Embedded newform 392.2.z.a.37.3

• Label: 392.2.z.a.37.3
• 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			37.3
Character	\(\chi\)	\(=\)	392.37
Dual form			392.2.z.a.53.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40118 + 0.191593i) q^{2} +(-2.22424 - 2.39716i) q^{3} +(1.92658 - 0.536910i) q^{4} +(0.266970 - 0.865496i) q^{5} +(3.57583 + 2.93269i) q^{6} +(2.06958 + 1.64828i) q^{7} +(-2.59661 + 1.12142i) q^{8} +(-0.574944 + 7.67209i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{16}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40118	+	0.191593i	−0.990781	+	0.135476i
							\(3\)	−2.22424	−	2.39716i	−1.28416	−	1.38400i	−0.883952	−	0.467578i	\(-0.845127\pi\)
−0.400213	−	0.916422i	\(-0.631064\pi\)
\(5\)	0.266970	−	0.865496i	0.119393	−	0.387062i	−0.876074	−	0.482178i	\(-0.839846\pi\)
							0.995466	+	0.0951160i	\(0.0303222\pi\)
\(7\)	2.06958	+	1.64828i	0.782228	+	0.622993i
							\(11\)	3.90949	−	0.292976i	1.17876	−	0.0883355i	0.529097	−	0.848562i	\(-0.322531\pi\)
0.649659	+	0.760226i	\(0.274912\pi\)
\(13\)	−3.01147	+	6.25339i	−0.835232	+	1.73438i	−0.173222	+	0.984883i	\(0.555418\pi\)
							−0.662010	+	0.749495i	\(0.730296\pi\)
\(17\)	1.60006	+	0.241171i	0.388073	+	0.0584925i	0.340181	−	0.940360i	\(-0.389512\pi\)
							0.0478916	+	0.998853i	\(0.484750\pi\)
\(19\)	0.834125	+	0.481582i	0.191361	+	0.110483i	0.592620	−	0.805482i	\(-0.298094\pi\)
							−0.401258	+	0.915965i	\(0.631427\pi\)
\(23\)	−0.973933	+	0.146797i	−0.203079	+	0.0306092i	−0.249794	−	0.968299i	\(-0.580363\pi\)
							0.0467149	+	0.998908i	\(0.485125\pi\)
\(29\)	3.94882	+	3.14908i	0.733277	+	0.584769i	0.917321	−	0.398149i	\(-0.130347\pi\)
							−0.184044	+	0.982918i	\(0.558919\pi\)
\(31\)	0.0366915	+	0.0635515i	0.00658999	+	0.0114142i	0.869302	−	0.494282i	\(-0.164569\pi\)
							−0.862712	+	0.505696i	\(0.831236\pi\)
\(37\)	−7.05610	+	2.76932i	−1.16002	+	0.455273i	−0.865763	−	0.500455i	\(-0.833166\pi\)
							−0.294253	+	0.955727i	\(0.595071\pi\)
\(41\)	−2.10841	−	9.23756i	−0.329279	−	1.44267i	−0.820508	−	0.571635i	\(-0.806309\pi\)
							0.491229	−	0.871030i	\(-0.336548\pi\)
\(43\)	7.08223	+	1.61647i	1.08003	+	0.246510i	0.725282	−	0.688452i	\(-0.241709\pi\)
							0.354749	+	0.934962i	\(0.384566\pi\)
\(47\)	2.44941	−	1.66998i	0.357284	−	0.243592i	−0.371354	−	0.928491i	\(-0.621106\pi\)
							0.728637	+	0.684900i	\(0.240154\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.37.3	✓	648	1.1	even	1	trivial
392.2.z.a.37.20	yes	648	8.5	even	2	inner
392.2.z.a.53.3	yes	648	392.53	even	42	inner
392.2.z.a.53.20	yes	648	49.4	even	21	inner