Embedded newform 392.2.z.a.109.20

• Label: 392.2.z.a.109.20
• 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			109.20
Character	\(\chi\)	\(=\)	392.109
Dual form			392.2.z.a.205.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.634241 - 1.26402i) q^{2} +(-0.0230268 - 0.152773i) q^{3} +(-1.19548 + 1.60338i) q^{4} +(-0.525668 - 0.206310i) q^{5} +(-0.178503 + 0.126001i) q^{6} +(-2.63610 - 0.225735i) q^{7} +(2.78492 + 0.494174i) q^{8} +(2.84391 - 0.877230i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.634241	−	1.26402i	−0.448476	−	0.893795i
							\(3\)	−0.0230268	−	0.152773i	−0.0132945	−	0.0882036i	0.981200	−	0.192996i	\(-0.0618204\pi\)
−0.994494	+	0.104792i	\(0.966582\pi\)
\(5\)	−0.525668	−	0.206310i	−0.235086	−	0.0922644i	0.244874	−	0.969555i	\(-0.421253\pi\)
							−0.479960	+	0.877291i	\(0.659349\pi\)
\(7\)	−2.63610	−	0.225735i	−0.996354	−	0.0853197i
							\(11\)	1.80388	−	5.84803i	0.543890	−	1.76325i	−0.0966457	−	0.995319i	\(-0.530811\pi\)
0.640535	−	0.767929i	\(-0.278712\pi\)
\(13\)	−6.52741	+	1.48984i	−1.81038	+	0.413207i	−0.987836	−	0.155500i	\(-0.950301\pi\)
							−0.822541	+	0.568706i	\(0.807444\pi\)
\(17\)	−4.09945	+	2.79496i	−0.994263	+	0.677877i	−0.947076	−	0.321009i	\(-0.895978\pi\)
							−0.0471870	+	0.998886i	\(0.515026\pi\)
\(19\)	−3.64219	+	2.10282i	−0.835575	+	0.482419i	−0.855758	−	0.517377i	\(-0.826909\pi\)
							0.0201826	+	0.999796i	\(0.493575\pi\)
\(23\)	−1.89465	−	1.29175i	−0.395062	−	0.269349i	0.349454	−	0.936954i	\(-0.386367\pi\)
							−0.744516	+	0.667605i	\(0.767320\pi\)
\(29\)	−2.65570	+	5.51461i	−0.493151	+	1.02404i	0.494762	+	0.869028i	\(0.335255\pi\)
							−0.987913	+	0.155010i	\(0.950459\pi\)
\(31\)	3.34270	−	5.78972i	0.600366	−	1.03986i	−0.392399	−	0.919795i	\(-0.628355\pi\)
							0.992765	−	0.120070i	\(-0.0383119\pi\)
\(37\)	5.30623	−	0.397647i	0.872339	−	0.0653728i	0.368967	−	0.929443i	\(-0.379712\pi\)
							0.503372	+	0.864070i	\(0.332093\pi\)
\(41\)	−4.66691	−	5.85212i	−0.728849	−	0.913948i	0.269953	−	0.962873i	\(-0.412992\pi\)
							−0.998802	+	0.0489255i	\(0.984420\pi\)
\(43\)	−1.84701	−	1.47294i	−0.281667	−	0.224622i	0.472457	−	0.881354i	\(-0.343367\pi\)
							−0.754124	+	0.656732i	\(0.771938\pi\)
\(47\)	2.94701	−	2.73443i	0.429866	−	0.398858i	−0.435346	−	0.900263i	\(-0.643374\pi\)
							0.865212	+	0.501406i	\(0.167184\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.109.20	✓	648
8.5	even	2	inner	392.2.z.a.109.44	yes	648
49.9	even	21	inner	392.2.z.a.205.44	yes	648
392.205	even	42	inner	392.2.z.a.205.20	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.109.20	✓	648	1.1	even	1	trivial
392.2.z.a.109.44	yes	648	8.5	even	2	inner
392.2.z.a.205.20	yes	648	392.205	even	42	inner
392.2.z.a.205.44	yes	648	49.9	even	21	inner