Newform orbit 840.2.dj.a

• Label: 840.2.dj.a
• Level: $840$
• Weight: $2$
• Character orbit: 840.dj
• Analytic conductor: $6.707$
• Analytic rank: $0$
• Dimension: $384$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	840.dj (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.70743376979\)
Analytic rank:	\(0\)
Dimension:	\(384\)
Relative dimension:	\(96\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 384 q + 24 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
67.1	−1.41408	−	0.0192981i	−0.258819	−	0.965926i	1.99926	+	0.0545782i	2.22573	−	0.214778i	0.347351	+	1.37089i	−1.50990	+	2.17261i	−2.82606	−	0.115760i	−0.866025	+	0.500000i	−3.15151	+	0.260761i
67.2	−1.41382	+	0.0334109i	0.258819	+	0.965926i	1.99777	−	0.0944738i	−2.23600	−	0.0172139i	−0.398196	−	1.35700i	−0.205532	+	2.63776i	−2.82132	+	0.200316i	−0.866025	+	0.500000i	3.16188	−	0.0503694i
67.3	−1.41188	−	0.0812078i	−0.258819	−	0.965926i	1.98681	+	0.229311i	1.69040	+	1.46374i	0.286981	+	1.38479i	2.19039	−	1.48397i	−2.78652	−	0.485105i	−0.866025	+	0.500000i	−2.26778	−	2.20390i
67.4	−1.39768	−	0.215604i	0.258819	+	0.965926i	1.90703	+	0.602691i	2.18489	−	0.475643i	−0.153490	−	1.40586i	−1.12127	−	2.39641i	−2.53548	−	1.25353i	−0.866025	+	0.500000i	−3.15634	+	0.193727i
67.5	−1.39332	−	0.242198i	−0.258819	−	0.965926i	1.88268	+	0.674918i	−0.367757	+	2.20562i	0.126673	+	1.40853i	−2.12477	+	1.57650i	−2.45971	−	1.39636i	−0.866025	+	0.500000i	1.04660	−	2.98406i
67.6	−1.38791	−	0.271509i	0.258819	+	0.965926i	1.85257	+	0.753658i	−0.498146	−	2.17987i	−0.0969588	−	1.41089i	2.58171	−	0.578616i	−2.36656	−	1.54900i	−0.866025	+	0.500000i	0.0995247	+	3.16071i
67.7	−1.38776	+	0.272231i	−0.258819	−	0.965926i	1.85178	−	0.755586i	−2.03276	−	0.931608i	0.622135	+	1.27002i	−2.39798	−	1.11791i	−2.36414	+	1.55269i	−0.866025	+	0.500000i	3.07460	+	0.739472i
67.8	−1.36225	+	0.379830i	0.258819	+	0.965926i	1.71146	−	1.03485i	0.307776	+	2.21479i	−0.719464	−	1.21753i	2.15031	+	1.54148i	−1.93837	+	2.05979i	−0.866025	+	0.500000i	−1.26051	−	2.90019i
67.9	−1.35272	−	0.412486i	−0.258819	−	0.965926i	1.65971	+	1.11596i	−1.64065	−	1.51930i	−0.0483206	+	1.41339i	1.93754	−	1.80165i	−1.78481	−	2.19419i	−0.866025	+	0.500000i	1.59265	+	2.73193i
67.10	−1.33895	−	0.455217i	0.258819	+	0.965926i	1.58555	+	1.21902i	−0.0786846	+	2.23468i	0.0931615	−	1.41114i	−0.594257	−	2.57815i	−1.56805	−	2.35398i	−0.866025	+	0.500000i	1.12262	−	2.95630i
67.11	−1.31447	+	0.521709i	0.258819	+	0.965926i	1.45564	−	1.37154i	0.864867	−	2.06204i	−0.844141	−	1.13465i	−2.57785	+	0.595553i	−1.19784	+	2.56226i	−0.866025	+	0.500000i	−0.0610528	+	3.16169i
