Newform orbit 400.2.be.a

• Label: 400.2.be.a
• Level: $400$
• Weight: $2$
• Character orbit: 400.be
• Analytic conductor: $3.194$
• Analytic rank: $0$
• Dimension: $464$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	400.be (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.19401608085\)
Analytic rank:	\(0\)
Dimension:	\(464\)
Relative dimension:	\(58\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 464 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 8 q^{5} - 6 q^{6} + 12 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
21.1	−1.41077	−	0.0986613i	2.12918	−	0.337229i	1.98053	+	0.278376i	0.968273	+	2.01555i	−3.03705	+	0.265684i		−	1.96518i	−2.76661	−	0.588127i	1.56651	−	0.508991i	−1.16715	−	2.93901i
21.2	−1.39702	−	0.219860i	−1.62017	+	0.256610i	1.90332	+	0.614297i	−1.31621	+	1.80765i	2.31983	−	0.00227828i			2.84840i	−2.52392	−	1.27665i	−0.294054	+	0.0955441i	2.23620	−	2.23593i
21.3	−1.38373	−	0.292059i	2.72156	−	0.431053i	1.82940	+	0.808261i	−1.03145	−	1.98396i	−3.89179	−	0.198398i			2.87694i	−2.29533	−	1.65271i	4.36791	−	1.41922i	0.847808	+	3.04651i
21.4	−1.37734	+	0.320840i	0.993702	−	0.157387i	1.79412	−	0.883811i	−2.20186	−	0.389615i	−1.31817	+	0.535595i			0.222360i	−2.18755	+	1.79293i	−1.89050	+	0.614260i	3.15771	−	0.169814i
21.5	−1.37167	+	0.344287i	−2.96765	+	0.470029i	1.76293	−	0.944493i	−1.94195	−	1.10853i	3.90880	−	1.66645i		−	4.09088i	−2.09298	+	1.90248i	5.73284	−	1.86271i	3.04536	+	0.851938i
21.6	−1.35531	+	0.403902i	0.307556	−	0.0487120i	1.67373	−	1.09482i	1.69021	+	1.46397i	−0.397158	+	0.190242i			1.72856i	−1.82622	+	2.15985i	−2.76095	+	0.897088i	−2.88205	−	1.30145i
21.7	−1.35172	−	0.415745i	0.141890	−	0.0224731i	1.65431	+	1.12395i	0.679981	−	2.13017i	−0.201139	−	0.0286125i		−	3.99719i	−1.76890	−	2.20704i	−2.83354	+	0.920674i	−1.80475	+	2.59670i
21.8	−1.34997	+	0.421416i	−1.51383	+	0.239767i	1.64482	−	1.13779i	0.647118	−	2.14038i	1.94258	−	0.961631i			3.05783i	−1.74096	+	2.22914i	−0.618970	+	0.201115i	0.0284042	+	3.16215i
21.9	−1.32191	−	0.502558i	−2.84258	+	0.450220i	1.49487	+	1.32867i	2.19785	+	0.411629i	3.98388	+	0.833413i		−	0.416200i	−1.30834	−	2.50764i	5.02439	−	1.63252i	−2.69849	−	1.64868i
21.10	−1.21160	−	0.729394i	0.742359	−	0.117578i	0.935969	+	1.76747i	2.23507	−	0.0667519i	−0.985206	−	0.399014i			3.12186i	0.155159	−	2.82417i	−2.31590	+	0.752481i	−2.75671	−	1.54937i
21.11	−1.12780	−	0.853266i	−1.06205	+	0.168212i	0.543875	+	1.92463i	−2.22886	−	0.179410i	1.34131	+	0.716502i		−	0.652480i	1.02884	−	2.63467i	−1.75351	+	0.569750i	2.36063	+	2.10415i
21.12	−1.10053	+	0.888164i	0.211293	−	0.0334655i	0.422329	−	1.95490i	−1.64970	+	1.50947i	−0.202811	+	0.224492i		−	3.25198i	1.27149	+	2.52652i	−2.80964	+	0.912909i	0.474885	−	3.12642i
21.13	−1.09297	+	0.897446i	3.32580	−	0.526756i	0.389180	−	1.96177i	−0.862133	+	2.06318i	−3.16228	+	3.56046i			3.15347i	1.33522	+	2.49343i	7.93033	−	2.57672i	−0.909308	−	3.02872i
21.14	−1.03320	+	0.965656i	−3.03259	+	0.480315i	0.135016	−	1.99544i	0.812902	+	2.08307i	2.66946	−	3.42470i			2.37186i	1.78741	+	2.19207i	6.11271	−	1.98614i	−2.85142	−	1.36725i
21.15	−0.981124	+	1.01853i	−1.08432	+	0.171740i	−0.0747909	−	1.99860i	2.17279	+	0.528170i	0.888935	−	1.27291i		−	3.35884i	2.10901	+	1.88470i	−1.70690	+	0.554607i	−2.66974	+	1.69485i
21.16	−0.919880	−	1.07416i	3.26199	−	0.516648i	−0.307643	+	1.97620i	2.23583	−	0.0328668i	−3.55560	−	3.02864i		−	1.19699i	2.40575	−	1.48741i	7.52046	−	2.44355i	−2.09200	−	2.37140i
21.17	−0.755128	−	1.19573i	2.38103	−	0.377117i	−0.859564	+	1.80587i	−2.20176	−	0.390215i	−2.24891	−	2.56230i		−	2.82512i	2.80842	−	0.335850i	2.67389	−	0.868801i	1.19601	+	2.92738i
21.18	−0.741429	+	1.20428i	−1.38564	+	0.219464i	−0.900567	−	1.78577i	0.522765	−	2.17410i	0.763059	−	1.83141i		−	1.14575i	2.81827	+	0.239491i	−0.981331	+	0.318854i	2.23063	+	2.24149i
21.19	−0.740418	−	1.20490i	−1.22547	+	0.194095i	−0.903562	+	1.78426i	0.515282	+	2.17589i	1.14122	+	1.33285i		−	3.96942i	2.81886	−	0.232396i	−1.38907	+	0.451337i	2.24020	−	2.23193i
21.20	−0.737939	−	1.20642i	0.372869	−	0.0590566i	−0.910891	+	1.78053i	−0.553147	−	2.16657i	−0.346402	−	0.406256i			3.39917i	2.82024	−	0.215005i	−2.71763	+	0.883010i	−2.20560	+	2.26612i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
16.e	even	4	1	inner
25.d	even	5	1	inner
400.be	even	20	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	400.2.be.a	✓	464
16.e	even	4	1	inner	400.2.be.a	✓	464
25.d	even	5	1	inner	400.2.be.a	✓	464
400.be	even	20	1	inner	400.2.be.a	✓	464

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
400.2.be.a	✓	464	1.a	even	1	1	trivial
400.2.be.a	✓	464	16.e	even	4	1	inner
400.2.be.a	✓	464	25.d	even	5	1	inner
400.2.be.a	✓	464	400.be	even	20	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(400, [\chi])\).