Newform orbit 420.2.n.a

• Label: 420.2.n.a
• Level: $420$
• Weight: $2$
• Character orbit: 420.n
• Analytic conductor: $3.354$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	420.n (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.35371688489\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{4} + 6 q^{6} + 4 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
71.1	−1.39371	−	0.239964i	−1.52556	−	0.820170i	1.88483	+	0.668880i		−	1.00000i	1.92937	+	1.50916i			1.00000i	−2.46640	−	1.38452i	1.65464	+	2.50243i	−0.239964	+	1.39371i
71.2	−1.39371	+	0.239964i	−1.52556	+	0.820170i	1.88483	−	0.668880i			1.00000i	1.92937	−	1.50916i		−	1.00000i	−2.46640	+	1.38452i	1.65464	−	2.50243i	−0.239964	−	1.39371i
71.3	−1.24051	−	0.679065i	−1.09154	−	1.34482i	1.07774	+	1.68478i			1.00000i	0.440848	+	2.40949i		−	1.00000i	−0.192876	−	2.82184i	−0.617080	+	2.93585i	0.679065	−	1.24051i
71.4	−1.24051	+	0.679065i	−1.09154	+	1.34482i	1.07774	−	1.68478i		−	1.00000i	0.440848	−	2.40949i			1.00000i	−0.192876	+	2.82184i	−0.617080	−	2.93585i	0.679065	+	1.24051i
71.5	−1.19442	−	0.757207i	0.667879	−	1.59810i	0.853276	+	1.80884i		−	1.00000i	−2.00782	+	1.40309i			1.00000i	0.350500	−	2.80663i	−2.10788	−	2.13468i	−0.757207	+	1.19442i
71.6	−1.19442	+	0.757207i	0.667879	+	1.59810i	0.853276	−	1.80884i			1.00000i	−2.00782	−	1.40309i		−	1.00000i	0.350500	+	2.80663i	−2.10788	+	2.13468i	−0.757207	−	1.19442i
71.7	−1.09123	−	0.899566i	0.826488	+	1.52214i	0.381563	+	1.96327i		−	1.00000i	0.467378	−	2.40449i			1.00000i	1.34971	−	2.48561i	−1.63383	+	2.51607i	−0.899566	+	1.09123i
71.8	−1.09123	+	0.899566i	0.826488	−	1.52214i	0.381563	−	1.96327i			1.00000i	0.467378	+	2.40449i		−	1.00000i	1.34971	+	2.48561i	−1.63383	−	2.51607i	−0.899566	−	1.09123i
71.9	−0.275453	−	1.38713i	−1.30386	+	1.14015i	−1.84825	+	0.764177i		−	1.00000i	1.94069	+	1.49456i			1.00000i	1.56912	+	2.35327i	0.400103	−	2.97320i	−1.38713	+	0.275453i
71.10	−0.275453	+	1.38713i	−1.30386	−	1.14015i	−1.84825	−	0.764177i			1.00000i	1.94069	−	1.49456i		−	1.00000i	1.56912	−	2.35327i	0.400103	+	2.97320i	−1.38713	−	0.275453i
71.11	−0.269719	−	1.38825i	0.0343495	−	1.73171i	−1.85450	+	0.748876i			1.00000i	−2.41332	+	0.419388i		−	1.00000i	1.53983	+	2.37254i	−2.99764	−	0.118967i	1.38825	−	0.269719i
71.12	−0.269719	+	1.38825i	0.0343495	+	1.73171i	−1.85450	−	0.748876i		−	1.00000i	−2.41332	−	0.419388i			1.00000i	1.53983	−	2.37254i	−2.99764	+	0.118967i	1.38825	+	0.269719i
71.13	0.0797590	−	1.41196i	1.52443	+	0.822258i	−1.98728	−	0.225233i			1.00000i	1.28258	−	2.08686i		−	1.00000i	−0.476524	+	2.78800i	1.64778	+	2.50695i	1.41196	+	0.0797590i
71.14	0.0797590	+	1.41196i	1.52443	−	0.822258i	−1.98728	+	0.225233i		−	1.00000i	1.28258	+	2.08686i			1.00000i	−0.476524	−	2.78800i	1.64778	−	2.50695i	1.41196	−	0.0797590i
71.15	0.637711	−	1.26227i	−1.30156	+	1.14278i	−1.18665	−	1.60993i			1.00000i	0.612484	+	2.37168i		−	1.00000i	−2.78890	+	0.471201i	0.388091	−	2.97479i	1.26227	+	0.637711i
71.16	0.637711	+	1.26227i	−1.30156	−	1.14278i	−1.18665	+	1.60993i		−	1.00000i	0.612484	−	2.37168i			1.00000i	−2.78890	−	0.471201i	0.388091	+	2.97479i	1.26227	−	0.637711i
71.17	0.859821	−	1.12281i	1.72451	+	0.161413i	−0.521414	−	1.93084i		−	1.00000i	1.66401	−	1.79752i			1.00000i	−2.61629	−	1.07472i	2.94789	+	0.556719i	−1.12281	−	0.859821i
71.18	0.859821	+	1.12281i	1.72451	−	0.161413i	−0.521414	+	1.93084i			1.00000i	1.66401	+	1.79752i		−	1.00000i	−2.61629	+	1.07472i	2.94789	−	0.556719i	−1.12281	+	0.859821i
71.19	1.09313	−	0.897253i	1.62224	−	0.606916i	0.389874	−	1.96163i			1.00000i	1.22876	−	2.11900i		−	1.00000i	−1.33390	−	2.49414i	2.26331	−	1.96912i	0.897253	+	1.09313i
71.20	1.09313	+	0.897253i	1.62224	+	0.606916i	0.389874	+	1.96163i		−	1.00000i	1.22876	+	2.11900i			1.00000i	−1.33390	+	2.49414i	2.26331	+	1.96912i	0.897253	−	1.09313i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
12.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	420.2.n.a	✓	24
3.b	odd	2	1		420.2.n.b	yes	24
4.b	odd	2	1		420.2.n.b	yes	24
12.b	even	2	1	inner	420.2.n.a	✓	24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
420.2.n.a	✓	24	1.a	even	1	1	trivial
420.2.n.a	✓	24	12.b	even	2	1	inner
420.2.n.b	yes	24	3.b	odd	2	1
420.2.n.b	yes	24	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T11^12 - 76*T11^10 + 100*T11^9 + 1829*T11^8 - 4844*T11^7 - 11202*T11^6 + 56128*T11^5 - 59808*T11^4 - 20768*T11^3 + 74272*T11^2 - 43136*T11 + 7552