Embedded newform 116.2.l.b.11.1

• Label: 116.2.l.b.11.1
• Level: $116$
• Weight: $2$
• Character: 116.11
• Analytic conductor: $0.926$
• Analytic rank: $0$
• Dimension: $144$
• 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			11.1
Character	\(\chi\)	\(=\)	116.11
Dual form			116.2.l.b.95.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38077 + 0.305727i) q^{2} +(2.41537 - 0.272147i) q^{3} +(1.81306 - 0.844280i) q^{4} +(0.786249 - 1.63266i) q^{5} +(-3.25188 + 1.11422i) q^{6} +(-2.13423 - 1.70199i) q^{7} +(-2.24530 + 1.72006i) q^{8} +(2.83518 - 0.647112i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 14 q^{4} - 28 q^{5} - 14 q^{6} - 12 q^{8} - 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(59\)	\(89\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{25}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.38077	+	0.305727i	−0.976353	+	0.216182i
							\(3\)	2.41537	−	0.272147i	1.39452	−	0.157124i	0.617484	−	0.786584i	\(-0.288152\pi\)
0.777033	+	0.629459i	\(0.216724\pi\)
\(5\)	0.786249	−	1.63266i	0.351621	−	0.730149i	−0.647879	−	0.761743i	\(-0.724344\pi\)
							0.999501	+	0.0315937i	\(0.0100583\pi\)
\(7\)	−2.13423	−	1.70199i	−0.806663	−	0.643292i	0.130788	−	0.991410i	\(-0.458249\pi\)
							−0.937451	+	0.348118i	\(0.886821\pi\)
\(11\)	−3.28303	+	2.06287i	−0.989872	+	0.621978i	−0.926612	−	0.376020i	\(-0.877293\pi\)
							−0.0632599	+	0.997997i	\(0.520150\pi\)
\(13\)	4.92506	+	1.12411i	1.36597	+	0.311773i	0.841776	−	0.539828i	\(-0.181511\pi\)
							0.524191	+	0.851601i	\(0.324368\pi\)
\(17\)	−1.25670	+	1.25670i	−0.304794	+	0.304794i	−0.842886	−	0.538092i	\(-0.819145\pi\)
							0.538092	+	0.842886i	\(0.319145\pi\)
\(19\)	−0.576157	+	5.11353i	−0.132179	+	1.17313i	0.736038	+	0.676941i	\(0.236695\pi\)
							−0.868217	+	0.496185i	\(0.834734\pi\)
\(23\)	2.02375	+	4.20237i	0.421982	+	0.876254i	0.998258	+	0.0589992i	\(0.0187909\pi\)
							−0.576276	+	0.817255i	\(0.695495\pi\)
\(29\)	−5.20494	−	1.38152i	−0.966533	−	0.256543i
							\(31\)	−3.32615	−	9.50559i	−0.597394	−	1.70725i	−0.701483	−	0.712687i	\(-0.747478\pi\)
0.104088	−	0.994568i	\(-0.466808\pi\)
\(37\)	0.203952	−	0.324588i	0.0335295	−	0.0533620i	−0.829539	−	0.558449i	\(-0.811397\pi\)
							0.863068	+	0.505087i	\(0.168540\pi\)
\(41\)	0.984322	+	0.984322i	0.153725	+	0.153725i	0.779780	−	0.626054i	\(-0.215331\pi\)
							−0.626054	+	0.779780i	\(0.715331\pi\)
\(43\)	5.31289	+	1.85906i	0.810209	+	0.283504i	0.703417	−	0.710777i	\(-0.251657\pi\)
							0.106791	+	0.994281i	\(0.465942\pi\)
\(47\)	−1.43724	−	2.28736i	−0.209643	−	0.333646i	0.725306	−	0.688427i	\(-0.241698\pi\)
							−0.934949	+	0.354781i	\(0.884555\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.2.l.b.11.1	✓	144
4.3	odd	2	inner	116.2.l.b.11.3	yes	144
29.8	odd	28	inner	116.2.l.b.95.3	yes	144
116.95	even	28	inner	116.2.l.b.95.1	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.l.b.11.1	✓	144	1.1	even	1	trivial
116.2.l.b.11.3	yes	144	4.3	odd	2	inner
116.2.l.b.95.1	yes	144	116.95	even	28	inner
116.2.l.b.95.3	yes	144	29.8	odd	28	inner