Newform orbit 420.2.bs.a

• Label: 420.2.bs.a
• Level: $420$
• Weight: $2$
• Character orbit: 420.bs
• Analytic conductor: $3.354$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	420.bs (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.35371688489\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(48\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 192 q + 24 q^{8}+O(q^{10}) \)

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	−1.41215	+	0.0763757i	0.965926	−	0.258819i	1.98833	−	0.215708i	−1.76165	−	1.37716i	−1.34426	+	0.439265i	−0.376856	−	2.61877i	−2.79135	+	0.456472i	0.866025	−	0.500000i	2.59290	+	1.81021i
67.2	−1.40730	+	0.139688i	−0.965926	+	0.258819i	1.96097	−	0.393166i	0.0268415	−	2.23591i	1.32319	−	0.499164i	−1.73890	+	1.99405i	−2.70475	+	0.827226i	0.866025	−	0.500000i	0.274556	+	3.15034i
67.3	−1.39146	−	0.252689i	−0.965926	+	0.258819i	1.87230	+	0.703212i	2.08338	+	0.812109i	1.40944	−	0.116056i	1.86913	+	1.87252i	−2.42752	−	1.45160i	0.866025	−	0.500000i	−2.69372	−	1.65646i
67.4	−1.34741	−	0.429520i	0.965926	−	0.258819i	1.63102	+	1.15748i	1.14403	+	1.92125i	−1.41267	−	0.0661494i	−2.62268	+	0.348654i	−1.70050	−	2.26016i	0.866025	−	0.500000i	−0.716263	−	3.08009i
67.5	−1.31927	−	0.509446i	−0.965926	+	0.258819i	1.48093	+	1.34419i	−1.61593	+	1.54556i	1.40617	+	0.150635i	1.02125	−	2.44071i	−1.26895	−	2.52780i	0.866025	−	0.500000i	2.91923	−	1.21578i
67.6	−1.31844	−	0.511590i	0.965926	−	0.258819i	1.47655	+	1.34900i	−0.724137	−	2.11557i	−1.40592	−	0.152922i	0.392782	+	2.61643i	−1.25660	−	2.53396i	0.866025	−	0.500000i	−0.127575	+	3.15970i
67.7	−1.30881	+	0.535750i	−0.965926	+	0.258819i	1.42594	−	1.40238i	−0.646016	+	2.14072i	1.12555	−	0.856238i	−2.64535	+	0.0463591i	−1.11496	+	2.59940i	0.866025	−	0.500000i	−0.301379	−	3.14788i
67.8	−1.18610	−	0.770168i	0.965926	−	0.258819i	0.813683	+	1.82700i	2.21062	−	0.336384i	−1.34502	−	0.436939i	1.93577	−	1.80355i	0.441983	−	2.79368i	0.866025	−	0.500000i	−2.88110	−	1.30356i
67.9	−1.15287	+	0.819073i	0.965926	−	0.258819i	0.658239	−	1.88858i	0.987882	+	2.00601i	−0.901599	+	1.08955i	0.948242	−	2.46999i	0.788016	+	2.71644i	0.866025	−	0.500000i	−2.78198	−	1.50353i
67.10	−1.14586	+	0.828864i	0.965926	−	0.258819i	0.625969	−	1.89952i	−2.19231	+	0.440188i	−0.892285	+	1.09719i	−0.218247	+	2.63673i	0.857171	+	2.69541i	0.866025	−	0.500000i	2.14722	−	2.32152i
67.11	−1.14380	+	0.831703i	−0.965926	+	0.258819i	0.616542	−	1.90260i	1.87095	−	1.22455i	0.889562	−	1.09940i	1.72261	−	2.00814i	0.877197	+	2.68896i	0.866025	−	0.500000i	−1.12152	+	2.95672i
