Embedded newform 116.2.l.b.95.2

• Label: 116.2.l.b.95.2
• Level: $116$
• Weight: $2$
• Character: 116.95
• 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			95.2
Character	\(\chi\)	\(=\)	116.95
Dual form			116.2.l.b.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33626 + 0.463036i) q^{2} +(0.950125 + 0.107053i) q^{3} +(1.57119 - 1.23748i) q^{4} +(-1.46565 - 3.04344i) q^{5} +(-1.31919 + 0.296891i) q^{6} +(2.38827 - 1.90458i) q^{7} +(-1.52653 + 2.38111i) q^{8} +(-2.03351 - 0.464135i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 14 q^{4} - 28 q^{5} - 14 q^{6} - 12 q^{8} - 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(59\)	\(89\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.33626	+	0.463036i	−0.944880	+	0.327416i
							\(3\)	0.950125	+	0.107053i	0.548555	+	0.0618073i	0.381892	−	0.924207i	\(-0.375272\pi\)
0.166662	+	0.986014i	\(0.446701\pi\)
\(5\)	−1.46565	−	3.04344i	−0.655457	−	1.36107i	−0.918167	−	0.396195i	\(-0.870331\pi\)
							0.262710	−	0.964875i	\(-0.415384\pi\)
\(7\)	2.38827	−	1.90458i	0.902681	−	0.719864i	−0.0577648	−	0.998330i	\(-0.518397\pi\)
							0.960446	+	0.278466i	\(0.0898259\pi\)
\(11\)	2.73923	+	1.72117i	0.825908	+	0.518952i	0.877422	−	0.479720i	\(-0.159262\pi\)
							−0.0515137	+	0.998672i	\(0.516405\pi\)
\(13\)	2.10361	−	0.480136i	0.583437	−	0.133166i	0.0793925	−	0.996843i	\(-0.474702\pi\)
							0.504044	+	0.863678i	\(0.331845\pi\)
\(17\)	0.808672	+	0.808672i	0.196132	+	0.196132i	0.798339	−	0.602208i	\(-0.205712\pi\)
							−0.602208	+	0.798339i	\(0.705712\pi\)
\(19\)	0.220671	+	1.95850i	0.0506253	+	0.449312i	0.993297	+	0.115590i	\(0.0368760\pi\)
							−0.942672	+	0.333722i	\(0.891695\pi\)
\(23\)	−1.95246	+	4.05432i	−0.407115	+	0.845383i	0.592104	+	0.805862i	\(0.298298\pi\)
							−0.999219	+	0.0395215i	\(0.987417\pi\)
\(29\)	0.900824	−	5.30929i	0.167279	−	0.985910i
							\(31\)	−2.96739	+	8.48031i	−0.532959	+	1.52311i	0.291150	+	0.956678i	\(0.405962\pi\)
−0.824109	+	0.566432i	\(0.808323\pi\)
\(37\)	−0.187035	−	0.297664i	−0.0307483	−	0.0489357i	0.830996	−	0.556278i	\(-0.187771\pi\)
							−0.861745	+	0.507342i	\(0.830628\pi\)
\(41\)	−7.20535	+	7.20535i	−1.12529	+	1.12529i	−0.134353	+	0.990934i	\(0.542896\pi\)
							−0.990934	+	0.134353i	\(0.957104\pi\)
\(43\)	5.16416	−	1.80702i	0.787527	−	0.275568i	0.0935972	−	0.995610i	\(-0.470163\pi\)
							0.693930	+	0.720042i	\(0.255878\pi\)
\(47\)	6.75914	−	10.7571i	0.985922	−	1.56909i	0.174474	−	0.984662i	\(-0.444178\pi\)
							0.811448	−	0.584424i	\(-0.198680\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.2.l.b.95.2	yes	144
4.3	odd	2	inner	116.2.l.b.95.5	yes	144
29.11	odd	28	inner	116.2.l.b.11.5	yes	144
116.11	even	28	inner	116.2.l.b.11.2	✓	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.l.b.11.2	✓	144	116.11	even	28	inner
116.2.l.b.11.5	yes	144	29.11	odd	28	inner
116.2.l.b.95.2	yes	144	1.1	even	1	trivial
116.2.l.b.95.5	yes	144	4.3	odd	2	inner