Embedded newform 116.2.l.b.39.3

• Label: 116.2.l.b.39.3
• 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			39.3
Character	\(\chi\)	\(=\)	116.39
Dual form			116.2.l.b.3.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.27836 - 0.604820i) q^{2} +(0.970355 - 2.77311i) q^{3} +(1.26839 + 1.54635i) q^{4} +(3.21355 + 0.733471i) q^{5} +(-2.91769 + 2.95814i) q^{6} +(0.923943 + 1.91859i) q^{7} +(-0.686186 - 2.74393i) q^{8} +(-4.40308 - 3.51134i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 14 q^{4} - 28 q^{5} - 14 q^{6} - 12 q^{8} - 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.27836	−	0.604820i	−0.903934	−	0.427672i
							\(3\)	0.970355	−	2.77311i	0.560234	−	1.60106i	−0.219952	−	0.975511i	\(-0.570590\pi\)
0.780186	−	0.625547i	\(-0.215124\pi\)
\(5\)	3.21355	+	0.733471i	1.43714	+	0.328018i	0.868959	−	0.494884i	\(-0.164790\pi\)
							0.568183	+	0.822902i	\(0.307647\pi\)
\(7\)	0.923943	+	1.91859i	0.349218	+	0.725158i	0.999401	−	0.0346198i	\(-0.0110220\pi\)
							−0.650183	+	0.759778i	\(0.725308\pi\)
\(11\)	−2.79503	−	0.314925i	−0.842734	−	0.0949533i	−0.319967	−	0.947429i	\(-0.603672\pi\)
							−0.522767	+	0.852475i	\(0.675100\pi\)
\(13\)	−3.03299	+	2.41873i	−0.841199	+	0.670834i	−0.946180	−	0.323640i	\(-0.895093\pi\)
							0.104981	+	0.994474i	\(0.466522\pi\)
\(17\)	0.781983	+	0.781983i	0.189659	+	0.189659i	0.795549	−	0.605890i	\(-0.207183\pi\)
							−0.605890	+	0.795549i	\(0.707183\pi\)
\(19\)	−5.77904	+	2.02217i	−1.32580	+	0.463918i	−0.898017	−	0.439962i	\(-0.854992\pi\)
							−0.427786	+	0.903880i	\(0.640706\pi\)
\(23\)	2.74425	−	0.626357i	0.572215	−	0.130604i	0.0733843	−	0.997304i	\(-0.476620\pi\)
							0.498831	+	0.866699i	\(0.333763\pi\)
\(29\)	−4.44365	−	3.04204i	−0.825165	−	0.564892i
							\(31\)	−0.884788	−	1.40813i	−0.158913	−	0.252908i	0.757772	−	0.652519i	\(-0.226288\pi\)
−0.916685	+	0.399611i	\(0.869145\pi\)
\(37\)	0.599039	+	5.31662i	0.0984815	+	0.874047i	0.941695	+	0.336468i	\(0.109232\pi\)
							−0.843213	+	0.537579i	\(0.819339\pi\)
\(41\)	5.28919	−	5.28919i	0.826032	−	0.826032i	−0.160933	−	0.986965i	\(-0.551450\pi\)
							0.986965	+	0.160933i	\(0.0514504\pi\)
\(43\)	3.76303	+	2.36447i	0.573857	+	0.360578i	0.787482	−	0.616337i	\(-0.211384\pi\)
							−0.213625	+	0.976916i	\(0.568527\pi\)
\(47\)	−0.430177	+	3.81793i	−0.0627478	+	0.556902i	0.922528	+	0.385930i	\(0.126119\pi\)
							−0.985276	+	0.170972i	\(0.945309\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.2.l.b.39.3	yes	144
4.3	odd	2	inner	116.2.l.b.39.8	yes	144
29.3	odd	28	inner	116.2.l.b.3.8	yes	144
116.3	even	28	inner	116.2.l.b.3.3	✓	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.l.b.3.3	✓	144	116.3	even	28	inner
116.2.l.b.3.8	yes	144	29.3	odd	28	inner
116.2.l.b.39.3	yes	144	1.1	even	1	trivial
116.2.l.b.39.8	yes	144	4.3	odd	2	inner