Embedded newform 116.2.l.b.39.12

• Label: 116.2.l.b.39.12
• Level: $116$
• Weight: $2$
• Character: 116.39
• 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			39.12
Character	\(\chi\)	\(=\)	116.39
Dual form			116.2.l.b.3.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.38106 + 0.304418i) q^{2} +(-0.677913 + 1.93736i) q^{3} +(1.81466 + 0.840840i) q^{4} +(-0.936960 - 0.213855i) q^{5} +(-1.52601 + 2.46925i) q^{6} +(-1.14800 - 2.38385i) q^{7} +(2.25019 + 1.71367i) q^{8} +(-0.948316 - 0.756257i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 14 q^{4} - 28 q^{5} - 14 q^{6} - 12 q^{8} - 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(59\)	\(89\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{23}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.38106	+	0.304418i	0.976558	+	0.215256i
							\(3\)	−0.677913	+	1.93736i	−0.391393	+	1.11854i	0.564655	+	0.825327i	\(0.309009\pi\)
−0.956048	+	0.293210i	\(0.905276\pi\)
\(5\)	−0.936960	−	0.213855i	−0.419021	−	0.0956389i	0.00781142	−	0.999969i	\(-0.497514\pi\)
							−0.426833	+	0.904331i	\(0.640371\pi\)
\(7\)	−1.14800	−	2.38385i	−0.433904	−	0.901010i	−0.997202	−	0.0747566i	\(-0.976182\pi\)
							0.563298	−	0.826254i	\(-0.309532\pi\)
\(11\)	−1.04051	−	0.117238i	−0.313726	−	0.0353484i	−0.0463023	−	0.998927i	\(-0.514744\pi\)
							−0.267424	+	0.963579i	\(0.586172\pi\)
\(13\)	3.43111	−	2.73622i	0.951618	−	0.758890i	−0.0189245	−	0.999821i	\(-0.506024\pi\)
							0.970542	+	0.240931i	\(0.0774528\pi\)
\(17\)	−2.30498	−	2.30498i	−0.559040	−	0.559040i	0.369994	−	0.929034i	\(-0.379360\pi\)
							−0.929034	+	0.369994i	\(0.879360\pi\)
\(19\)	0.0184759	−	0.00646499i	0.00423866	−	0.00148317i	−0.328159	−	0.944623i	\(-0.606428\pi\)
							0.332398	+	0.943139i	\(0.392142\pi\)
\(23\)	2.77679	−	0.633783i	0.579000	−	0.132153i	0.0770160	−	0.997030i	\(-0.475461\pi\)
							0.501984	+	0.864877i	\(0.332604\pi\)
\(29\)	−1.11694	+	5.26806i	−0.207410	+	0.978254i
							\(31\)	−5.46908	−	8.70399i	−0.982276	−	1.56328i	−0.816681	−	0.577089i	\(-0.804189\pi\)
−0.165594	−	0.986194i	\(-0.552954\pi\)
\(37\)	1.09847	+	9.74922i	0.180588	+	1.60276i	0.677876	+	0.735176i	\(0.262901\pi\)
							−0.497288	+	0.867586i	\(0.665671\pi\)
\(41\)	−7.23701	+	7.23701i	−1.13023	+	1.13023i	−0.140092	+	0.990138i	\(0.544740\pi\)
							−0.990138	+	0.140092i	\(0.955260\pi\)
\(43\)	3.00595	+	1.88876i	0.458403	+	0.288033i	0.741371	−	0.671096i	\(-0.234176\pi\)
							−0.282968	+	0.959129i	\(0.591319\pi\)
\(47\)	0.163506	−	1.45116i	0.0238498	−	0.211673i	−0.976135	−	0.217164i	\(-0.930319\pi\)
							0.999985	+	0.00549105i	\(0.00174786\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.l.b.3.9	✓	144	29.3	odd	28	inner
116.2.l.b.3.12	yes	144	116.3	even	28	inner
116.2.l.b.39.9	yes	144	4.3	odd	2	inner
116.2.l.b.39.12	yes	144	1.1	even	1	trivial