67.12	−1.11863	−	0.865252i	−0.965926	+	0.258819i	0.502677	+	1.93580i	0.661512	−	2.13598i	1.30446	+	0.546246i	−0.641304	−	2.56685i	1.11264	−	2.60039i	0.866025	−	0.500000i	−2.58815	+	1.81700i
67.13	−0.944970	+	1.05216i	0.965926	−	0.258819i	−0.214064	−	1.98851i	0.628815	−	2.14583i	−0.640453	+	1.26088i	−2.64550	+	0.0366227i	2.29451	+	1.65385i	0.866025	−	0.500000i	1.66354	+	2.68936i
67.14	−0.894495	−	1.09539i	−0.965926	+	0.258819i	−0.399756	+	1.95964i	−1.53386	−	1.62704i	1.14752	+	0.826552i	2.29297	+	1.31995i	2.50415	−	1.31500i	0.866025	−	0.500000i	−0.410208	+	3.13556i
67.15	−0.839505	−	1.13808i	0.965926	−	0.258819i	−0.590463	+	1.91085i	−1.54280	+	1.61857i	−1.10546	−	0.882023i	−1.81066	−	1.92912i	2.67040	−	0.932174i	0.866025	−	0.500000i	3.13725	+	0.397037i
67.16	−0.636910	−	1.26267i	0.965926	−	0.258819i	−1.18869	+	1.60842i	−2.21865	+	0.278560i	−0.942011	−	1.05481i	2.37377	+	1.16842i	2.78800	+	0.476514i	0.866025	−	0.500000i	1.76481	+	2.62401i
67.17	−0.587527	+	1.28640i	−0.965926	+	0.258819i	−1.30962	−	1.51158i	−0.609020	+	2.15153i	0.234564	−	1.39463i	2.63246	−	0.264857i	2.71393	−	0.796599i	0.866025	−	0.500000i	−2.40991	−	2.04752i
67.18	−0.552378	+	1.30188i	−0.965926	+	0.258819i	−1.38976	−	1.43825i	−2.22757	−	0.194712i	0.196606	−	1.40048i	−0.358372	+	2.62137i	2.64010	−	1.01483i	0.866025	−	0.500000i	1.48395	−	2.79247i
67.19	−0.433665	−	1.34608i	−0.965926	+	0.258819i	−1.62387	+	1.16750i	1.64640	+	1.51306i	0.767280	+	1.18797i	−2.02130	−	1.70715i	2.27576	+	1.67956i	0.866025	−	0.500000i	1.32272	−	2.87235i
67.20	−0.408960	+	1.35379i	−0.965926	+	0.258819i	−1.66550	−	1.10729i	2.21557	+	0.302064i	0.0446377	−	1.41351i	−2.09035	−	1.62186i	2.18017	−	1.80191i	0.866025	−	0.500000i	−1.31501	+	2.87589i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.c	odd	4	1	inner
7.c	even	3	1	inner
20.e	even	4	1	inner
28.g	odd	6	1	inner
35.l	odd	12	1	inner
140.w	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	420.2.bs.a	✓	192
4.b	odd	2	1	inner	420.2.bs.a	✓	192
5.c	odd	4	1	inner	420.2.bs.a	✓	192
7.c	even	3	1	inner	420.2.bs.a	✓	192
20.e	even	4	1	inner	420.2.bs.a	✓	192
28.g	odd	6	1	inner	420.2.bs.a	✓	192
35.l	odd	12	1	inner	420.2.bs.a	✓	192
140.w	even	12	1	inner	420.2.bs.a	✓	192

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
420.2.bs.a	✓	192	1.a	even	1	1	trivial
420.2.bs.a	✓	192	4.b	odd	2	1	inner
420.2.bs.a	✓	192	5.c	odd	4	1	inner
420.2.bs.a	✓	192	7.c	even	3	1	inner
420.2.bs.a	✓	192	20.e	even	4	1	inner
420.2.bs.a	✓	192	28.g	odd	6	1	inner
420.2.bs.a	✓	192	35.l	odd	12	1	inner
420.2.bs.a	✓	192	140.w	even	12	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(420, [\chi])\).