Embedded newform 392.2.z.a.261.46

• Label: 392.2.z.a.261.46
• Level: $392$
• Weight: $2$
• Character: 392.261
• 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			261.46
Character	\(\chi\)	\(=\)	392.261
Dual form			392.2.z.a.389.46

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.16974 + 0.794808i) q^{2} +(1.44692 - 0.108432i) q^{3} +(0.736560 + 1.85943i) q^{4} +(-0.623207 - 0.914077i) q^{5} +(1.77869 + 1.02319i) q^{6} +(2.16640 + 1.51879i) q^{7} +(-0.616310 + 2.76046i) q^{8} +(-0.884679 + 0.133344i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{10}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.16974	+	0.794808i	0.827128	+	0.562014i
							\(3\)	1.44692	−	0.108432i	0.835378	−	0.0626030i	0.349831	−	0.936813i	\(-0.386239\pi\)
0.485548	+	0.874210i	\(0.338620\pi\)
\(5\)	−0.623207	−	0.914077i	−0.278707	−	0.408788i	0.661205	−	0.750205i	\(-0.270045\pi\)
							−0.939912	+	0.341418i	\(0.889093\pi\)
\(7\)	2.16640	+	1.51879i	0.818822	+	0.574047i
							\(11\)	0.746459	−	4.95243i	0.225066	−	1.49321i	−0.535822	−	0.844331i	\(-0.679998\pi\)
0.760888	−	0.648883i	\(-0.224764\pi\)
\(13\)	2.84155	+	2.26606i	0.788105	+	0.628493i	0.932559	−	0.361019i	\(-0.117571\pi\)
							−0.144453	+	0.989512i	\(0.546142\pi\)
\(17\)	−4.03326	+	1.24410i	−0.978209	+	0.301737i	−0.742322	−	0.670043i	\(-0.766276\pi\)
							−0.235887	+	0.971781i	\(0.575799\pi\)
\(19\)	0.400263	+	0.231092i	0.0918267	+	0.0530162i	0.545210	−	0.838299i	\(-0.316450\pi\)
							−0.453383	+	0.891316i	\(0.649783\pi\)
\(23\)	−4.33439	−	1.33698i	−0.903783	−	0.278780i	−0.192178	−	0.981360i	\(-0.561555\pi\)
							−0.711605	+	0.702580i	\(0.752031\pi\)
\(29\)	4.34685	+	0.992141i	0.807190	+	0.184236i	0.606153	−	0.795348i	\(-0.292712\pi\)
							0.201037	+	0.979584i	\(0.435569\pi\)
\(31\)	0.283877	+	0.491690i	0.0509859	+	0.0883101i	0.890392	−	0.455195i	\(-0.150430\pi\)
							−0.839406	+	0.543505i	\(0.817097\pi\)
\(37\)	5.64307	−	6.08178i	0.927714	−	0.999838i	−0.0722826	−	0.997384i	\(-0.523028\pi\)
							0.999997	−	0.00245410i	\(-0.000781165\pi\)
\(41\)	−11.0023	−	5.29841i	−1.71826	−	0.827472i	−0.989813	−	0.142374i	\(-0.954526\pi\)
							−0.728451	−	0.685098i	\(-0.759759\pi\)
\(43\)	−0.917963	−	1.90617i	−0.139988	−	0.290688i	0.819174	−	0.573545i	\(-0.194432\pi\)
							−0.959162	+	0.282857i	\(0.908718\pi\)
\(47\)	3.02054	+	7.69620i	0.440591	+	1.12261i	0.962902	+	0.269851i	\(0.0869746\pi\)
							−0.522311	+	0.852755i	\(0.674930\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.261.46	yes	648
8.5	even	2	inner	392.2.z.a.261.24	✓	648
49.46	even	21	inner	392.2.z.a.389.24	yes	648
392.389	even	42	inner	392.2.z.a.389.46	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.261.24	✓	648	8.5	even	2	inner
392.2.z.a.261.46	yes	648	1.1	even	1	trivial
392.2.z.a.389.24	yes	648	49.46	even	21	inner
392.2.z.a.389.46	yes	648	392.389	even	42	inner