Embedded newform 392.2.z.a.53.16

• Label: 392.2.z.a.53.16
• Level: $392$
• Weight: $2$
• Character: 392.53
• 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			53.16
Character	\(\chi\)	\(=\)	392.53
Dual form			392.2.z.a.37.16

$q$-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 \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{5}{21}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.37.16	✓	648	392.37	even	42	inner
392.2.z.a.37.37	yes	648	49.37	even	21	inner
392.2.z.a.53.16	yes	648	1.1	even	1	trivial
392.2.z.a.53.37	yes	648	8.5	even	2	inner