0.378681i	−1.02856	−	2.22307i	−1.15905	−	2.37836i	1.13903	+	2.58894i	−2.69237	−	1.32330i	1.90585	−	0.805684i
11.14	−1.10334	−	0.884670i	1.72780	+	0.121317i	0.434720	+	1.95218i	2.01953	+	0.541132i	−1.79902	−	1.66238i	2.33730	+	1.23976i	1.24739	−	2.53851i	2.97056	+	0.419221i	−1.74951	−	2.38367i
11.15	−1.02071	−	0.978849i	0.0677712	+	1.73072i	0.0837109	+	1.99825i	3.51084	+	0.940727i	1.62494	−	1.83291i	−0.174511	−	2.63999i	1.87054	−	2.12158i	−2.99081	+	0.234586i	−2.66273	−	4.39679i
11.16	−0.950619	+	1.04705i	1.24919	−	1.19980i	−0.192647	−	1.99070i	0.322006	+	0.0862812i	0.0687593	+	2.44852i	1.62394	+	2.08874i	2.26751	+	1.69069i	0.120939	−	2.99756i	−0.396446	+	0.255137i
11.17	−0.934536	+	1.06143i	1.64436	+	0.544135i	−0.253284	−	1.98390i	−3.28878	−	0.881227i	−2.11428	+	1.23686i	−2.48766	+	0.900855i	2.34248	+	1.58518i	2.40783	+	1.78951i	4.00885	−	2.66729i
11.18	−0.863239	−	1.12019i	1.08000	−	1.35410i	−0.509638	+	1.93398i	−1.29824	−	0.347862i	−2.44915	−	0.0408893i	−2.01473	−	1.71490i	2.60636	−	1.09860i	−0.667192	−	2.92487i	0.731019	+	1.75456i
11.19	−0.860221	+	1.12251i	−1.55800	+	0.756720i	−0.520041	−	1.93121i	1.89562	+	0.507929i	0.490803	−	2.39982i	−0.588039	+	2.57958i	2.61514	+	1.07751i	1.85475	−	2.35795i	−2.20080	+	1.69091i
11.20	−0.674285	−	1.24312i	1.47021	+	0.915682i	−1.09068	+	1.67643i	−3.56829	−	0.956121i	0.146958	−	2.44508i	2.17230	−	1.51033i	2.81943	+	0.225451i	1.32305	+	2.69250i	1.21747	+	5.08050i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.c	even	3	1	inner
16.f	odd	4	1	inner
21.h	odd	6	1	inner
48.k	even	4	1	inner
112.u	odd	12	1	inner
336.bu	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	336.2.bu.a	✓	240
3.b	odd	2	1	inner	336.2.bu.a	✓	240
7.c	even	3	1	inner	336.2.bu.a	✓	240
16.f	odd	4	1	inner	336.2.bu.a	✓	240
21.h	odd	6	1	inner	336.2.bu.a	✓	240
48.k	even	4	1	inner	336.2.bu.a	✓	240
112.u	odd	12	1	inner	336.2.bu.a	✓	240
336.bu	even	12	1	inner	336.2.bu.a	✓	240

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
336.2.bu.a	✓	240	1.a	even	1	1	trivial
336.2.bu.a	✓	240	3.b	odd	2	1	inner
336.2.bu.a	✓	240	7.c	even	3	1	inner
336.2.bu.a	✓	240	16.f	odd	4	1	inner
336.2.bu.a	✓	240	21.h	odd	6	1	inner
336.2.bu.a	✓	240	48.k	even	4	1	inner
336.2.bu.a	✓	240	112.u	odd	12	1	inner
336.2.bu.a	✓	240	336.bu	even	12	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(336, [\chi])\).