Embedded newform 416.2.bg.a.77.9

• Label: 416.2.bg.a.77.9
• Level: $416$
• Weight: $2$
• Character: 416.77
• Analytic conductor: $3.322$
• Analytic rank: $0$
• Dimension: $216$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.bg (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.32177672409\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(54\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			77.9
Character	\(\chi\)	\(=\)	416.77
Dual form			416.2.bg.a.389.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.21243 - 0.728017i) q^{2} +(0.979432 - 2.36456i) q^{3} +(0.939982 + 1.76534i) q^{4} +(-0.107364 - 0.259200i) q^{5} +(-2.90893 + 2.15382i) q^{6} +(-0.641321 - 0.641321i) q^{7} +(0.145536 - 2.82468i) q^{8} +(-2.51053 - 2.51053i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 8 q^{3} - 8 q^{4} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/416\mathbb{Z}\right)^\times\).

\(n\)	\(261\)	\(287\)	\(353\)
\(\chi(n)\)	\(e\left(\frac{7}{8}\right)\)	\(1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.21243	−	0.728017i	−0.857319	−	0.514786i
							\(3\)	0.979432	−	2.36456i	0.565475	−	1.36518i	−0.339858	−	0.940477i	\(-0.610379\pi\)
0.905334	−	0.424701i	\(-0.139621\pi\)
\(5\)	−0.107364	−	0.259200i	−0.0480147	−	0.115918i	0.898052	−	0.439889i	\(-0.144982\pi\)
							−0.946067	+	0.323971i	\(0.894982\pi\)
\(7\)	−0.641321	−	0.641321i	−0.242397	−	0.242397i	0.575444	−	0.817841i	\(-0.304829\pi\)
							−0.817841	+	0.575444i	\(0.804829\pi\)
\(11\)	−4.84119	+	2.00529i	−1.45967	+	0.604617i	−0.964478	−	0.264164i	\(-0.914904\pi\)
							−0.495196	+	0.868781i	\(0.664904\pi\)
\(13\)	−3.34354	−	1.34936i	−0.927330	−	0.374245i
							\(17\)		−	0.785651i		−	0.190548i	−0.995451	−	0.0952741i	\(-0.969627\pi\)
0.995451	−	0.0952741i	\(-0.0303728\pi\)
\(19\)	2.42279	−	5.84913i	0.555826	−	1.34188i	−0.357218	−	0.934021i	\(-0.616275\pi\)
							0.913044	−	0.407862i	\(-0.133725\pi\)
\(23\)	−5.58065	−	5.58065i	−1.16365	−	1.16365i	−0.983671	−	0.179976i	\(-0.942398\pi\)
							−0.179976	−	0.983671i	\(-0.557602\pi\)
\(29\)	−2.46506	+	5.95118i	−0.457750	+	1.10511i	0.511556	+	0.859250i	\(0.329069\pi\)
							−0.969306	+	0.245857i	\(0.920931\pi\)
\(31\)		−	5.28049i		−	0.948403i	−0.880416	−	0.474202i	\(-0.842737\pi\)
							0.880416	−	0.474202i	\(-0.157263\pi\)
\(37\)	2.66511	+	6.43414i	0.438141	+	1.05777i	0.976590	+	0.215108i	\(0.0690104\pi\)
							−0.538449	+	0.842658i	\(0.680990\pi\)
\(41\)	−1.04573	+	1.04573i	−0.163315	+	0.163315i	−0.784033	−	0.620719i	\(-0.786841\pi\)
							0.620719	+	0.784033i	\(0.286841\pi\)
\(43\)	1.97137	+	4.75931i	0.300631	+	0.725788i	0.999940	+	0.0109681i	\(0.00349132\pi\)
							−0.699309	+	0.714820i	\(0.746509\pi\)
\(47\)	6.33634			0.924250			0.462125	−	0.886815i	\(-0.347087\pi\)
							0.462125	+	0.886815i	\(0.347087\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.bg.a.77.9	✓	216
13.12	even	2	inner	416.2.bg.a.77.46	yes	216
32.5	even	8	inner	416.2.bg.a.389.46	yes	216
416.389	even	8	inner	416.2.bg.a.389.9	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.bg.a.77.9	✓	216	1.1	even	1	trivial
416.2.bg.a.77.46	yes	216	13.12	even	2	inner
416.2.bg.a.389.9	yes	216	416.389	even	8	inner
416.2.bg.a.389.46	yes	216	32.5	even	8	inner