Embedded newform 116.2.l.b.11.10

• Label: 116.2.l.b.11.10
• 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			11.10
Character	\(\chi\)	\(=\)	116.11
Dual form			116.2.l.b.95.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.523166 - 1.31389i) q^{2} +(2.69956 - 0.304168i) q^{3} +(-1.45260 - 1.37476i) q^{4} +(-1.20112 + 2.49416i) q^{5} +(1.01268 - 3.70605i) q^{6} +(-0.625709 - 0.498986i) q^{7} +(-2.56623 + 1.18932i) q^{8} +(4.27033 - 0.974675i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 14 q^{4} - 28 q^{5} - 14 q^{6} - 12 q^{8} - 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.523166	−	1.31389i	0.369934	−	0.929058i
							\(3\)	2.69956	−	0.304168i	1.55859	−	0.175611i	0.709825	−	0.704378i	\(-0.248774\pi\)
0.848767	+	0.528767i	\(0.177345\pi\)
\(5\)	−1.20112	+	2.49416i	−0.537159	+	1.11542i	0.439024	+	0.898475i	\(0.355324\pi\)
							−0.976183	+	0.216947i	\(0.930390\pi\)
\(7\)	−0.625709	−	0.498986i	−0.236496	−	0.188599i	0.498070	−	0.867137i	\(-0.334042\pi\)
							−0.734565	+	0.678538i	\(0.762614\pi\)
\(11\)	−4.00072	+	2.51382i	−1.20626	+	0.757945i	−0.976545	−	0.215313i	\(-0.930923\pi\)
							−0.229717	+	0.973258i	\(0.573780\pi\)
\(13\)	−2.59950	−	0.593318i	−0.720971	−	0.164557i	−0.153735	−	0.988112i	\(-0.549130\pi\)
							−0.567236	+	0.823555i	\(0.691987\pi\)
\(17\)	4.12057	−	4.12057i	0.999386	−	0.999386i	−0.000613870	−	1.00000i	\(-0.500195\pi\)
							1.00000		0.000613870i	\(0.000195401\pi\)
\(19\)	0.647599	−	5.74760i	0.148569	−	1.31859i	−0.669217	−	0.743067i	\(-0.733370\pi\)
							0.817786	−	0.575522i	\(-0.195201\pi\)
\(23\)	1.26320	+	2.62305i	0.263394	+	0.546944i	0.990160	−	0.139942i	\(-0.0446915\pi\)
							−0.726765	+	0.686886i	\(0.758977\pi\)
\(29\)	5.26763	−	1.11894i	0.978175	−	0.207782i
							\(31\)	0.0981689	+	0.280550i	0.0176316	+	0.0503884i	0.952363	−	0.304967i	\(-0.0986455\pi\)
−0.934731	+	0.355356i	\(0.884360\pi\)
\(37\)	−1.42715	+	2.27130i	−0.234623	+	0.373400i	−0.943196	−	0.332238i	\(-0.892196\pi\)
							0.708573	+	0.705638i	\(0.249339\pi\)
\(41\)	6.66062	+	6.66062i	1.04021	+	1.04021i	0.999157	+	0.0410580i	\(0.0130729\pi\)
							0.0410580	+	0.999157i	\(0.486927\pi\)
\(43\)	1.02284	+	0.357909i	0.155982	+	0.0545806i	0.407141	−	0.913365i	\(-0.366526\pi\)
							−0.251158	+	0.967946i	\(0.580812\pi\)
\(47\)	1.28181	+	2.04000i	0.186972	+	0.297564i	0.927036	−	0.374972i	\(-0.122348\pi\)
							−0.740064	+	0.672536i	\(0.765205\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.2.l.b.11.10	✓	144
4.3	odd	2	inner	116.2.l.b.11.12	yes	144
29.8	odd	28	inner	116.2.l.b.95.12	yes	144
116.95	even	28	inner	116.2.l.b.95.10	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.l.b.11.10	✓	144	1.1	even	1	trivial
116.2.l.b.11.12	yes	144	4.3	odd	2	inner
116.2.l.b.95.10	yes	144	116.95	even	28	inner
116.2.l.b.95.12	yes	144	29.8	odd	28	inner