Embedded newform 392.2.z.a.109.29

• Label: 392.2.z.a.109.29
• 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			109.29
Character	\(\chi\)	\(=\)	392.109
Dual form			392.2.z.a.205.29

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0200803 - 1.41407i) q^{2} +(-0.0214053 - 0.142015i) q^{3} +(-1.99919 + 0.0567899i) q^{4} +(1.13882 + 0.446956i) q^{5} +(-0.200389 + 0.0331203i) q^{6} +(0.813208 + 2.51768i) q^{7} +(0.120449 + 2.82586i) q^{8} +(2.84701 - 0.878186i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0200803	−	1.41407i	−0.0141989	−	0.999899i
							\(3\)	−0.0214053	−	0.142015i	−0.0123583	−	0.0819923i	0.981801	−	0.189910i	\(-0.0608197\pi\)
−0.994160	+	0.107918i	\(0.965582\pi\)
\(5\)	1.13882	+	0.446956i	0.509298	+	0.199885i	0.606054	−	0.795423i	\(-0.292752\pi\)
							−0.0967563	+	0.995308i	\(0.530847\pi\)
\(7\)	0.813208	+	2.51768i	0.307364	+	0.951592i
							\(11\)	−0.161802	+	0.524550i	−0.0487852	+	0.158158i	−0.976688	−	0.214665i	\(-0.931134\pi\)
0.927902	+	0.372823i	\(0.121610\pi\)
\(13\)	2.40919	−	0.549882i	0.668190	−	0.152510i	0.125048	−	0.992151i	\(-0.460092\pi\)
							0.543142	+	0.839641i	\(0.317235\pi\)
\(17\)	1.10075	−	0.750482i	0.266972	−	0.182019i	−0.422434	−	0.906394i	\(-0.638824\pi\)
							0.689406	+	0.724375i	\(0.257872\pi\)
\(19\)	0.491203	−	0.283596i	0.112690	−	0.0650614i	−0.442596	−	0.896721i	\(-0.645942\pi\)
							0.555285	+	0.831660i	\(0.312609\pi\)
\(23\)	1.26229	+	0.860612i	0.263205	+	0.179450i	0.687731	−	0.725966i	\(-0.258607\pi\)
							−0.424526	+	0.905416i	\(0.639559\pi\)
\(29\)	−0.690841	+	1.43455i	−0.128286	+	0.266388i	−0.955212	−	0.295922i	\(-0.904373\pi\)
							0.826926	+	0.562310i	\(0.190087\pi\)
\(31\)	0.121176	−	0.209883i	0.0217638	−	0.0376961i	−0.854938	−	0.518730i	\(-0.826405\pi\)
							0.876702	+	0.481034i	\(0.159738\pi\)
\(37\)	4.95589	−	0.371393i	0.814743	−	0.0610566i	0.339162	−	0.940728i	\(-0.389857\pi\)
							0.475582	+	0.879672i	\(0.342238\pi\)
\(41\)	−5.21212	−	6.53579i	−0.813996	−	1.02072i	−0.999276	−	0.0380334i	\(-0.987891\pi\)
							0.185280	−	0.982686i	\(-0.440681\pi\)
\(43\)	5.20829	+	4.15347i	0.794257	+	0.633399i	0.934196	−	0.356761i	\(-0.116119\pi\)
							−0.139938	+	0.990160i	\(0.544690\pi\)
\(47\)	−2.83225	+	2.62794i	−0.413126	+	0.383325i	−0.859173	−	0.511684i	\(-0.829022\pi\)
							0.446048	+	0.895009i	\(0.352831\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.109.29	✓	648
8.5	even	2	inner	392.2.z.a.109.54	yes	648
49.9	even	21	inner	392.2.z.a.205.54	yes	648
392.205	even	42	inner	392.2.z.a.205.29	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.109.29	✓	648	1.1	even	1	trivial
392.2.z.a.109.54	yes	648	8.5	even	2	inner
392.2.z.a.205.29	yes	648	392.205	even	42	inner
392.2.z.a.205.54	yes	648	49.9	even	21	inner