Embedded newform 116.2.l.b.11.12

• Label: 116.2.l.b.11.12
• 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			11.12
Character	\(\chi\)	\(=\)	116.11
Dual form			116.2.l.b.95.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.22822 + 0.701049i) q^{2} +(-2.69956 + 0.304168i) q^{3} +(1.01706 + 1.72209i) q^{4} +(-1.20112 + 2.49416i) q^{5} +(-3.52890 - 1.51894i) q^{6} +(0.625709 + 0.498986i) q^{7} +(0.0419068 + 2.82812i) q^{8} +(4.27033 - 0.974675i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 14 q^{4} - 28 q^{5} - 14 q^{6} - 12 q^{8} - 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.22822	+	0.701049i	0.868484	+	0.495717i
							\(3\)	−2.69956	+	0.304168i	−1.55859	+	0.175611i	−0.848767	−	0.528767i	\(-0.822655\pi\)
−0.709825	+	0.704378i	\(0.751226\pi\)
\(5\)	−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	0.734565	−	0.678538i	\(-0.237386\pi\)
							−0.498070	+	0.867137i	\(0.665958\pi\)
\(11\)	4.00072	−	2.51382i	1.20626	−	0.757945i	0.229717	−	0.973258i	\(-0.426220\pi\)
							0.976545	+	0.215313i	\(0.0690771\pi\)
\(13\)	−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	\(0.266630\pi\)
							−0.817786	+	0.575522i	\(0.804799\pi\)
\(23\)	−1.26320	−	2.62305i	−0.263394	−	0.546944i	0.726765	−	0.686886i	\(-0.241023\pi\)
							−0.990160	+	0.139942i	\(0.955309\pi\)
\(29\)	5.26763	−	1.11894i	0.978175	−	0.207782i
							\(31\)	−0.0981689	−	0.280550i	−0.0176316	−	0.0503884i	0.934731	−	0.355356i	\(-0.115640\pi\)
−0.952363	+	0.304967i	\(0.901354\pi\)
\(37\)	−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	0.251158	−	0.967946i	\(-0.419188\pi\)
							−0.407141	+	0.913365i	\(0.633474\pi\)
\(47\)	−1.28181	−	2.04000i	−0.186972	−	0.297564i	0.740064	−	0.672536i	\(-0.234795\pi\)
							−0.927036	+	0.374972i	\(0.877652\pi\)

(See \(a_n\) instead)

Twists

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

    

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