67.12	−1.31097	+	0.530434i	−0.258819	−	0.965926i	1.43728	−	1.39076i	−1.06883	+	1.96408i	0.851663	+	1.12901i	2.59441	+	0.518672i	−1.14652	+	2.58563i	−0.866025	+	0.500000i	0.359390	−	3.14179i
67.13	−1.30781	+	0.538179i	0.258819	+	0.965926i	1.42073	−	1.40767i	−2.22955	−	0.170648i	−0.858327	−	1.12395i	0.923812	−	2.47923i	−1.10046	+	2.60557i	−0.866025	+	0.500000i	3.00766	−	0.976721i
67.14	−1.27612	+	0.609523i	−0.258819	−	0.965926i	1.25696	−	1.55565i	1.98820	+	1.02326i	0.919038	+	1.07488i	−0.742316	−	2.53948i	−0.655835	+	2.75134i	−0.866025	+	0.500000i	−3.16088	−	0.0939436i
67.15	−1.22888	−	0.699897i	0.258819	+	0.965926i	1.02029	+	1.72018i	1.30129	+	1.81842i	0.357992	−	1.36815i	1.49266	+	2.18448i	−0.0498606	−	2.82799i	−0.866025	+	0.500000i	−0.326425	−	3.14539i
67.16	−1.20530	−	0.739767i	0.258819	+	0.965926i	0.905490	+	1.78328i	0.248009	−	2.22227i	0.402606	−	1.35569i	−0.660700	+	2.56193i	0.227825	−	2.81924i	−0.866025	+	0.500000i	−1.94289	+	2.49503i
67.17	−1.19136	+	0.762012i	0.258819	+	0.965926i	0.838675	−	1.81566i	1.67469	−	1.48169i	−1.04439	−	0.953542i	2.57507	+	0.607466i	0.384392	+	2.80219i	−0.866025	+	0.500000i	−0.866099	+	3.04136i
67.18	−1.18076	−	0.778329i	−0.258819	−	0.965926i	0.788408	+	1.83805i	1.09264	−	1.95093i	−0.446204	+	1.34198i	−0.274484	−	2.63147i	0.499681	−	2.78394i	−0.866025	+	0.500000i	−2.80862	+	1.45316i
67.19	−1.14884	−	0.824726i	−0.258819	−	0.965926i	0.639654	+	1.89495i	2.02758	−	0.942830i	−0.499283	+	1.32315i	1.69518	+	2.03134i	0.827958	−	2.70453i	−0.866025	+	0.500000i	−3.10693	−	0.589039i
67.20	−1.11658	−	0.867894i	0.258819	+	0.965926i	0.493520	+	1.93815i	−1.81636	−	1.30416i	0.549328	−	1.30316i	−2.24581	−	1.39869i	1.13105	−	2.59243i	−0.866025	+	0.500000i	0.896246	+	3.03261i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
7.c	even	3	1	inner
8.d	odd	2	1	inner
35.l	odd	12	1	inner
40.k	even	4	1	inner
56.k	odd	6	1	inner
280.br	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	840.2.dj.a	✓	384
5.c	odd	4	1	inner	840.2.dj.a	✓	384
7.c	even	3	1	inner	840.2.dj.a	✓	384
8.d	odd	2	1	inner	840.2.dj.a	✓	384
35.l	odd	12	1	inner	840.2.dj.a	✓	384
40.k	even	4	1	inner	840.2.dj.a	✓	384
56.k	odd	6	1	inner	840.2.dj.a	✓	384
280.br	even	12	1	inner	840.2.dj.a	✓	384

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
840.2.dj.a	✓	384	1.a	even	1	1	trivial
840.2.dj.a	✓	384	5.c	odd	4	1	inner
840.2.dj.a	✓	384	7.c	even	3	1	inner
840.2.dj.a	✓	384	8.d	odd	2	1	inner
840.2.dj.a	✓	384	35.l	odd	12	1	inner
840.2.dj.a	✓	384	40.k	even	4	1	inner
840.2.dj.a	✓	384	56.k	odd	6	1	inner
840.2.dj.a	✓	384	280.br	even	12	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(840, [\chi])\).