Embedded newform 392.2.z.a.109.40

• Label: 392.2.z.a.109.40
• 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			109.40
Character	\(\chi\)	\(=\)	392.109
Dual form			392.2.z.a.205.40

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.804354 - 1.16319i) q^{2} +(0.111463 + 0.739509i) q^{3} +(-0.706028 - 1.87124i) q^{4} +(2.34300 + 0.919559i) q^{5} +(0.949846 + 0.465174i) q^{6} +(-2.33721 - 1.23994i) q^{7} +(-2.74450 - 0.683890i) q^{8} +(2.33227 - 0.719410i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.804354	−	1.16319i	0.568764	−	0.822501i
							\(3\)	0.111463	+	0.739509i	0.0643532	+	0.426956i	0.997689	+	0.0679503i	\(0.0216459\pi\)
−0.933335	+	0.359005i	\(0.883116\pi\)
\(5\)	2.34300	+	0.919559i	1.04782	+	0.411239i	0.825895	−	0.563824i	\(-0.190670\pi\)
							0.221926	+	0.975064i	\(0.428766\pi\)
\(7\)	−2.33721	−	1.23994i	−0.883383	−	0.468652i
							\(11\)	0.561732	−	1.82109i	0.169369	−	0.549080i	−0.830601	−	0.556868i	\(-0.812003\pi\)
0.999970	+	0.00778832i	\(0.00247912\pi\)
\(13\)	3.11539	−	0.711067i	0.864053	−	0.197215i	0.232547	−	0.972585i	\(-0.425294\pi\)
							0.631506	+	0.775371i	\(0.282437\pi\)
\(17\)	2.96125	−	2.01894i	0.718208	−	0.489666i	−0.148229	−	0.988953i	\(-0.547357\pi\)
							0.866436	+	0.499287i	\(0.166405\pi\)
\(19\)	−0.698383	+	0.403212i	−0.160220	+	0.0925031i	−0.577966	−	0.816061i	\(-0.696153\pi\)
							0.417746	+	0.908564i	\(0.362820\pi\)
\(23\)	2.36810	+	1.61454i	0.493783	+	0.336656i	0.784474	−	0.620161i	\(-0.212933\pi\)
							−0.290691	+	0.956817i	\(0.593885\pi\)
\(29\)	−3.29643	+	6.84510i	−0.612131	+	1.27110i	0.332541	+	0.943089i	\(0.392094\pi\)
							−0.944673	+	0.328015i	\(0.893620\pi\)
\(31\)	1.00577	−	1.74205i	0.180642	−	0.312881i	−0.761457	−	0.648215i	\(-0.775516\pi\)
							0.942099	+	0.335334i	\(0.108849\pi\)
\(37\)	−10.4356	+	0.782040i	−1.71560	+	0.128567i	−0.895909	−	0.444238i	\(-0.853474\pi\)
							−0.819692	+	0.572805i	\(0.805855\pi\)
\(41\)	1.38952	+	1.74241i	0.217007	+	0.272118i	0.878405	−	0.477917i	\(-0.158608\pi\)
							−0.661398	+	0.750035i	\(0.730037\pi\)
\(43\)	−5.31893	−	4.24170i	−0.811128	−	0.646853i	0.127479	−	0.991841i	\(-0.459312\pi\)
							−0.938607	+	0.344988i	\(0.887883\pi\)
\(47\)	−2.73303	+	2.53588i	−0.398654	+	0.369897i	−0.853872	−	0.520483i	\(-0.825752\pi\)
							0.455218	+	0.890380i	\(0.349561\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.109.40	✓	648
8.5	even	2	inner	392.2.z.a.109.45	yes	648
49.9	even	21	inner	392.2.z.a.205.45	yes	648
392.205	even	42	inner	392.2.z.a.205.40	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.109.40	✓	648	1.1	even	1	trivial
392.2.z.a.109.45	yes	648	8.5	even	2	inner
392.2.z.a.205.40	yes	648	392.205	even	42	inner
392.2.z.a.205.45	yes	648	49.9	even	21	inner