Embedded newform 392.2.z.a.109.21

• Label: 392.2.z.a.109.21
• 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			109.21
Character	\(\chi\)	\(=\)	392.109
Dual form			392.2.z.a.205.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.624156 + 1.26903i) q^{2} +(-0.371894 - 2.46736i) q^{3} +(-1.22086 - 1.58414i) q^{4} +(-1.53857 - 0.603845i) q^{5} +(3.36326 + 1.06807i) q^{6} +(2.53791 - 0.747686i) q^{7} +(2.77232 - 0.560553i) q^{8} +(-3.08282 + 0.950925i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.624156	+	1.26903i	−0.441345	+	0.897338i
							\(3\)	−0.371894	−	2.46736i	−0.214713	−	1.42453i	−0.793262	−	0.608881i	\(-0.791619\pi\)
0.578549	−	0.815648i	\(-0.303619\pi\)
\(5\)	−1.53857	−	0.603845i	−0.688071	−	0.270048i	−0.00456136	−	0.999990i	\(-0.501452\pi\)
							−0.683509	+	0.729942i	\(0.739547\pi\)
\(7\)	2.53791	−	0.747686i	0.959238	−	0.282599i
							\(11\)	0.767062	−	2.48676i	0.231278	−	0.749785i	−0.763323	−	0.646017i	\(-0.776434\pi\)
0.994601	−	0.103768i	\(-0.0330901\pi\)
\(13\)	−6.17046	+	1.40837i	−1.71138	+	0.390610i	−0.962325	−	0.271902i	\(-0.912348\pi\)
							−0.749051	+	0.662512i	\(0.769490\pi\)
\(17\)	1.72613	−	1.17685i	0.418647	−	0.285429i	−0.335619	−	0.941998i	\(-0.608945\pi\)
							0.754266	+	0.656569i	\(0.227993\pi\)
\(19\)	−1.99327	+	1.15081i	−0.457287	+	0.264015i	−0.710903	−	0.703290i	\(-0.751714\pi\)
							0.253616	+	0.967305i	\(0.418380\pi\)
\(23\)	−4.07160	−	2.77597i	−0.848987	−	0.578830i	0.0587959	−	0.998270i	\(-0.481274\pi\)
							−0.907783	+	0.419441i	\(0.862226\pi\)
\(29\)	0.406906	−	0.844948i	0.0755605	−	0.156903i	−0.859767	−	0.510687i	\(-0.829391\pi\)
							0.935327	+	0.353784i	\(0.115105\pi\)
\(31\)	−3.84883	+	6.66636i	−0.691270	+	1.19731i	0.280153	+	0.959955i	\(0.409615\pi\)
							−0.971422	+	0.237359i	\(0.923718\pi\)
\(37\)	5.68048	−	0.425693i	0.933864	−	0.0699835i	0.400901	−	0.916122i	\(-0.368697\pi\)
							0.532964	+	0.846138i	\(0.321078\pi\)
\(41\)	−6.63484	−	8.31983i	−1.03619	−	1.29934i	−0.953054	−	0.302801i	\(-0.902078\pi\)
							−0.0831344	−	0.996538i	\(-0.526493\pi\)
\(43\)	4.22854	+	3.37215i	0.644846	+	0.514247i	0.890426	−	0.455129i	\(-0.150407\pi\)
							−0.245580	+	0.969376i	\(0.578978\pi\)
\(47\)	−3.14157	+	2.91495i	−0.458245	+	0.425189i	−0.875217	−	0.483730i	\(-0.839282\pi\)
							0.416972	+	0.908919i	\(0.363091\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.109.21	yes	648
8.5	even	2	inner	392.2.z.a.109.5	✓	648
49.9	even	21	inner	392.2.z.a.205.5	yes	648
392.205	even	42	inner	392.2.z.a.205.21	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.109.5	✓	648	8.5	even	2	inner
392.2.z.a.109.21	yes	648	1.1	even	1	trivial
392.2.z.a.205.5	yes	648	49.9	even	21	inner
392.2.z.a.205.21	yes	648	392.205	even	42	inner