Embedded newform 392.2.z.a.37.6

• Label: 392.2.z.a.37.6
• 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			37.6
Character	\(\chi\)	\(=\)	392.37
Dual form			392.2.z.a.53.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37066 + 0.348278i) q^{2} +(1.48945 + 1.60524i) q^{3} +(1.75740 - 0.954740i) q^{4} +(-1.25029 + 4.05335i) q^{5} +(-2.60059 - 1.68150i) q^{6} +(2.55536 - 0.685667i) q^{7} +(-2.07629 + 1.92069i) q^{8} +(-0.134159 + 1.79023i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37066	+	0.348278i	−0.969201	+	0.246270i
							\(3\)	1.48945	+	1.60524i	0.859933	+	0.926787i	0.997973	−	0.0636398i	\(-0.0202709\pi\)
−0.138040	+	0.990427i	\(0.544080\pi\)
\(5\)	−1.25029	+	4.05335i	−0.559148	+	1.81271i	0.0231418	+	0.999732i	\(0.492633\pi\)
							−0.582290	+	0.812981i	\(0.697843\pi\)
\(7\)	2.55536	−	0.685667i	0.965835	−	0.259158i
							\(11\)	5.31750	−	0.398492i	1.60329	−	0.120150i	0.757196	−	0.653188i	\(-0.226569\pi\)
0.846091	+	0.533038i	\(0.178950\pi\)
\(13\)	−1.33482	+	2.77179i	−0.370213	+	0.768756i	−0.999968	−	0.00803866i	\(-0.997441\pi\)
							0.629755	+	0.776794i	\(0.283155\pi\)
\(17\)	0.221290	+	0.0333541i	0.0536707	+	0.00808955i	0.175823	−	0.984422i	\(-0.443741\pi\)
							−0.122152	+	0.992511i	\(0.538980\pi\)
\(19\)	−4.52324	−	2.61149i	−1.03770	−	0.599118i	−0.118521	−	0.992952i	\(-0.537815\pi\)
							−0.919182	+	0.393834i	\(0.871148\pi\)
\(23\)	2.76583	−	0.416882i	0.576716	−	0.0869259i	0.145794	−	0.989315i	\(-0.453426\pi\)
							0.430922	+	0.902389i	\(0.358188\pi\)
\(29\)	−1.75874	−	1.40254i	−0.326589	−	0.260446i	0.446444	−	0.894811i	\(-0.352690\pi\)
							−0.773033	+	0.634365i	\(0.781261\pi\)
\(31\)	−0.285976	−	0.495326i	−0.0513629	−	0.0889631i	0.839201	−	0.543822i	\(-0.183023\pi\)
							−0.890564	+	0.454858i	\(0.849690\pi\)
\(37\)	2.64765	−	1.03913i	0.435271	−	0.170831i	−0.137580	−	0.990491i	\(-0.543932\pi\)
							0.572851	+	0.819659i	\(0.305837\pi\)
\(41\)	−0.686642	−	3.00838i	−0.107235	−	0.469829i	−0.999820	−	0.0189500i	\(-0.993968\pi\)
							0.892585	−	0.450879i	\(-0.148889\pi\)
\(43\)	−7.43650	−	1.69733i	−1.13406	−	0.258841i	−0.386014	−	0.922493i	\(-0.626148\pi\)
							−0.748041	+	0.663652i	\(0.769006\pi\)
\(47\)	−9.37630	+	6.39265i	−1.36767	+	0.932464i	−0.367677	+	0.929954i	\(0.619847\pi\)
							−0.999997	+	0.00251019i	\(0.999201\pi\)