Embedded newform 392.2.z.a.37.9

• Label: 392.2.z.a.37.9
• 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			37.9
Character	\(\chi\)	\(=\)	392.37
Dual form			392.2.z.a.53.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.24570 - 0.669503i) q^{2} +(-1.67046 - 1.80033i) q^{3} +(1.10353 + 1.66800i) q^{4} +(0.687205 - 2.22786i) q^{5} +(0.875568 + 3.36105i) q^{6} +(-0.555114 - 2.58686i) q^{7} +(-0.257941 - 2.81664i) q^{8} +(-0.226554 + 3.02315i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.24570	−	0.669503i	−0.880842	−	0.473410i
							\(3\)	−1.67046	−	1.80033i	−0.964442	−	1.03942i	−0.999199	−	0.0400251i	\(-0.987256\pi\)
0.0347570	−	0.999396i	\(-0.488934\pi\)
\(5\)	0.687205	−	2.22786i	0.307327	−	0.996331i	−0.661781	−	0.749697i	\(-0.730199\pi\)
							0.969109	−	0.246634i	\(-0.0793246\pi\)
\(7\)	−0.555114	−	2.58686i	−0.209814	−	0.977741i
							\(11\)	2.46014	−	0.184362i	0.741761	−	0.0555873i	0.301511	−	0.953463i	\(-0.402509\pi\)
0.440250	+	0.897875i	\(0.354890\pi\)
\(13\)	2.57364	−	5.34423i	0.713801	−	1.48222i	−0.155447	−	0.987844i	\(-0.549682\pi\)
							0.869247	−	0.494378i	\(-0.164604\pi\)
\(17\)	2.20562	+	0.332443i	0.534941	+	0.0806294i	0.410957	−	0.911655i	\(-0.365195\pi\)
							0.123984	+	0.992284i	\(0.460433\pi\)
\(19\)	−5.27066	−	3.04302i	−1.20917	−	0.698116i	−0.246594	−	0.969119i	\(-0.579311\pi\)
							−0.962579	+	0.271003i	\(0.912645\pi\)
\(23\)	−0.371013	+	0.0559211i	−0.0773615	+	0.0116604i	−0.187609	−	0.982244i	\(-0.560074\pi\)
							0.110247	+	0.993904i	\(0.464836\pi\)
\(29\)	5.53026	+	4.41023i	1.02694	+	0.818960i	0.983648	−	0.180103i	\(-0.0576430\pi\)
							0.0432951	+	0.999062i	\(0.486214\pi\)
\(31\)	2.19337	+	3.79902i	0.393940	+	0.682324i	0.992965	−	0.118406i	\(-0.0377784\pi\)
							−0.599025	+	0.800730i	\(0.704445\pi\)
\(37\)	−10.3721	+	4.07075i	−1.70516	+	0.669226i	−0.999237	−	0.0390611i	\(-0.987563\pi\)
							−0.705924	+	0.708287i	\(0.749468\pi\)
\(41\)	0.514176	+	2.25275i	0.0803008	+	0.351821i	0.999077	−	0.0429584i	\(-0.0136783\pi\)
							−0.918776	+	0.394779i	\(0.870821\pi\)
\(43\)	−9.21818	−	2.10399i	−1.40576	−	0.320855i	−0.548677	−	0.836035i	\(-0.684868\pi\)
							−0.857082	+	0.515179i	\(0.827725\pi\)
\(47\)	1.61763	−	1.10288i	0.235955	−	0.160871i	−0.439570	−	0.898208i	\(-0.644869\pi\)
							0.675525	+	0.737337i	\(0.263917\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.37.9	✓	648
8.5	even	2	inner	392.2.z.a.37.30	yes	648
49.4	even	21	inner	392.2.z.a.53.30	yes	648
392.53	even	42	inner	392.2.z.a.53.9	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.37.9	✓	648	1.1	even	1	trivial
392.2.z.a.37.30	yes	648	8.5	even	2	inner
392.2.z.a.53.9	yes	648	392.53	even	42	inner
392.2.z.a.53.30	yes	648	49.4	even	21	inner