Embedded newform 116.2.l.b.95.10

• Label: 116.2.l.b.95.10
• Level: $116$
• Weight: $2$
• Character: 116.95
• 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			95.10
Character	\(\chi\)	\(=\)	116.95
Dual form			116.2.l.b.11.10

$q$-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 \(\left(\mathbb{Z}/116\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.l.b.11.10	✓	144	116.11	even	28	inner
116.2.l.b.11.12	yes	144	29.11	odd	28	inner
116.2.l.b.95.10	yes	144	1.1	even	1	trivial
116.2.l.b.95.12	yes	144	4.3	odd	2	inner