Embedded newform 392.2.z.a.109.39

• Label: 392.2.z.a.109.39
• 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			109.39
Character	\(\chi\)	\(=\)	392.109
Dual form			392.2.z.a.205.39

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.784103 + 1.17694i) q^{2} +(-0.264046 - 1.75183i) q^{3} +(-0.770366 + 1.84568i) q^{4} +(-3.38580 - 1.32883i) q^{5} +(1.85476 - 1.68438i) q^{6} +(-0.357778 + 2.62145i) q^{7} +(-2.77630 + 0.540529i) q^{8} +(-0.132482 + 0.0408653i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.784103	+	1.17694i	0.554444	+	0.832221i
							\(3\)	−0.264046	−	1.75183i	−0.152447	−	1.01142i	−0.925955	−	0.377634i	\(-0.876738\pi\)
0.773507	−	0.633787i	\(-0.218500\pi\)
\(5\)	−3.38580	−	1.32883i	−1.51417	−	0.594270i	−0.544520	−	0.838748i	\(-0.683288\pi\)
							−0.969655	+	0.244478i	\(0.921383\pi\)
\(7\)	−0.357778	+	2.62145i	−0.135227	+	0.990815i
							\(11\)	−0.946542	+	3.06861i	−0.285393	+	0.925222i	0.693432	+	0.720522i	\(0.256098\pi\)
−0.978825	+	0.204700i	\(0.934378\pi\)
\(13\)	−4.21081	+	0.961090i	−1.16787	+	0.266558i	−0.762126	−	0.647429i	\(-0.775844\pi\)
							−0.405743	+	0.913987i	\(0.632987\pi\)
\(17\)	−3.19299	+	2.17694i	−0.774414	+	0.527987i	−0.884838	−	0.465898i	\(-0.845731\pi\)
							0.110424	+	0.993885i	\(0.464779\pi\)
\(19\)	2.19780	−	1.26890i	0.504209	−	0.291105i	−0.226241	−	0.974071i	\(-0.572644\pi\)
							0.730450	+	0.682966i	\(0.239310\pi\)
\(23\)	−6.54375	−	4.46145i	−1.36447	−	0.930277i	−0.364466	−	0.931217i	\(-0.618748\pi\)
							−1.00000		0.000940129i	\(0.999701\pi\)
\(29\)	−3.71732	+	7.71910i	−0.690289	+	1.43340i	0.200828	+	0.979627i	\(0.435637\pi\)
							−0.891117	+	0.453774i	\(0.850077\pi\)
\(31\)	1.03774	−	1.79741i	0.186383	−	0.322825i	−0.757659	−	0.652651i	\(-0.773657\pi\)
							0.944042	+	0.329826i	\(0.106990\pi\)
\(37\)	9.23936	−	0.692394i	1.51894	−	0.113829i	0.711042	−	0.703149i	\(-0.248223\pi\)
							0.807899	+	0.589320i	\(0.200604\pi\)
\(41\)	5.03910	+	6.31884i	0.786976	+	0.986837i	0.999952	+	0.00977915i	\(0.00311285\pi\)
							−0.212976	+	0.977057i	\(0.568316\pi\)
\(43\)	−0.212516	−	0.169476i	−0.0324084	−	0.0258448i	0.607152	−	0.794586i	\(-0.292312\pi\)
							−0.639560	+	0.768741i	\(0.720883\pi\)
\(47\)	2.62394	−	2.43466i	0.382740	−	0.355131i	−0.465216	−	0.885197i	\(-0.654023\pi\)
							0.847956	+	0.530066i	\(0.177833\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.109.39	yes	648
8.5	even	2	inner	392.2.z.a.109.14	✓	648
49.9	even	21	inner	392.2.z.a.205.14	yes	648
392.205	even	42	inner	392.2.z.a.205.39	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.109.14	✓	648	8.5	even	2	inner
392.2.z.a.109.39	yes	648	1.1	even	1	trivial
392.2.z.a.205.14	yes	648	49.9	even	21	inner
392.2.z.a.205.39	yes	648	392.205	even	42	inner