Embedded newform 116.2.l.b.19.3

• Label: 116.2.l.b.19.3
• Level: $116$
• Weight: $2$
• Character: 116.19
• 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			19.3
Character	\(\chi\)	\(=\)	116.19
Dual form			116.2.l.b.55.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.21130 - 0.729892i) q^{2} +(-3.10478 - 1.08641i) q^{3} +(0.934514 + 1.76824i) q^{4} +(-1.05692 - 0.241234i) q^{5} +(2.96787 + 3.58212i) q^{6} +(1.22512 + 2.54399i) q^{7} +(0.158647 - 2.82397i) q^{8} +(6.11385 + 4.87564i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 14 q^{4} - 28 q^{5} - 14 q^{6} - 12 q^{8} - 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(59\)	\(89\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{9}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.21130	−	0.729892i	−0.856521	−	0.516112i
							\(3\)	−3.10478	−	1.08641i	−1.79254	−	0.627238i	−0.999845	−	0.0176237i	\(-0.994390\pi\)
−0.792698	−	0.609614i	\(-0.791324\pi\)
\(5\)	−1.05692	−	0.241234i	−0.472668	−	0.107883i	−0.0204500	−	0.999791i	\(-0.506510\pi\)
							−0.452218	+	0.891908i	\(0.649367\pi\)
\(7\)	1.22512	+	2.54399i	0.463053	+	0.961539i	0.993501	+	0.113822i	\(0.0363093\pi\)
							−0.530448	+	0.847717i	\(0.677976\pi\)
\(11\)	−0.0781135	+	0.693276i	−0.0235521	+	0.209031i	−0.999977	−	0.00682816i	\(-0.997827\pi\)
							0.976425	+	0.215859i	\(0.0692551\pi\)
\(13\)	−2.28840	+	1.82494i	−0.634688	+	0.506147i	−0.887163	−	0.461457i	\(-0.847327\pi\)
							0.252475	+	0.967603i	\(0.418755\pi\)
\(17\)	−3.47784	+	3.47784i	−0.843501	+	0.843501i	−0.989312	−	0.145812i	\(-0.953421\pi\)
							0.145812	+	0.989312i	\(0.453421\pi\)
\(19\)	1.15827	+	3.31015i	0.265726	+	0.759401i	0.997006	+	0.0773240i	\(0.0246376\pi\)
							−0.731280	+	0.682077i	\(0.761077\pi\)
\(23\)	2.55212	−	0.582505i	0.532154	−	0.121461i	0.0520035	−	0.998647i	\(-0.483439\pi\)
							0.480150	+	0.877186i	\(0.340582\pi\)
\(29\)	−2.03414	+	4.98621i	−0.377730	+	0.925916i
							\(31\)	−1.98815	+	1.24924i	−0.357083	+	0.224370i	−0.698611	−	0.715501i	\(-0.746198\pi\)
0.341528	+	0.939871i	\(0.389055\pi\)
\(37\)	9.64708	−	1.08697i	1.58597	−	0.178696i	0.725510	−	0.688212i	\(-0.241604\pi\)
							0.860461	+	0.509516i	\(0.170176\pi\)
\(41\)	0.357195	+	0.357195i	0.0557844	+	0.0557844i	0.734449	−	0.678664i	\(-0.237441\pi\)
							−0.678664	+	0.734449i	\(0.737441\pi\)
\(43\)	−5.65258	+	8.99604i	−0.862011	+	1.37188i	0.0654081	+	0.997859i	\(0.479165\pi\)
							−0.927419	+	0.374024i	\(0.877978\pi\)
\(47\)	−4.77923	−	0.538490i	−0.697123	−	0.0785469i	−0.243711	−	0.969848i	\(-0.578365\pi\)
							−0.453412	+	0.891301i	\(0.649793\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.2.l.b.19.3	✓	144
4.3	odd	2	inner	116.2.l.b.19.6	yes	144
29.26	odd	28	inner	116.2.l.b.55.6	yes	144
116.55	even	28	inner	116.2.l.b.55.3	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.l.b.19.3	✓	144	1.1	even	1	trivial
116.2.l.b.19.6	yes	144	4.3	odd	2	inner
116.2.l.b.55.3	yes	144	116.55	even	28	inner
116.2.l.b.55.6	yes	144	29.26	odd	28	inner