Embedded newform 392.2.z.a.109.4

• Label: 392.2.z.a.109.4
• 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			109.4
Character	\(\chi\)	\(=\)	392.109
Dual form			392.2.z.a.205.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34815 - 0.427180i) q^{2} +(-0.281677 - 1.86880i) q^{3} +(1.63503 + 1.15181i) q^{4} +(2.47215 + 0.970246i) q^{5} +(-0.418573 + 2.63976i) q^{6} +(-2.31027 - 1.28943i) q^{7} +(-1.71225 - 2.25127i) q^{8} +(-0.546363 + 0.168531i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34815	−	0.427180i	−0.953288	−	0.302062i
							\(3\)	−0.281677	−	1.86880i	−0.162626	−	1.07895i	−0.910018	−	0.414570i	\(-0.863932\pi\)
0.747391	−	0.664384i	\(-0.231306\pi\)
\(5\)	2.47215	+	0.970246i	1.10558	+	0.433907i	0.846737	−	0.532012i	\(-0.178564\pi\)
							0.258841	+	0.965920i	\(0.416659\pi\)
\(7\)	−2.31027	−	1.28943i	−0.873202	−	0.487359i
							\(11\)	0.464619	−	1.50626i	0.140088	−	0.454154i	−0.858156	−	0.513389i	\(-0.828390\pi\)
0.998244	+	0.0592347i	\(0.0188660\pi\)
\(13\)	0.149412	−	0.0341022i	0.0414394	−	0.00945826i	−0.201751	−	0.979437i	\(-0.564663\pi\)
							0.243190	+	0.969979i	\(0.421806\pi\)
\(17\)	0.505930	−	0.344937i	0.122706	−	0.0836595i	−0.500412	−	0.865787i	\(-0.666818\pi\)
							0.623118	+	0.782128i	\(0.285866\pi\)
\(19\)	6.85602	−	3.95832i	1.57288	−	0.908102i	0.577064	−	0.816699i	\(-0.304198\pi\)
							0.995814	−	0.0914030i	\(-0.0291351\pi\)
\(23\)	−2.07435	−	1.41427i	−0.432533	−	0.294896i	0.327414	−	0.944881i	\(-0.393823\pi\)
							−0.759947	+	0.649985i	\(0.774775\pi\)
\(29\)	1.12566	−	2.33746i	0.209030	−	0.434055i	−0.769924	−	0.638135i	\(-0.779706\pi\)
							0.978954	+	0.204080i	\(0.0654203\pi\)
\(31\)	−2.17356	+	3.76472i	−0.390383	+	0.676164i	−0.992500	−	0.122244i	\(-0.960991\pi\)
							0.602117	+	0.798408i	\(0.294324\pi\)
\(37\)	0.622332	−	0.0466374i	0.102311	−	0.00766713i	−0.0234763	−	0.999724i	\(-0.507473\pi\)
							0.125787	+	0.992057i	\(0.459854\pi\)
\(41\)	−5.51088	−	6.91042i	−0.860654	−	1.07923i	−0.996082	−	0.0884390i	\(-0.971812\pi\)
							0.135427	−	0.990787i	\(-0.456759\pi\)
\(43\)	−3.95280	−	3.15225i	−0.602796	−	0.480714i	0.273900	−	0.961758i	\(-0.411686\pi\)
							−0.876696	+	0.481045i	\(0.840258\pi\)
\(47\)	−4.42087	+	4.10196i	−0.644850	+	0.598333i	−0.933272	−	0.359170i	\(-0.883060\pi\)
							0.288422	+	0.957503i	\(0.406869\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.109.4	✓	648
8.5	even	2	inner	392.2.z.a.109.30	yes	648
49.9	even	21	inner	392.2.z.a.205.30	yes	648
392.205	even	42	inner	392.2.z.a.205.4	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.109.4	✓	648	1.1	even	1	trivial
392.2.z.a.109.30	yes	648	8.5	even	2	inner
392.2.z.a.205.4	yes	648	392.205	even	42	inner
392.2.z.a.205.30	yes	648	49.9	even	21	inner