Newform orbit 407.2.w.a

• Label: 407.2.w.a
• Level: $407$
• Weight: $2$
• Character orbit: 407.w
• Analytic conductor: $3.250$
• Analytic rank: $0$
• Dimension: $288$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 407 = 11 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	407.w (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.24991136227\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(36\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 288 q - 10 q^{2} - 4 q^{5} - 10 q^{6} - 20 q^{7} - 30 q^{8} + 48 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
6.1	−1.18969	+	2.33490i	0.219360	−	0.301924i	−2.86081	−	3.93757i	0.188593	−	0.0960929i	0.443990	+	0.871379i	−1.24766	−	1.71725i	7.42079	−	1.17534i	0.884012	+	2.72071i			0.554666i
6.2	−1.16481	+	2.28607i	−0.779069	+	1.07230i	−2.69375	−	3.70763i	2.16898	−	1.10515i	−1.54387	−	3.03002i	1.71324	+	2.35808i	6.54536	−	1.03668i	0.384180	+	1.18238i			6.24571i
6.3	−1.15239	+	2.26169i	0.353942	−	0.487160i	−2.61166	−	3.59464i	−3.60892	+	1.83884i	0.693924	+	1.36190i	0.872917	+	1.20147i	6.12540	−	0.970168i	0.815001	+	2.50832i		−	10.2813i
6.4	−1.05213	+	2.06491i	−1.58857	+	2.18647i	−1.98133	−	2.72707i	1.11673	−	0.569002i	−2.84351	−	5.58070i	−1.06814	−	1.47017i	3.13782	−	0.496983i	−1.33008	−	4.09355i			2.90461i
6.5	−0.965911	+	1.89571i	1.06935	−	1.47184i	−1.48515	−	2.04414i	3.14707	−	1.60351i	1.75728	+	3.44885i	−2.96353	−	4.07895i	1.10680	−	0.175299i	−0.0957430	−	0.294667i			7.51477i
6.6	−0.960840	+	1.88576i	1.15881	−	1.59496i	−1.45729	−	2.00579i	−0.238868	+	0.121709i	1.89428	+	3.71773i	1.41498	+	1.94755i	1.00189	−	0.158684i	−0.274014	−	0.843328i		−	0.567390i
6.7	−0.894341	+	1.75524i	−1.82703	+	2.51469i	−1.10546	−	1.52153i	−2.18504	+	1.11333i	−2.77991	−	5.45588i	1.99395	+	2.74444i	−0.232083	+	0.0367583i	−2.05859	−	6.33569i		−	4.83098i
6.8	−0.823636	+	1.61648i	−1.05617	+	1.45370i	−0.759049	−	1.04474i	−3.07532	+	1.56696i	−1.47997	−	2.90460i	−1.62887	−	2.24194i	−1.26978	+	0.201113i	−0.0706860	−	0.217549i		−	6.26179i
6.9	−0.789474	+	1.54943i	−0.0950687	+	0.130851i	−0.601895	−	0.828437i	−1.10384	+	0.562432i	−0.127690	−	0.250606i	−1.09346	−	1.50503i	−1.67633	+	0.265504i	0.918967	+	2.82829i		−	2.15434i
6.10	−0.774619	+	1.52027i	1.33338	−	1.83524i	−0.535631	−	0.737233i	3.09495	−	1.57696i	1.75720	+	3.44871i	2.24582	+	3.09110i	−1.83477	+	0.290599i	−0.663145	−	2.04095i			5.92672i
6.11	−0.647246	+	1.27029i	1.79474	−	2.47025i	−0.0191424	−	0.0263472i	−1.55833	+	0.794009i	1.97630	+	3.87870i	−2.30725	−	3.17565i	−2.77040	+	0.438788i	−1.95398	−	6.01374i		−	2.49345i
6.12	−0.521637	+	1.02377i	−1.34658	+	1.85340i	0.399570	+	0.549962i	2.91337	−	1.48444i	−1.19504	−	2.34539i	0.542954	+	0.747312i	−3.04118	+	0.481676i	−0.694786	−	2.13833i			3.75696i
6.13	−0.490713	+	0.963079i	0.110843	−	0.152562i	0.488849	+	0.672843i	0.909269	−	0.463296i	0.0925372	+	0.181614i	0.667449	+	0.918664i	−3.02305	+	0.478804i	0.916062	+	2.81935i			1.10304i
6.14	−0.367987	+	0.722216i	1.43531	−	1.97553i	0.789389	+	1.08650i	−3.24351	+	1.65265i	0.898584	+	1.76357i	1.95892	+	2.69622i	−2.67634	+	0.423890i	−0.915560	−	2.81780i		−	2.95066i
6.15	−0.337371	+	0.662127i	−0.673641	+	0.927187i	0.850977	+	1.17127i	−1.42462	+	0.725880i	−0.386649	−	0.758842i	2.13314	+	2.93602i	−2.53057	+	0.400803i	0.521167	+	1.60399i		−	1.18817i
6.16	−0.288514	+	0.566240i	−1.34658	+	1.85341i	0.938183	+	1.29130i	0.890074	−	0.453515i	−0.660966	−	1.29722i	−2.94662	−	4.05568i	−2.25723	+	0.357510i	−0.694790	−	2.13834i			0.634841i
6.17	−0.203622	+	0.399630i	0.701594	−	0.965662i	1.05733	+	1.45529i	−1.56666	+	0.798252i	0.243048	+	0.477008i	−1.57412	−	2.16659i	−1.68286	+	0.266539i	0.486783	+	1.49816i		−	0.788625i
6.18	−0.0869002	+	0.170551i	0.398389	−	0.548335i	1.15403	+	1.58839i	2.59159	−	1.32048i	0.0588992	+	0.115596i	−0.717866	−	0.988058i	−0.749303	+	0.118678i	0.785093	+	2.41627i			0.556748i
6.19	−0.0558070	+	0.109527i	1.69671	−	2.33532i	1.16669	+	1.60581i	1.79824	−	0.916251i	0.161094	+	0.316164i	0.354286	+	0.487633i	−0.483814	+	0.0766285i	−1.64785	−	5.07158i			0.248090i
6.20	0.162489	−	0.318903i	−1.59518	+	2.19558i	1.10027	+	1.51440i	−2.35398	+	1.19941i	0.440978	+	0.865467i	0.732921	+	1.00878i	1.36874	−	0.216787i	−1.34892	−	4.15155i			0.945584i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.d	odd	10	1	inner
37.d	odd	4	1	inner
407.w	even	20	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	407.2.w.a	✓	288
11.d	odd	10	1	inner	407.2.w.a	✓	288
37.d	odd	4	1	inner	407.2.w.a	✓	288
407.w	even	20	1	inner	407.2.w.a	✓	288

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
407.2.w.a	✓	288	1.a	even	1	1	trivial
407.2.w.a	✓	288	11.d	odd	10	1	inner
407.2.w.a	✓	288	37.d	odd	4	1	inner
407.2.w.a	✓	288	407.w	even	20	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(407, [\chi])\).