Embedded newform 392.2.z.a.261.34

• Label: 392.2.z.a.261.34
• 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			261.34
Character	\(\chi\)	\(=\)	392.261
Dual form			392.2.z.a.389.34

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.502291 - 1.32201i) q^{2} +(-1.81458 + 0.135984i) q^{3} +(-1.49541 - 1.32807i) q^{4} +(-2.08228 - 3.05414i) q^{5} +(-0.731676 + 2.46719i) q^{6} +(2.47660 - 0.930847i) q^{7} +(-2.50684 + 1.30986i) q^{8} +(0.307720 - 0.0463814i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.502291	−	1.32201i	0.355174	−	0.934800i
							\(3\)	−1.81458	+	0.135984i	−1.04765	+	0.0785104i	−0.587414	−	0.809287i	\(-0.699854\pi\)
−0.460235	+	0.887797i	\(0.652235\pi\)
\(5\)	−2.08228	−	3.05414i	−0.931223	−	1.36585i	−0.930673	−	0.365853i	\(-0.880777\pi\)
							−0.000550069	−	1.00000i	\(-0.500175\pi\)
\(7\)	2.47660	−	0.930847i	0.936065	−	0.351827i
							\(11\)	−0.312422	+	2.07278i	−0.0941986	+	0.624967i	0.891059	+	0.453888i	\(0.149963\pi\)
−0.985257	+	0.171079i	\(0.945275\pi\)
\(13\)	0.719862	+	0.574071i	0.199654	+	0.159219i	0.718213	−	0.695823i	\(-0.244960\pi\)
							−0.518560	+	0.855041i	\(0.673532\pi\)
\(17\)	−3.72062	+	1.14766i	−0.902382	+	0.278348i	−0.711020	−	0.703172i	\(-0.751767\pi\)
							−0.191362	+	0.981520i	\(0.561290\pi\)
\(19\)	−2.81306	−	1.62412i	−0.645359	−	0.372598i	0.141317	−	0.989964i	\(-0.454866\pi\)
							−0.786676	+	0.617366i	\(0.788200\pi\)
\(23\)	0.979481	+	0.302130i	0.204236	+	0.0629984i	0.395185	−	0.918601i	\(-0.370680\pi\)
							−0.190949	+	0.981600i	\(0.561157\pi\)
\(29\)	1.18383	+	0.270201i	0.219831	+	0.0501751i	0.331018	−	0.943624i	\(-0.392608\pi\)
							−0.111186	+	0.993800i	\(0.535465\pi\)
\(31\)	−0.920516	−	1.59438i	−0.165330	−	0.286359i	0.771443	−	0.636299i	\(-0.219535\pi\)
							−0.936772	+	0.349940i	\(0.886202\pi\)
\(37\)	−5.85553	+	6.31076i	−0.962643	+	1.03748i	0.0366291	+	0.999329i	\(0.488338\pi\)
							−0.999272	+	0.0381533i	\(0.987852\pi\)
\(41\)	−10.6278	−	5.11808i	−1.65978	−	0.799309i	−0.998803	−	0.0489055i	\(-0.984427\pi\)
							−0.660980	−	0.750404i	\(-0.729859\pi\)
\(43\)	−0.114663	−	0.238100i	−0.0174859	−	0.0363099i	0.892043	−	0.451951i	\(-0.149272\pi\)
							−0.909529	+	0.415641i	\(0.863557\pi\)
\(47\)	−1.68120	−	4.28363i	−0.245228	−	0.624832i	0.754268	−	0.656567i	\(-0.227992\pi\)
							−0.999496	+	0.0317349i	\(0.989897\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.261.34	yes	648
8.5	even	2	inner	392.2.z.a.261.8	✓	648
49.46	even	21	inner	392.2.z.a.389.8	yes	648
392.389	even	42	inner	392.2.z.a.389.34	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.261.8	✓	648	8.5	even	2	inner
392.2.z.a.261.34	yes	648	1.1	even	1	trivial
392.2.z.a.389.8	yes	648	49.46	even	21	inner
392.2.z.a.389.34	yes	648	392.389	even	42	inner