Embedded newform 116.2.l.b.31.2

• Label: 116.2.l.b.31.2
• Level: $116$
• Weight: $2$
• Character: 116.31
• 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			31.2
Character	\(\chi\)	\(=\)	116.31
Dual form			116.2.l.b.15.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.16318 - 0.804373i) q^{2} +(0.564706 - 0.898725i) q^{3} +(0.705968 + 1.87126i) q^{4} +(1.84963 - 1.47503i) q^{5} +(-1.37976 + 0.591143i) q^{6} +(-0.762672 + 0.174075i) q^{7} +(0.684023 - 2.74447i) q^{8} +(0.812838 + 1.68788i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 14 q^{4} - 28 q^{5} - 14 q^{6} - 12 q^{8} - 28 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.16318	−	0.804373i	−0.822491	−	0.568778i
							\(3\)	0.564706	−	0.898725i	0.326033	−	0.518879i	−0.642982	−	0.765882i	\(-0.722303\pi\)
0.969015	+	0.247003i	\(0.0794457\pi\)
\(5\)	1.84963	−	1.47503i	0.827181	−	0.659655i	−0.115516	−	0.993306i	\(-0.536852\pi\)
							0.942697	+	0.333651i	\(0.108281\pi\)
\(7\)	−0.762672	+	0.174075i	−0.288263	+	0.0657941i	−0.364207	−	0.931318i	\(-0.618660\pi\)
							0.0759443	+	0.997112i	\(0.475803\pi\)
\(11\)	−1.41065	−	4.03140i	−0.425326	−	1.21551i	−0.934846	−	0.355054i	\(-0.884463\pi\)
							0.509520	−	0.860459i	\(-0.329823\pi\)
\(13\)	1.14835	−	2.38457i	0.318495	−	0.661361i	−0.678843	−	0.734284i	\(-0.737518\pi\)
							0.997338	+	0.0729223i	\(0.0232325\pi\)
\(17\)	−0.864876	+	0.864876i	−0.209763	+	0.209763i	−0.804167	−	0.594404i	\(-0.797388\pi\)
							0.594404	+	0.804167i	\(0.297388\pi\)
\(19\)	−3.53894	+	2.22366i	−0.811889	+	0.510143i	−0.872845	−	0.487997i	\(-0.837728\pi\)
							0.0609567	+	0.998140i	\(0.480585\pi\)
\(23\)	5.08465	+	4.05487i	1.06022	+	0.845499i	0.988393	−	0.151918i	\(-0.0485448\pi\)
							0.0718295	+	0.997417i	\(0.477116\pi\)
\(29\)	1.27027	+	5.23320i	0.235882	+	0.971782i
							\(31\)	−0.840478	+	7.45945i	−0.150954	+	1.33976i	0.658503	+	0.752578i	\(0.271190\pi\)
−0.809458	+	0.587178i	\(0.800239\pi\)
\(37\)	−3.11215	−	1.08899i	−0.511634	−	0.179028i	0.0621005	−	0.998070i	\(-0.480220\pi\)
							−0.573734	+	0.819042i	\(0.694506\pi\)
\(41\)	−6.22325	−	6.22325i	−0.971908	−	0.971908i	0.0277078	−	0.999616i	\(-0.491179\pi\)
							−0.999616	+	0.0277078i	\(0.991179\pi\)
\(43\)	−2.71002	+	0.305346i	−0.413274	+	0.0465648i	−0.316154	−	0.948708i	\(-0.602392\pi\)
							−0.0971199	+	0.995273i	\(0.530963\pi\)
\(47\)	5.74030	−	2.00862i	0.837309	−	0.292987i	0.122641	−	0.992451i	\(-0.460863\pi\)
							0.714668	+	0.699464i	\(0.246578\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.2.l.b.31.2	yes	144
4.3	odd	2	inner	116.2.l.b.31.1	yes	144
29.15	odd	28	inner	116.2.l.b.15.1	✓	144
116.15	even	28	inner	116.2.l.b.15.2	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.l.b.15.1	✓	144	29.15	odd	28	inner
116.2.l.b.15.2	yes	144	116.15	even	28	inner
116.2.l.b.31.1	yes	144	4.3	odd	2	inner
116.2.l.b.31.2	yes	144	1.1	even	1	trivial