Embedded newform 392.2.z.a.109.48

• Label: 392.2.z.a.109.48
• 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			109.48
Character	\(\chi\)	\(=\)	392.109
Dual form			392.2.z.a.205.48

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.25069 - 0.660137i) q^{2} +(0.456722 + 3.03015i) q^{3} +(1.12844 - 1.65125i) q^{4} +(-0.412315 - 0.161822i) q^{5} +(2.57153 + 3.48827i) q^{6} +(-0.534913 + 2.59111i) q^{7} +(0.321272 - 2.81012i) q^{8} +(-6.10651 + 1.88361i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.25069	−	0.660137i	0.884370	−	0.466787i
							\(3\)	0.456722	+	3.03015i	0.263688	+	1.74946i	0.592968	+	0.805226i	\(0.297956\pi\)
−0.329279	+	0.944233i	\(0.606806\pi\)
\(5\)	−0.412315	−	0.161822i	−0.184393	−	0.0723689i	0.271348	−	0.962481i	\(-0.412531\pi\)
							−0.455741	+	0.890112i	\(0.650626\pi\)
\(7\)	−0.534913	+	2.59111i	−0.202178	+	0.979349i
							\(11\)	−1.10688	+	3.58842i	−0.333737	+	1.08195i	0.620828	+	0.783947i	\(0.286797\pi\)
−0.954565	+	0.298003i	\(0.903680\pi\)
\(13\)	5.45238	−	1.24447i	1.51222	−	0.345154i	0.615636	−	0.788031i	\(-0.288899\pi\)
							0.896582	+	0.442877i	\(0.146042\pi\)
\(17\)	5.67383	−	3.86835i	1.37611	−	0.938214i	0.376175	−	0.926549i	\(-0.377239\pi\)
							0.999932	−	0.0116648i	\(-0.00371312\pi\)
\(19\)	−5.31713	+	3.06985i	−1.21983	+	0.704271i	−0.964882	−	0.262683i	\(-0.915393\pi\)
							−0.254951	+	0.966954i	\(0.582059\pi\)
\(23\)	0.585962	+	0.399502i	0.122181	+	0.0833019i	0.622869	−	0.782326i	\(-0.285967\pi\)
							−0.500687	+	0.865628i	\(0.666919\pi\)
\(29\)	0.317230	−	0.658735i	0.0589082	−	0.122324i	−0.869436	−	0.494045i	\(-0.835518\pi\)
							0.928345	+	0.371721i	\(0.121232\pi\)
\(31\)	2.47635	−	4.28917i	0.444766	−	0.770358i	−0.553270	−	0.833002i	\(-0.686620\pi\)
							0.998036	+	0.0626445i	\(0.0199534\pi\)
\(37\)	4.96708	−	0.372231i	0.816583	−	0.0611945i	0.340113	−	0.940385i	\(-0.389535\pi\)
							0.476471	+	0.879190i	\(0.341916\pi\)
\(41\)	1.96736	+	2.46699i	0.307250	+	0.385279i	0.911352	−	0.411628i	\(-0.135040\pi\)
							−0.604102	+	0.796907i	\(0.706468\pi\)
\(43\)	−3.07973	−	2.45600i	−0.469654	−	0.374537i	0.359874	−	0.933001i	\(-0.382820\pi\)
							−0.829529	+	0.558464i	\(0.811391\pi\)
\(47\)	7.46210	−	6.92382i	1.08846	−	1.00994i	0.0885557	−	0.996071i	\(-0.471775\pi\)
							0.999904	−	0.0138716i	\(-0.00441560\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.109.48	yes	648
8.5	even	2	inner	392.2.z.a.109.37	✓	648
49.9	even	21	inner	392.2.z.a.205.37	yes	648
392.205	even	42	inner	392.2.z.a.205.48	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.109.37	✓	648	8.5	even	2	inner
392.2.z.a.109.48	yes	648	1.1	even	1	trivial
392.2.z.a.205.37	yes	648	49.9	even	21	inner
392.2.z.a.205.48	yes	648	392.205	even	42	inner