Embedded newform 116.2.l.b.79.3

• Label: 116.2.l.b.79.3
• Level: $116$
• Weight: $2$
• Character: 116.79
• 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			79.3
Character	\(\chi\)	\(=\)	116.79
Dual form			116.2.l.b.47.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18126 - 0.777580i) q^{2} +(-0.0533353 + 0.473364i) q^{3} +(0.790739 + 1.83704i) q^{4} +(-1.20802 - 2.50849i) q^{5} +(0.431081 - 0.517693i) q^{6} +(2.02500 - 1.61489i) q^{7} +(0.494382 - 2.78489i) q^{8} +(2.70355 + 0.617069i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 14 q^{4} - 28 q^{5} - 14 q^{6} - 12 q^{8} - 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(59\)	\(89\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{17}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18126	−	0.777580i	−0.835275	−	0.549832i
							\(3\)	−0.0533353	+	0.473364i	−0.0307932	+	0.273297i	0.968875	+	0.247549i	\(0.0796252\pi\)
−0.999668	+	0.0257477i	\(0.991803\pi\)
\(5\)	−1.20802	−	2.50849i	−0.540245	−	1.12183i	−0.975189	−	0.221373i	\(-0.928946\pi\)
							0.434944	−	0.900457i	\(-0.356768\pi\)
\(7\)	2.02500	−	1.61489i	0.765379	−	0.610369i	−0.161003	−	0.986954i	\(-0.551473\pi\)
							0.926382	+	0.376584i	\(0.122902\pi\)
\(11\)	2.40720	−	3.83104i	0.725799	−	1.15510i	−0.256275	−	0.966604i	\(-0.582495\pi\)
							0.982074	−	0.188498i	\(-0.0603620\pi\)
\(13\)	−5.03606	+	1.14945i	−1.39675	+	0.318800i	−0.853641	−	0.520862i	\(-0.825611\pi\)
							−0.543111	+	0.839661i	\(0.682754\pi\)
\(17\)	2.07849	−	2.07849i	0.504108	−	0.504108i	−0.408604	−	0.912712i	\(-0.633984\pi\)
							0.912712	+	0.408604i	\(0.133984\pi\)
\(19\)	1.85436	−	0.208936i	0.425418	−	0.0479331i	0.103340	−	0.994646i	\(-0.467047\pi\)
							0.322079	+	0.946713i	\(0.395618\pi\)
\(23\)	−3.78278	+	7.85502i	−0.788764	+	1.63789i	−0.0187794	+	0.999824i	\(0.505978\pi\)
							−0.769985	+	0.638062i	\(0.779736\pi\)
\(29\)	−2.79831	+	4.60103i	−0.519633	+	0.854390i
							\(31\)	4.30248	+	1.50550i	0.772748	+	0.270396i	0.687714	−	0.725982i	\(-0.258614\pi\)
0.0850339	+	0.996378i	\(0.472900\pi\)
\(37\)	2.03563	−	1.27907i	0.334655	−	0.210278i	−0.354212	−	0.935165i	\(-0.615251\pi\)
							0.688868	+	0.724887i	\(0.258108\pi\)
\(41\)	−0.419501	−	0.419501i	−0.0655151	−	0.0655151i	0.673590	−	0.739105i	\(-0.264751\pi\)
							−0.739105	+	0.673590i	\(0.764751\pi\)
\(43\)	1.14761	+	3.27968i	0.175009	+	0.500146i	0.997658	−	0.0684034i	\(-0.0217905\pi\)
							−0.822649	+	0.568550i	\(0.807505\pi\)
\(47\)	−8.48404	−	5.33087i	−1.23752	−	0.777588i	−0.255585	−	0.966787i	\(-0.582268\pi\)
							−0.981939	+	0.189199i	\(0.939411\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.2.l.b.79.3	yes	144
4.3	odd	2	inner	116.2.l.b.79.12	yes	144
29.18	odd	28	inner	116.2.l.b.47.12	yes	144
116.47	even	28	inner	116.2.l.b.47.3	✓	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.l.b.47.3	✓	144	116.47	even	28	inner
116.2.l.b.47.12	yes	144	29.18	odd	28	inner
116.2.l.b.79.3	yes	144	1.1	even	1	trivial
116.2.l.b.79.12	yes	144	4.3	odd	2	inner