Embedded newform 392.2.z.a.37.7

• Label: 392.2.z.a.37.7
• 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			37.7
Character	\(\chi\)	\(=\)	392.37
Dual form			392.2.z.a.53.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34654 - 0.432240i) q^{2} +(1.94533 + 2.09656i) q^{3} +(1.62634 + 1.16406i) q^{4} +(0.117162 - 0.379831i) q^{5} +(-1.71324 - 3.66395i) q^{6} +(-1.64975 + 2.06841i) q^{7} +(-1.68677 - 2.27042i) q^{8} +(-0.387093 + 5.16539i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34654	−	0.432240i	−0.952147	−	0.305640i
							\(3\)	1.94533	+	2.09656i	1.12314	+	1.21045i	0.975170	+	0.221459i	\(0.0710820\pi\)
0.147965	+	0.988993i	\(0.452728\pi\)
\(5\)	0.117162	−	0.379831i	0.0523965	−	0.169865i	−0.925593	−	0.378521i	\(-0.876433\pi\)
							0.977989	+	0.208656i	\(0.0669089\pi\)
\(7\)	−1.64975	+	2.06841i	−0.623545	+	0.781787i
							\(11\)	0.215544	−	0.0161528i	0.0649890	−	0.00487025i	−0.0421959	−	0.999109i	\(-0.513435\pi\)
0.107185	+	0.994239i	\(0.465816\pi\)
\(13\)	−1.59721	+	3.31664i	−0.442986	+	0.919869i	0.553236	+	0.833024i	\(0.313393\pi\)
							−0.996222	+	0.0868447i	\(0.972322\pi\)
\(17\)	7.04217	+	1.06144i	1.70798	+	0.257436i	0.929124	−	0.369769i	\(-0.120563\pi\)
							0.778854	+	0.627205i	\(0.215801\pi\)
\(19\)	−4.90746	−	2.83332i	−1.12585	−	0.650008i	−0.182960	−	0.983120i	\(-0.558568\pi\)
							−0.942887	+	0.333112i	\(0.891901\pi\)
\(23\)	−3.02488	+	0.455928i	−0.630732	+	0.0950675i	−0.456625	−	0.889659i	\(-0.650942\pi\)
							−0.174107	+	0.984727i	\(0.555704\pi\)
\(29\)	−2.06221	−	1.64456i	−0.382943	−	0.305387i	0.413032	−	0.910717i	\(-0.364470\pi\)
							−0.795975	+	0.605330i	\(0.793041\pi\)
\(31\)	−1.78007	−	3.08317i	−0.319709	−	0.553753i	0.660718	−	0.750634i	\(-0.270252\pi\)
							−0.980427	+	0.196881i	\(0.936919\pi\)
\(37\)	1.46563	−	0.575217i	0.240948	−	0.0945651i	−0.241794	−	0.970327i	\(-0.577736\pi\)
							0.482742	+	0.875762i	\(0.339641\pi\)
\(41\)	2.27967	+	9.98790i	0.356025	+	1.55985i	0.762994	+	0.646406i	\(0.223729\pi\)
							−0.406968	+	0.913442i	\(0.633414\pi\)
\(43\)	−1.28322	−	0.292887i	−0.195689	−	0.0446648i	0.123553	−	0.992338i	\(-0.460571\pi\)
							−0.319242	+	0.947673i	\(0.603428\pi\)
\(47\)	10.2926	−	7.01738i	1.50133	−	1.02359i	0.515302	−	0.857008i	\(-0.327680\pi\)
							0.986028	−	0.166581i	\(-0.0532727\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.37.7	✓	648
8.5	even	2	inner	392.2.z.a.37.27	yes	648
49.4	even	21	inner	392.2.z.a.53.27	yes	648
392.53	even	42	inner	392.2.z.a.53.7	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.37.7	✓	648	1.1	even	1	trivial
392.2.z.a.37.27	yes	648	8.5	even	2	inner
392.2.z.a.53.7	yes	648	392.53	even	42	inner
392.2.z.a.53.27	yes	648	49.4	even	21	inner