Embedded newform 116.2.l.b.11.7

• Label: 116.2.l.b.11.7
• 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			11.7
Character	\(\chi\)	\(=\)	116.11
Dual form			116.2.l.b.95.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.490339 - 1.32649i) q^{2} +(-2.34863 + 0.264627i) q^{3} +(-1.51913 + 1.30086i) q^{4} +(-0.588667 + 1.22238i) q^{5} +(1.50265 + 2.98566i) q^{6} +(2.56130 + 2.04257i) q^{7} +(2.47046 + 1.37725i) q^{8} +(2.52123 - 0.575454i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 14 q^{4} - 28 q^{5} - 14 q^{6} - 12 q^{8} - 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.490339	−	1.32649i	−0.346722	−	0.937968i
							\(3\)	−2.34863	+	0.264627i	−1.35598	+	0.152782i	−0.759846	−	0.650103i	\(-0.774726\pi\)
−0.596133	+	0.802885i	\(0.703297\pi\)
\(5\)	−0.588667	+	1.22238i	−0.263260	+	0.546665i	−0.990137	−	0.140103i	\(-0.955257\pi\)
							0.726877	+	0.686768i	\(0.240971\pi\)
\(7\)	2.56130	+	2.04257i	0.968081	+	0.772019i	0.973669	−	0.227966i	\(-0.0732074\pi\)
							−0.00558836	+	0.999984i	\(0.501779\pi\)
\(11\)	−5.07822	+	3.19086i	−1.53114	+	0.962080i	−0.538285	+	0.842763i	\(0.680928\pi\)
							−0.992856	+	0.119317i	\(0.961930\pi\)
\(13\)	−1.29716	−	0.296069i	−0.359769	−	0.0821149i	0.0388160	−	0.999246i	\(-0.487641\pi\)
							−0.398585	+	0.917131i	\(0.630499\pi\)
\(17\)	−3.79071	+	3.79071i	−0.919383	+	0.919383i	−0.996984	−	0.0776011i	\(-0.975274\pi\)
							0.0776011	+	0.996984i	\(0.475274\pi\)
\(19\)	0.151626	−	1.34572i	0.0347854	−	0.308729i	−0.964257	−	0.264970i	\(-0.914638\pi\)
							0.999042	−	0.0437590i	\(-0.0139334\pi\)
\(23\)	1.29280	+	2.68452i	0.269567	+	0.559762i	0.991178	−	0.132537i	\(-0.0423123\pi\)
							−0.721611	+	0.692299i	\(0.756598\pi\)
\(29\)	2.93553	−	4.51472i	0.545114	−	0.838362i
							\(31\)	−1.44988	−	4.14352i	−0.260406	−	0.744199i	−0.997646	−	0.0685678i	\(-0.978157\pi\)
0.737240	−	0.675631i	\(-0.236129\pi\)
\(37\)	−3.02614	+	4.81608i	−0.497495	+	0.791759i	−0.997227	−	0.0744214i	\(-0.976289\pi\)
							0.499732	+	0.866180i	\(0.333432\pi\)
\(41\)	−4.55671	−	4.55671i	−0.711638	−	0.711638i	0.255239	−	0.966878i	\(-0.417846\pi\)
							−0.966878	+	0.255239i	\(0.917846\pi\)
\(43\)	6.65934	+	2.33020i	1.01554	+	0.355353i	0.786186	−	0.617990i	\(-0.212053\pi\)
							0.229354	+	0.973343i	\(0.426339\pi\)
\(47\)	−0.260350	−	0.414345i	−0.0379760	−	0.0604384i	0.827197	−	0.561912i	\(-0.189934\pi\)
							−0.865173	+	0.501474i	\(0.832791\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.2.l.b.11.7	✓	144
4.3	odd	2	inner	116.2.l.b.11.9	yes	144
29.8	odd	28	inner	116.2.l.b.95.9	yes	144
116.95	even	28	inner	116.2.l.b.95.7	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.l.b.11.7	✓	144	1.1	even	1	trivial
116.2.l.b.11.9	yes	144	4.3	odd	2	inner
116.2.l.b.95.7	yes	144	116.95	even	28	inner
116.2.l.b.95.9	yes	144	29.8	odd	28	inner