Embedded newform 31.2.g.a.9.2

• Label: 31.2.g.a.9.2
• Level: $31$
• Weight: $2$
• Character: 31.9
• 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 \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.247536246266\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(2\) over \(\Q(\zeta_{15})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial:	\( x^{16} + 19x^{14} + 140x^{12} + 511x^{10} + 979x^{8} + 956x^{6} + 410x^{4} + 44x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			9.2
Root			\(-0.176392i\) of defining polynomial
Character	\(\chi\)	\(=\)	31.9
Dual form			31.2.g.a.7.2

$q$-expansion

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

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{15}\right)\)

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

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	31.2.g.a.9.2	yes	16
3.2	odd	2		279.2.y.c.226.1		16
4.3	odd	2		496.2.bg.c.257.2		16
5.2	odd	4		775.2.ck.a.474.3		32
5.3	odd	4		775.2.ck.a.474.2		32
5.4	even	2		775.2.bl.a.226.1		16
31.2	even	5		961.2.c.j.439.5		16
31.3	odd	30		961.2.g.n.732.1		16
31.4	even	5		961.2.g.k.547.2		16
31.5	even	3		961.2.d.o.628.3		16
31.6	odd	6		961.2.g.j.448.2		16
31.7	even	15	inner	31.2.g.a.7.2	✓	16
31.8	even	5		961.2.g.t.235.1		16
31.9	even	15		961.2.d.p.388.2		16
31.10	even	15		961.2.a.i.1.5		8
31.11	odd	30		961.2.d.n.531.3		16
31.12	odd	30		961.2.c.i.521.5		16
31.13	odd	30		961.2.d.q.374.2		16
31.14	even	15		961.2.g.s.338.1		16
31.15	odd	10		961.2.g.m.816.1		16
31.16	even	5		961.2.g.s.816.1		16
31.17	odd	30		961.2.g.m.338.1		16
31.18	even	15		961.2.d.p.374.2		16
31.19	even	15		961.2.c.j.521.5		16
31.20	even	15		961.2.d.o.531.3		16
31.21	odd	30		961.2.a.j.1.5		8
31.22	odd	30		961.2.d.q.388.2		16
31.23	odd	10		961.2.g.n.235.1		16
31.24	odd	30		961.2.g.l.844.2		16
31.25	even	3		961.2.g.k.448.2		16
31.26	odd	6		961.2.d.n.628.3		16
31.27	odd	10		961.2.g.j.547.2		16
31.28	even	15		961.2.g.t.732.1		16
31.29	odd	10		961.2.c.i.439.5		16
31.30	odd	2		961.2.g.l.846.2		16
93.38	odd	30		279.2.y.c.100.1		16
93.41	odd	30		8649.2.a.bf.1.4		8
93.83	even	30		8649.2.a.be.1.4		8
124.7	odd	30		496.2.bg.c.193.2		16
155.7	odd	60		775.2.ck.a.224.2		32
155.38	odd	60		775.2.ck.a.224.3		32
155.69	even	30		775.2.bl.a.751.1		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.2.g.a.7.2	✓	16	31.7	even	15	inner
31.2.g.a.9.2	yes	16	1.1	even	1	trivial
279.2.y.c.100.1		16	93.38	odd	30
279.2.y.c.226.1		16	3.2	odd	2
496.2.bg.c.193.2		16	124.7	odd	30
496.2.bg.c.257.2		16	4.3	odd	2
775.2.bl.a.226.1		16	5.4	even	2
775.2.bl.a.751.1		16	155.69	even	30
775.2.ck.a.224.2		32	155.7	odd	60
775.2.ck.a.224.3		32	155.38	odd	60
775.2.ck.a.474.2		32	5.3	odd	4
775.2.ck.a.474.3		32	5.2	odd	4
961.2.a.i.1.5		8	31.10	even	15
961.2.a.j.1.5		8	31.21	odd	30
961.2.c.i.439.5		16	31.29	odd	10
961.2.c.i.521.5		16	31.12	odd	30
961.2.c.j.439.5		16	31.2	even	5
961.2.c.j.521.5		16	31.19	even	15
961.2.d.n.531.3		16	31.11	odd	30
961.2.d.n.628.3		16	31.26	odd	6
961.2.d.o.531.3		16	31.20	even	15
961.2.d.o.628.3		16	31.5	even	3
961.2.d.p.374.2		16	31.18	even	15
961.2.d.p.388.2		16	31.9	even	15
961.2.d.q.374.2		16	31.13	odd	30
961.2.d.q.388.2		16	31.22	odd	30
961.2.g.j.448.2		16	31.6	odd	6
961.2.g.j.547.2		16	31.27	odd	10
961.2.g.k.448.2		16	31.25	even	3
961.2.g.k.547.2		16	31.4	even	5
961.2.g.l.844.2		16	31.24	odd	30
961.2.g.l.846.2		16	31.30	odd	2
961.2.g.m.338.1		16	31.17	odd	30
961.2.g.m.816.1		16	31.15	odd	10
961.2.g.n.235.1		16	31.23	odd	10
961.2.g.n.732.1		16	31.3	odd	30
961.2.g.s.338.1		16	31.14	even	15
961.2.g.s.816.1		16	31.16	even	5
961.2.g.t.235.1		16	31.8	even	5
961.2.g.t.732.1		16	31.28	even	15
8649.2.a.be.1.4		8	93.83	even	30
8649.2.a.bf.1.4		8	93.41	odd	30