Embedded Newform 31.2.g.a.7.2

• Label: 31.2.g.a.7.2
• Level: 31
• Weight: 2
• Character: 31.7
• Analytic conductor: 0.248
• Analytic rank: 0
• Dimension: 16
• CM: No
• Inner twists: 2

Newspace parameters

Level:	\( N \)	=	\( 31 \)
Weight:	\( k \)	=	\( 2 \)
Character orbit:	\([\chi]\)	=	31.g (of order \(15\) and degree \(8\))

Newform invariants

Self dual:	No
Analytic conductor:	\(0.247536246266\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(2\) over \(\Q(\zeta_{15})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			7.2
Root			\(0.176392i\)
Character	\(\chi\)	=	31.7
Dual form			31.2.g.a.9.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.380762 + 1.17187i) q^{2} +(-2.02963 + 0.431412i) q^{3} +(0.389745 - 0.283166i) q^{4} +(0.772811 - 1.33855i) q^{5} +(-1.27836 - 2.21419i) q^{6} +(-3.47491 - 1.54713i) q^{7} +(2.47393 + 1.79742i) q^{8} +(1.19265 - 0.531003i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16q - 6q^{2} - 12q^{3} - 14q^{4} - 3q^{5} + 11q^{6} + 2q^{7} + 17q^{8} - 10q^{9} + O(q^{10}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/31\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{14}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.380762	+	1.17187i	0.269240	+	0.828634i	0.990686	+	0.136164i	\(0.0434774\pi\)
							−0.721447	+	0.692470i	\(0.756523\pi\)
\(3\)	−2.02963	+	0.431412i	−1.17181	+	0.249076i	−0.752390	−	0.658718i	\(-0.771099\pi\)
							−0.419418	+	0.907793i	\(0.637766\pi\)
\(5\)	0.772811	−	1.33855i	0.345612	−	0.598617i	−0.639853	−	0.768497i	\(-0.721005\pi\)
							0.985465	+	0.169880i	\(0.0543381\pi\)
\(7\)	−3.47491	−	1.54713i	−1.31339	−	0.584759i	−0.373943	−	0.927452i	\(-0.621995\pi\)
							−0.939448	+	0.342692i	\(0.888661\pi\)
\(11\)	0.393186	+	3.74092i	0.118550	+	1.12793i	0.878432	+	0.477868i	\(0.158590\pi\)
							−0.759882	+	0.650061i	\(0.774743\pi\)
\(13\)	−1.76320	−	1.95823i	−0.489024	−	0.543116i	0.447240	−	0.894414i	\(-0.352407\pi\)
							−0.936264	+	0.351298i	\(0.885740\pi\)
\(17\)	−0.394526	+	3.75366i	−0.0956866	+	0.910397i	0.836390	+	0.548135i	\(0.184662\pi\)
							−0.932076	+	0.362262i	\(0.882005\pi\)
\(19\)	4.08057	−	4.53193i	0.936147	−	1.03970i	−0.0629842	−	0.998015i	\(-0.520062\pi\)
							0.999131	−	0.0416817i	\(-0.0132716\pi\)
\(23\)	−0.736082	−	0.534795i	−0.153484	−	0.111512i	0.508393	−	0.861125i	\(-0.330240\pi\)
							−0.661877	+	0.749613i	\(0.730240\pi\)
\(29\)	−2.10397	−	6.47535i	−0.390697	−	1.20244i	−0.932262	−	0.361784i	\(-0.882168\pi\)
							0.541565	−	0.840659i	\(-0.317832\pi\)
\(31\)	−3.88819	+	3.98522i	−0.698339	+	0.715767i
							\(37\)	0.907032	+	1.57103i	0.149115	+	0.258275i	0.930901	−	0.365272i	\(-0.119024\pi\)
−0.781786	+	0.623547i	\(0.785691\pi\)
\(41\)	−0.329777	−	0.0700963i	−0.0515026	−	0.0109472i	0.182088	−	0.983282i	\(-0.441714\pi\)
							−0.233591	+	0.972335i	\(0.575048\pi\)
\(43\)	2.59890	−	2.88637i	0.396329	−	0.440167i	−0.511644	−	0.859198i	\(-0.670963\pi\)
							0.907972	+	0.419030i	\(0.137630\pi\)
\(47\)	−0.367467	+	1.13095i	−0.0536005	+	0.164965i	−0.974273	−	0.225370i	\(-0.927641\pi\)
							0.920673	+	0.390335i	\(0.127641\pi\)