Embedded newform 392.2.z.a.53.20

• Label: 392.2.z.a.53.20
• Level: $392$
• Weight: $2$
• Character: 392.53
• 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			53.20
Character	\(\chi\)	\(=\)	392.53
Dual form			392.2.z.a.37.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.690255 - 1.23432i) q^{2} +(2.22424 - 2.39716i) q^{3} +(-1.04710 + 1.70399i) q^{4} +(-0.266970 - 0.865496i) q^{5} +(-4.49415 - 1.09077i) q^{6} +(2.06958 - 1.64828i) q^{7} +(2.82604 + 0.116261i) q^{8} +(-0.574944 - 7.67209i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{5}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.690255	−	1.23432i	−0.488084	−	0.872797i
							\(3\)	2.22424	−	2.39716i	1.28416	−	1.38400i	0.400213	−	0.916422i	\(-0.368936\pi\)
0.883952	−	0.467578i	\(-0.154873\pi\)
\(5\)	−0.266970	−	0.865496i	−0.119393	−	0.387062i	0.876074	−	0.482178i	\(-0.160154\pi\)
							−0.995466	+	0.0951160i	\(0.969678\pi\)
\(7\)	2.06958	−	1.64828i	0.782228	−	0.622993i
							\(11\)	−3.90949	−	0.292976i	−1.17876	−	0.0883355i	−0.529097	−	0.848562i	\(-0.677469\pi\)
−0.649659	+	0.760226i	\(0.725088\pi\)
\(13\)	3.01147	+	6.25339i	0.835232	+	1.73438i	0.662010	+	0.749495i	\(0.269704\pi\)
							0.173222	+	0.984883i	\(0.444582\pi\)
\(17\)	1.60006	−	0.241171i	0.388073	−	0.0584925i	0.0478916	−	0.998853i	\(-0.484750\pi\)
							0.340181	+	0.940360i	\(0.389512\pi\)
\(19\)	−0.834125	+	0.481582i	−0.191361	+	0.110483i	−0.592620	−	0.805482i	\(-0.701906\pi\)
							0.401258	+	0.915965i	\(0.368573\pi\)
\(23\)	−0.973933	−	0.146797i	−0.203079	−	0.0306092i	0.0467149	−	0.998908i	\(-0.485125\pi\)
							−0.249794	+	0.968299i	\(0.580363\pi\)
\(29\)	−3.94882	+	3.14908i	−0.733277	+	0.584769i	−0.917321	−	0.398149i	\(-0.869653\pi\)
							0.184044	+	0.982918i	\(0.441081\pi\)
\(31\)	0.0366915	−	0.0635515i	0.00658999	−	0.0114142i	−0.862712	−	0.505696i	\(-0.831236\pi\)
							0.869302	+	0.494282i	\(0.164569\pi\)
\(37\)	7.05610	+	2.76932i	1.16002	+	0.455273i	0.865763	−	0.500455i	\(-0.166834\pi\)
							0.294253	+	0.955727i	\(0.404929\pi\)
\(41\)	−2.10841	+	9.23756i	−0.329279	+	1.44267i	0.491229	+	0.871030i	\(0.336548\pi\)
							−0.820508	+	0.571635i	\(0.806309\pi\)
\(43\)	−7.08223	+	1.61647i	−1.08003	+	0.246510i	−0.725282	−	0.688452i	\(-0.758291\pi\)
							−0.354749	+	0.934962i	\(0.615434\pi\)
\(47\)	2.44941	+	1.66998i	0.357284	+	0.243592i	0.728637	−	0.684900i	\(-0.240154\pi\)
							−0.371354	+	0.928491i	\(0.621106\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.53.20	yes	648
8.5	even	2	inner	392.2.z.a.53.3	yes	648
49.37	even	21	inner	392.2.z.a.37.3	✓	648
392.37	even	42	inner	392.2.z.a.37.20	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.37.3	✓	648	49.37	even	21	inner
392.2.z.a.37.20	yes	648	392.37	even	42	inner
392.2.z.a.53.3	yes	648	8.5	even	2	inner
392.2.z.a.53.20	yes	648	1.1	even	1	trivial