Embedded newform 116.2.l.b.27.8

• Label: 116.2.l.b.27.8
• Level: $116$
• Weight: $2$
• Character: 116.27
• Analytic conductor: $0.926$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 116 = 2^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	116.l (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.926264663447\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(12\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			27.8
Character	\(\chi\)	\(=\)	116.27
Dual form			116.2.l.b.43.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.417620 + 1.35115i) q^{2} +(1.28319 + 0.806282i) q^{3} +(-1.65119 + 1.12853i) q^{4} +(1.86371 - 1.48626i) q^{5} +(-0.553518 + 2.07050i) q^{6} +(-0.787001 + 0.179628i) q^{7} +(-2.21438 - 1.75970i) q^{8} +(-0.305164 - 0.633680i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 14 q^{4} - 28 q^{5} - 14 q^{6} - 12 q^{8} - 28 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/116\mathbb{Z}\right)^\times\).

\(n\)	\(59\)	\(89\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{15}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.417620	+	1.35115i	0.295302	+	0.955404i
							\(3\)	1.28319	+	0.806282i	0.740850	+	0.465507i	0.848843	−	0.528645i	\(-0.177300\pi\)
−0.107993	+	0.994152i	\(0.534442\pi\)
\(5\)	1.86371	−	1.48626i	0.833478	−	0.664677i	−0.110794	−	0.993843i	\(-0.535339\pi\)
							0.944272	+	0.329167i	\(0.106768\pi\)
\(7\)	−0.787001	+	0.179628i	−0.297458	+	0.0678930i	−0.368646	−	0.929570i	\(-0.620178\pi\)
							0.0711871	+	0.997463i	\(0.477321\pi\)
\(11\)	−3.32550	+	1.16364i	−1.00268	+	0.350851i	−0.781177	−	0.624310i	\(-0.785380\pi\)
							−0.221498	+	0.975161i	\(0.571095\pi\)
\(13\)	0.140123	−	0.290969i	0.0388632	−	0.0807003i	−0.880633	−	0.473799i	\(-0.842882\pi\)
							0.919496	+	0.393098i	\(0.128597\pi\)
\(17\)	−1.00149	−	1.00149i	−0.242898	−	0.242898i	0.575150	−	0.818048i	\(-0.304944\pi\)
							−0.818048	+	0.575150i	\(0.804944\pi\)
\(19\)	−0.955286	−	1.52033i	−0.219158	−	0.348787i	0.718991	−	0.695019i	\(-0.244604\pi\)
							−0.938149	+	0.346232i	\(0.887461\pi\)
\(23\)	6.35218	+	5.06569i	1.32452	+	1.05627i	0.993641	+	0.112597i	\(0.0359171\pi\)
							0.330880	+	0.943673i	\(0.392654\pi\)
\(29\)	4.54129	−	2.89426i	0.843296	−	0.537450i
							\(31\)	−3.77766	−	0.425640i	−0.678488	−	0.0764472i	−0.234006	−	0.972235i	\(-0.575184\pi\)
−0.444482	+	0.895788i	\(0.646612\pi\)
\(37\)	−3.52641	+	10.0779i	−0.579739	+	1.65680i	0.162536	+	0.986703i	\(0.448033\pi\)
							−0.742275	+	0.670096i	\(0.766253\pi\)
\(41\)	0.178656	−	0.178656i	0.0279013	−	0.0279013i	−0.693018	−	0.720920i	\(-0.743720\pi\)
							0.720920	+	0.693018i	\(0.243720\pi\)
\(43\)	−0.783440	−	6.95322i	−0.119473	−	1.06036i	−0.900198	−	0.435481i	\(-0.856578\pi\)
							0.780725	−	0.624875i	\(-0.214850\pi\)
\(47\)	−3.05870	−	8.74125i	−0.446157	−	1.27504i	−0.919126	−	0.393964i	\(-0.871104\pi\)
							0.472969	−	0.881079i	\(-0.343182\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.2.l.b.27.8	yes	144
4.3	odd	2	inner	116.2.l.b.27.4	✓	144
29.14	odd	28	inner	116.2.l.b.43.4	yes	144
116.43	even	28	inner	116.2.l.b.43.8	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.l.b.27.4	✓	144	4.3	odd	2	inner
116.2.l.b.27.8	yes	144	1.1	even	1	trivial
116.2.l.b.43.4	yes	144	29.14	odd	28	inner
116.2.l.b.43.8	yes	144	116.43	even	28	inner