Embedded newform 392.2.x.a.29.34

• Label: 392.2.x.a.29.34
• Level: $392$
• Weight: $2$
• Character: 392.29
• Analytic conductor: $3.130$
• Analytic rank: $0$
• Dimension: $324$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	392.x (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.13013575923\)
Analytic rank:	\(0\)
Dimension:	\(324\)
Relative dimension:	\(54\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			29.34
Character	\(\chi\)	\(=\)	392.29
Dual form			392.2.x.a.365.34

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.466545 + 1.33504i) q^{2} +(-2.88706 + 0.658953i) q^{3} +(-1.56467 + 1.24571i) q^{4} +(3.84764 - 0.878198i) q^{5} +(-2.22667 - 3.54691i) q^{6} +(0.848300 - 2.50607i) q^{7} +(-2.39307 - 1.50772i) q^{8} +(5.19799 - 2.50322i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q - 5 q^{2} - 5 q^{4} - 9 q^{6} - 12 q^{7} + q^{8} + 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{3}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.466545	+	1.33504i	0.329897	+	0.944017i
							\(3\)	−2.88706	+	0.658953i	−1.66685	+	0.380447i	−0.948881	−	0.315634i	\(-0.897783\pi\)
−0.717964	+	0.696080i	\(0.754926\pi\)
\(5\)	3.84764	−	0.878198i	1.72072	−	0.392742i	0.755696	−	0.654922i	\(-0.227299\pi\)
							0.965019	+	0.262180i	\(0.0844415\pi\)
\(7\)	0.848300	−	2.50607i	0.320627	−	0.947205i
							\(11\)	1.89912	−	3.94357i	0.572607	−	1.18903i	−0.390672	−	0.920530i	\(-0.627757\pi\)
0.963279	−	0.268501i	\(-0.0865284\pi\)
\(13\)	0.131405	−	0.272865i	0.0364452	−	0.0756793i	−0.881951	−	0.471341i	\(-0.843770\pi\)
							0.918397	+	0.395661i	\(0.129485\pi\)
\(17\)	−1.57348	+	1.97308i	−0.381625	+	0.478542i	−0.935131	−	0.354303i	\(-0.884718\pi\)
							0.553506	+	0.832845i	\(0.313290\pi\)
\(19\)		−	2.44592i		−	0.561132i	−0.959835	−	0.280566i	\(-0.909478\pi\)
							0.959835	−	0.280566i	\(-0.0905221\pi\)
\(23\)	4.89760	+	6.14140i	1.02122	+	1.28057i	0.959270	+	0.282490i	\(0.0911605\pi\)
							0.0619500	+	0.998079i	\(0.480268\pi\)
\(29\)	1.87737	+	1.49715i	0.348618	+	0.278014i	0.782106	−	0.623146i	\(-0.214146\pi\)
							−0.433487	+	0.901160i	\(0.642717\pi\)
\(31\)	−7.01959			−1.26076			−0.630378	−	0.776288i	\(-0.717100\pi\)
							−0.630378	+	0.776288i	\(0.717100\pi\)
\(37\)	−3.32598	−	2.65238i	−0.546788	−	0.436049i	0.310733	−	0.950497i	\(-0.399425\pi\)
							−0.857521	+	0.514448i	\(0.827997\pi\)
\(41\)	−0.832581	−	3.64778i	−0.130027	−	0.569687i	−0.997402	−	0.0720318i	\(-0.977052\pi\)
							0.867375	−	0.497655i	\(-0.165805\pi\)
\(43\)	4.36912	+	0.997222i	0.666284	+	0.152075i	0.542268	−	0.840205i	\(-0.317566\pi\)
							0.124016	+	0.992280i	\(0.460423\pi\)
\(47\)	2.60341	+	1.25374i	0.379747	+	0.182876i	0.614013	−	0.789296i	\(-0.289554\pi\)
							−0.234266	+	0.972173i	\(0.575269\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.x.a.29.34	✓	324
8.5	even	2	inner	392.2.x.a.29.49	yes	324
49.22	even	7	inner	392.2.x.a.365.49	yes	324
392.365	even	14	inner	392.2.x.a.365.34	yes	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.x.a.29.34	✓	324	1.1	even	1	trivial
392.2.x.a.29.49	yes	324	8.5	even	2	inner
392.2.x.a.365.34	yes	324	392.365	even	14	inner
392.2.x.a.365.49	yes	324	49.22	even	7	inner