Embedded newform 392.2.z.a.109.19

• Label: 392.2.z.a.109.19
• Level: $392$
• Weight: $2$
• Character: 392.109
• 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			109.19
Character	\(\chi\)	\(=\)	392.109
Dual form			392.2.z.a.205.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.637755 + 1.26225i) q^{2} +(-0.420455 - 2.78954i) q^{3} +(-1.18654 - 1.61001i) q^{4} +(3.23171 + 1.26835i) q^{5} +(3.78923 + 1.24832i) q^{6} +(0.157885 + 2.64104i) q^{7} +(2.78895 - 0.470915i) q^{8} +(-4.73801 + 1.46148i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{20}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.637755	+	1.26225i	−0.450961	+	0.892544i
							\(3\)	−0.420455	−	2.78954i	−0.242750	−	1.61054i	−0.695245	−	0.718772i	\(-0.744704\pi\)
0.452496	−	0.891767i	\(-0.350534\pi\)
\(5\)	3.23171	+	1.26835i	1.44526	+	0.567224i	0.952932	−	0.303184i	\(-0.0980495\pi\)
							0.492332	+	0.870408i	\(0.336145\pi\)
\(7\)	0.157885	+	2.64104i	0.0596749	+	0.998218i
							\(11\)	−1.48356	+	4.80959i	−0.447310	+	1.45014i	0.398751	+	0.917059i	\(0.369444\pi\)
−0.846061	+	0.533085i	\(0.821032\pi\)
\(13\)	5.09843	−	1.16368i	1.41405	−	0.322748i	0.553813	−	0.832641i	\(-0.313172\pi\)
							0.860238	+	0.509894i	\(0.170315\pi\)
\(17\)	1.52900	−	1.04246i	0.370838	−	0.252833i	−0.363533	−	0.931581i	\(-0.618430\pi\)
							0.734371	+	0.678748i	\(0.237477\pi\)
\(19\)	3.89482	−	2.24868i	0.893533	−	0.515882i	0.0184367	−	0.999830i	\(-0.494131\pi\)
							0.875097	+	0.483948i	\(0.160798\pi\)
\(23\)	−3.05737	−	2.08448i	−0.637506	−	0.434645i	0.200959	−	0.979600i	\(-0.435594\pi\)
							−0.838465	+	0.544955i	\(0.816547\pi\)
\(29\)	−1.89806	+	3.94137i	−0.352462	+	0.731894i	−0.999534	−	0.0305348i	\(-0.990279\pi\)
							0.647072	+	0.762429i	\(0.275993\pi\)
\(31\)	2.30081	−	3.98512i	0.413238	−	0.715749i	−0.582004	−	0.813186i	\(-0.697731\pi\)
							0.995242	+	0.0974373i	\(0.0310646\pi\)
\(37\)	5.27107	−	0.395012i	0.866558	−	0.0649396i	0.365972	−	0.930626i	\(-0.380737\pi\)
							0.500587	+	0.865686i	\(0.333118\pi\)
\(41\)	−0.683972	−	0.857674i	−0.106818	−	0.133946i	0.725548	−	0.688171i	\(-0.241586\pi\)
							−0.832367	+	0.554225i	\(0.813015\pi\)
\(43\)	−5.18978	−	4.13871i	−0.791434	−	0.631148i	0.142012	−	0.989865i	\(-0.454643\pi\)
							−0.933446	+	0.358717i	\(0.883214\pi\)
\(47\)	−3.45770	+	3.20827i	−0.504357	+	0.467975i	−0.890837	−	0.454323i	\(-0.849881\pi\)
							0.386480	+	0.922298i	\(0.373691\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.109.19	yes	648
8.5	even	2	inner	392.2.z.a.109.7	✓	648
49.9	even	21	inner	392.2.z.a.205.7	yes	648
392.205	even	42	inner	392.2.z.a.205.19	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.109.7	✓	648	8.5	even	2	inner
392.2.z.a.109.19	yes	648	1.1	even	1	trivial
392.2.z.a.205.7	yes	648	49.9	even	21	inner
392.2.z.a.205.19	yes	648	392.205	even	42	inner