Embedded newform 116.2.l.b.11.4

• Label: 116.2.l.b.11.4
• 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			11.4
Character	\(\chi\)	\(=\)	116.11
Dual form			116.2.l.b.95.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.24228 - 0.675831i) q^{2} +(-1.48114 + 0.166885i) q^{3} +(1.08651 + 1.67914i) q^{4} +(0.727418 - 1.51050i) q^{5} +(1.95278 + 0.793685i) q^{6} +(-3.42876 - 2.73435i) q^{7} +(-0.214928 - 2.82025i) q^{8} +(-0.758851 + 0.173203i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 14 q^{4} - 28 q^{5} - 14 q^{6} - 12 q^{8} - 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.24228	−	0.675831i	−0.878423	−	0.477884i
							\(3\)	−1.48114	+	0.166885i	−0.855138	+	0.0963509i	−0.528639	−	0.848847i	\(-0.677298\pi\)
−0.326499	+	0.945198i	\(0.605869\pi\)
\(5\)	0.727418	−	1.51050i	0.325311	−	0.675516i	−0.672608	−	0.739999i	\(-0.734826\pi\)
							0.997919	+	0.0644837i	\(0.0205401\pi\)
\(7\)	−3.42876	−	2.73435i	−1.29595	−	1.03349i	−0.996854	−	0.0792629i	\(-0.974743\pi\)
							−0.299096	−	0.954223i	\(-0.596685\pi\)
\(11\)	1.20249	−	0.755577i	0.362566	−	0.227815i	−0.338412	−	0.940998i	\(-0.609890\pi\)
							0.700978	+	0.713183i	\(0.252747\pi\)
\(13\)	−5.83372	−	1.33151i	−1.61798	−	0.369294i	−0.684809	−	0.728722i	\(-0.740115\pi\)
							−0.933173	+	0.359428i	\(0.882972\pi\)
\(17\)	3.39448	−	3.39448i	0.823283	−	0.823283i	−0.163294	−	0.986577i	\(-0.552212\pi\)
							0.986577	+	0.163294i	\(0.0522120\pi\)
\(19\)	0.339529	−	3.01340i	0.0778933	−	0.691322i	−0.892709	−	0.450634i	\(-0.851198\pi\)
							0.970602	−	0.240689i	\(-0.0773733\pi\)
\(23\)	2.08623	+	4.33210i	0.435009	+	0.903305i	0.997091	+	0.0762234i	\(0.0242862\pi\)
							−0.562082	+	0.827081i	\(0.689999\pi\)
\(29\)	−2.66387	−	4.68015i	−0.494668	−	0.869082i
							\(31\)	0.727961	+	2.08039i	0.130746	+	0.373650i	0.990582	−	0.136920i	\(-0.0437203\pi\)
−0.859836	+	0.510570i	\(0.829435\pi\)
\(37\)	−1.98657	+	3.16161i	−0.326591	+	0.519766i	−0.969151	−	0.246469i	\(-0.920730\pi\)
							0.642560	+	0.766236i	\(0.277872\pi\)
\(41\)	−0.423896	−	0.423896i	−0.0662014	−	0.0662014i	0.673231	−	0.739432i	\(-0.264906\pi\)
							−0.739432	+	0.673231i	\(0.764906\pi\)
\(43\)	0.259529	+	0.0908133i	0.0395779	+	0.0138489i	0.349995	−	0.936751i	\(-0.386183\pi\)
							−0.310418	+	0.950600i	\(0.600469\pi\)
\(47\)	0.934739	+	1.48763i	0.136346	+	0.216993i	0.907917	−	0.419150i	\(-0.137672\pi\)
							−0.771571	+	0.636143i	\(0.780529\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.2.l.b.11.4	✓	144
4.3	odd	2	inner	116.2.l.b.11.6	yes	144
29.8	odd	28	inner	116.2.l.b.95.6	yes	144
116.95	even	28	inner	116.2.l.b.95.4	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.l.b.11.4	✓	144	1.1	even	1	trivial
116.2.l.b.11.6	yes	144	4.3	odd	2	inner
116.2.l.b.95.4	yes	144	116.95	even	28	inner
116.2.l.b.95.6	yes	144	29.8	odd	28	inner