Embedded newform 392.2.z.a.37.16

• Label: 392.2.z.a.37.16
• Level: $392$
• Weight: $2$
• Character: 392.37
• Analytic conductor: $3.130$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	392.z (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.13013575923\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			37.16
Character	\(\chi\)	\(=\)	392.37
Dual form			392.2.z.a.53.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.804026 - 1.16342i) q^{2} +(0.264186 + 0.284725i) q^{3} +(-0.707086 + 1.87084i) q^{4} +(-1.09949 + 3.56447i) q^{5} +(0.118842 - 0.536285i) q^{6} +(-1.95202 - 1.78595i) q^{7} +(2.74508 - 0.681563i) q^{8} +(0.212916 - 2.84117i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{16}{21}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.804026	−	1.16342i	−0.568532	−	0.822661i
							\(3\)	0.264186	+	0.284725i	0.152528	+	0.164386i	0.804693	−	0.593691i	\(-0.202330\pi\)
−0.652165	+	0.758077i	\(0.726139\pi\)
\(5\)	−1.09949	+	3.56447i	−0.491709	+	1.59408i	0.281256	+	0.959633i	\(0.409249\pi\)
							−0.772965	+	0.634448i	\(0.781227\pi\)
\(7\)	−1.95202	−	1.78595i	−0.737795	−	0.675025i
							\(11\)	−2.33927	+	0.175304i	−0.705317	+	0.0528562i	−0.422559	−	0.906335i	\(-0.638868\pi\)
−0.282757	+	0.959191i	\(0.591249\pi\)
\(13\)	−0.768475	+	1.59576i	−0.213137	+	0.442583i	−0.979938	−	0.199301i	\(-0.936133\pi\)
							0.766802	+	0.641884i	\(0.221847\pi\)
\(17\)	0.541435	+	0.0816083i	0.131317	+	0.0197929i	0.214372	−	0.976752i	\(-0.431230\pi\)
							−0.0830546	+	0.996545i	\(0.526468\pi\)
\(19\)	−5.36222	−	3.09588i	−1.23018	−	0.710244i	−0.263111	−	0.964765i	\(-0.584749\pi\)
							−0.967067	+	0.254522i	\(0.918082\pi\)
\(23\)	−7.14795	+	1.07738i	−1.49045	+	0.224649i	−0.843188	−	0.537618i	\(-0.819324\pi\)
							−0.647262	+	0.762267i	\(0.724086\pi\)
\(29\)	−1.29591	−	1.03345i	−0.240644	−	0.191907i	0.495740	−	0.868471i	\(-0.334897\pi\)
							−0.736384	+	0.676564i	\(0.763468\pi\)
\(31\)	1.61796	+	2.80239i	0.290594	+	0.503324i	0.973950	−	0.226761i	\(-0.0728137\pi\)
							−0.683356	+	0.730085i	\(0.739480\pi\)
\(37\)	−3.28417	+	1.28894i	−0.539915	+	0.211901i	−0.619598	−	0.784919i	\(-0.712704\pi\)
							0.0796831	+	0.996820i	\(0.474609\pi\)
\(41\)	1.27620	+	5.59139i	0.199309	+	0.873229i	0.971349	+	0.237656i	\(0.0763790\pi\)
							−0.772041	+	0.635573i	\(0.780764\pi\)
\(43\)	−7.00458	−	1.59875i	−1.06819	−	0.243807i	−0.347936	−	0.937518i	\(-0.613117\pi\)
							−0.720253	+	0.693711i	\(0.755974\pi\)
\(47\)	7.73954	−	5.27673i	1.12893	−	0.769690i	0.153096	−	0.988211i	\(-0.451076\pi\)
							0.975832	+	0.218521i	\(0.0701232\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.37.16	✓	648
8.5	even	2	inner	392.2.z.a.37.37	yes	648
49.4	even	21	inner	392.2.z.a.53.37	yes	648
392.53	even	42	inner	392.2.z.a.53.16	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.37.16	✓	648	1.1	even	1	trivial
392.2.z.a.37.37	yes	648	8.5	even	2	inner
392.2.z.a.53.16	yes	648	392.53	even	42	inner
392.2.z.a.53.37	yes	648	49.4	even	21	inner