Embedded newform 392.2.z.a.109.15

• Label: 392.2.z.a.109.15
• 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			109.15
Character	\(\chi\)	\(=\)	392.109
Dual form			392.2.z.a.205.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.897288 - 1.09310i) q^{2} +(-0.131109 - 0.869851i) q^{3} +(-0.389748 + 1.96166i) q^{4} +(-0.989897 - 0.388506i) q^{5} +(-0.833194 + 0.923822i) q^{6} +(2.48187 - 0.916698i) q^{7} +(2.49401 - 1.33414i) q^{8} +(2.12727 - 0.656175i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.897288	−	1.09310i	−0.634478	−	0.772941i
							\(3\)	−0.131109	−	0.869851i	−0.0756958	−	0.502209i	−0.994265	−	0.106948i	\(-0.965892\pi\)
0.918569	−	0.395261i	\(-0.129346\pi\)
\(5\)	−0.989897	−	0.388506i	−0.442695	−	0.173745i	0.133515	−	0.991047i	\(-0.457374\pi\)
							−0.576210	+	0.817302i	\(0.695469\pi\)
\(7\)	2.48187	−	0.916698i	0.938058	−	0.346479i
							\(11\)	−0.522073	+	1.69252i	−0.157411	+	0.510314i	−0.999582	−	0.0288961i	\(-0.990801\pi\)
0.842171	+	0.539210i	\(0.181277\pi\)
\(13\)	0.782831	−	0.178676i	0.217118	−	0.0495558i	−0.112578	−	0.993643i	\(-0.535911\pi\)
							0.329696	+	0.944087i	\(0.393054\pi\)
\(17\)	−0.935755	+	0.637987i	−0.226954	+	0.154735i	−0.671459	−	0.741042i	\(-0.734332\pi\)
							0.444505	+	0.895776i	\(0.353380\pi\)
\(19\)	6.50293	−	3.75447i	1.49187	−	0.861334i	0.491918	−	0.870642i	\(-0.336296\pi\)
							0.999957	+	0.00930734i	\(0.00296266\pi\)
\(23\)	−4.67431	−	3.18689i	−0.974662	−	0.664513i	−0.0324298	−	0.999474i	\(-0.510325\pi\)
							−0.942232	+	0.334961i	\(0.891277\pi\)
\(29\)	0.367178	−	0.762454i	0.0681833	−	0.141584i	−0.864092	−	0.503334i	\(-0.832106\pi\)
							0.932275	+	0.361750i	\(0.117821\pi\)
\(31\)	4.28471	−	7.42134i	0.769557	−	1.33291i	−0.168246	−	0.985745i	\(-0.553810\pi\)
							0.937803	−	0.347167i	\(-0.112856\pi\)
\(37\)	−8.15444	+	0.611091i	−1.34058	+	0.100463i	−0.725621	−	0.688095i	\(-0.758447\pi\)
							−0.614961	+	0.788558i	\(0.710828\pi\)
\(41\)	3.46635	+	4.34666i	0.541352	+	0.678834i	0.974989	−	0.222253i	\(-0.0713412\pi\)
							−0.433636	+	0.901088i	\(0.642770\pi\)
\(43\)	7.41324	+	5.91186i	1.13051	+	0.901551i	0.995999	−	0.0893614i	\(-0.0284826\pi\)
							0.134510	+	0.990912i	\(0.457054\pi\)
\(47\)	−2.23058	+	2.06968i	−0.325364	+	0.301894i	−0.825894	−	0.563826i	\(-0.809329\pi\)
							0.500530	+	0.865719i	\(0.333139\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.109.15	✓	648
8.5	even	2	inner	392.2.z.a.109.41	yes	648
49.9	even	21	inner	392.2.z.a.205.41	yes	648
392.205	even	42	inner	392.2.z.a.205.15	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.109.15	✓	648	1.1	even	1	trivial
392.2.z.a.109.41	yes	648	8.5	even	2	inner
392.2.z.a.205.15	yes	648	392.205	even	42	inner
392.2.z.a.205.41	yes	648	49.9	even	21	inner