Embedded newform 116.2.l.b.3.9

• Label: 116.2.l.b.3.9
• Level: $116$
• Weight: $2$
• Character: 116.3
• 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			3.9
Character	\(\chi\)	\(=\)	116.3
Dual form			116.2.l.b.39.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.324949 + 1.37638i) q^{2} +(0.677913 + 1.93736i) q^{3} +(-1.78882 + 0.894502i) q^{4} +(-0.936960 + 0.213855i) q^{5} +(-2.44625 + 1.56261i) q^{6} +(1.14800 - 2.38385i) q^{7} +(-1.81244 - 2.17142i) q^{8} +(-0.948316 + 0.756257i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 14 q^{4} - 28 q^{5} - 14 q^{6} - 12 q^{8} - 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(59\)	\(89\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.324949	+	1.37638i	0.229773	+	0.973244i
							\(3\)	0.677913	+	1.93736i	0.391393	+	1.11854i	0.956048	+	0.293210i	\(0.0947235\pi\)
−0.564655	+	0.825327i	\(0.690991\pi\)
\(5\)	−0.936960	+	0.213855i	−0.419021	+	0.0956389i	−0.426833	−	0.904331i	\(-0.640371\pi\)
							0.00781142	+	0.999969i	\(0.497514\pi\)
\(7\)	1.14800	−	2.38385i	0.433904	−	0.901010i	−0.563298	−	0.826254i	\(-0.690468\pi\)
							0.997202	−	0.0747566i	\(-0.0238180\pi\)
\(11\)	1.04051	−	0.117238i	0.313726	−	0.0353484i	0.0463023	−	0.998927i	\(-0.485256\pi\)
							0.267424	+	0.963579i	\(0.413828\pi\)
\(13\)	3.43111	+	2.73622i	0.951618	+	0.758890i	0.970542	−	0.240931i	\(-0.0774528\pi\)
							−0.0189245	+	0.999821i	\(0.506024\pi\)
\(17\)	−2.30498	+	2.30498i	−0.559040	+	0.559040i	−0.929034	−	0.369994i	\(-0.879360\pi\)
							0.369994	+	0.929034i	\(0.379360\pi\)
\(19\)	−0.0184759	−	0.00646499i	−0.00423866	−	0.00148317i	0.328159	−	0.944623i	\(-0.393572\pi\)
							−0.332398	+	0.943139i	\(0.607858\pi\)
\(23\)	−2.77679	−	0.633783i	−0.579000	−	0.132153i	−0.0770160	−	0.997030i	\(-0.524539\pi\)
							−0.501984	+	0.864877i	\(0.667396\pi\)
\(29\)	−1.11694	−	5.26806i	−0.207410	−	0.978254i
							\(31\)	5.46908	−	8.70399i	0.982276	−	1.56328i	0.165594	−	0.986194i	\(-0.447046\pi\)
0.816681	−	0.577089i	\(-0.195811\pi\)
\(37\)	1.09847	−	9.74922i	0.180588	−	1.60276i	−0.497288	−	0.867586i	\(-0.665671\pi\)
							0.677876	−	0.735176i	\(-0.262901\pi\)
\(41\)	−7.23701	−	7.23701i	−1.13023	−	1.13023i	−0.990138	−	0.140092i	\(-0.955260\pi\)
							−0.140092	−	0.990138i	\(-0.544740\pi\)
\(43\)	−3.00595	+	1.88876i	−0.458403	+	0.288033i	−0.741371	−	0.671096i	\(-0.765824\pi\)
							0.282968	+	0.959129i	\(0.408681\pi\)
\(47\)	−0.163506	−	1.45116i	−0.0238498	−	0.211673i	0.976135	−	0.217164i	\(-0.0696807\pi\)
							−0.999985	+	0.00549105i	\(0.998252\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.2.l.b.3.9	✓	144
4.3	odd	2	inner	116.2.l.b.3.12	yes	144
29.10	odd	28	inner	116.2.l.b.39.12	yes	144
116.39	even	28	inner	116.2.l.b.39.9	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.l.b.3.9	✓	144	1.1	even	1	trivial
116.2.l.b.3.12	yes	144	4.3	odd	2	inner
116.2.l.b.39.9	yes	144	116.39	even	28	inner
116.2.l.b.39.12	yes	144	29.10	odd	28	inner