Embedded newform 31.2.g.a.9.1

• Label: 31.2.g.a.9.1
• 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			9.1
Root			\(2.16544i\) of defining polynomial
Character	\(\chi\)	\(=\)	31.9
Dual form			31.2.g.a.7.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.571745 + 1.75965i) q^{2} +(-0.488442 - 0.103822i) q^{3} +(-1.15144 - 0.836573i) q^{4} +(-0.603681 - 1.04561i) q^{5} +(0.461954 - 0.800128i) q^{6} +(3.41030 - 1.51837i) q^{7} +(-0.863288 + 0.627215i) q^{8} +(-2.51284 - 1.11879i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{2} - 12 q^{3} - 14 q^{4} - 3 q^{5} + 11 q^{6} + 2 q^{7} + 17 q^{8} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.571745	+	1.75965i	−0.404285	+	1.24426i	0.517206	+	0.855861i	\(0.326972\pi\)
							−0.921491	+	0.388400i	\(0.873028\pi\)
\(3\)	−0.488442	−	0.103822i	−0.282002	−	0.0599414i	0.0647390	−	0.997902i	\(-0.479379\pi\)
							−0.346741	+	0.937961i	\(0.612712\pi\)
\(5\)	−0.603681	−	1.04561i	−0.269974	−	0.467609i	0.698881	−	0.715238i	\(-0.253682\pi\)
							−0.968855	+	0.247629i	\(0.920349\pi\)
\(7\)	3.41030	−	1.51837i	1.28897	−	0.573888i	0.356220	−	0.934402i	\(-0.384065\pi\)
							0.932754	+	0.360514i	\(0.117399\pi\)
\(11\)	0.194058	−	1.84634i	0.0585107	−	0.556692i	−0.925521	−	0.378697i	\(-0.876372\pi\)
							0.984031	−	0.177995i	\(-0.0569611\pi\)
\(13\)	−3.46909	+	3.85281i	−0.962152	+	1.06858i	0.0354505	+	0.999371i	\(0.488713\pi\)
							−0.997602	+	0.0692065i	\(0.977953\pi\)
\(17\)	0.592413	+	5.63643i	0.143681	+	1.36703i	0.794252	+	0.607589i	\(0.207863\pi\)
							−0.650571	+	0.759446i	\(0.725470\pi\)
\(19\)	−0.962535	−	1.06900i	−0.220821	−	0.245246i	0.622548	−	0.782582i	\(-0.286098\pi\)
							−0.843369	+	0.537335i	\(0.819431\pi\)
\(23\)	2.86762	−	2.08345i	0.597940	−	0.434429i	−0.247207	−	0.968963i	\(-0.579513\pi\)
							0.845147	+	0.534534i	\(0.179513\pi\)
\(29\)	0.424157	−	1.30542i	0.0787641	−	0.242411i	−0.903919	−	0.427703i	\(-0.859323\pi\)
							0.982684	+	0.185292i	\(0.0593230\pi\)
\(31\)	−1.81847	+	5.26243i	−0.326607	+	0.945160i
							\(37\)	−2.25141	+	3.89955i	−0.370129	+	0.641082i	−0.989585	−	0.143949i	\(-0.954020\pi\)
0.619456	+	0.785031i	\(0.287353\pi\)
\(41\)	4.61599	−	0.981158i	0.720896	−	0.153231i	0.167168	−	0.985928i	\(-0.446538\pi\)
							0.553728	+	0.832697i	\(0.313205\pi\)
\(43\)	4.38897	+	4.87445i	0.669312	+	0.743347i	0.978180	−	0.207758i	\(-0.0666167\pi\)
							−0.308868	+	0.951105i	\(0.599950\pi\)
\(47\)	−1.30682	−	4.02199i	−0.190620	−	0.586667i	0.809380	−	0.587285i	\(-0.199803\pi\)
							−1.00000		0.000618100i	\(0.999803\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	31.2.g.a.9.1	yes	16
3.2	odd	2		279.2.y.c.226.2		16
4.3	odd	2		496.2.bg.c.257.1		16
5.2	odd	4		775.2.ck.a.474.1		32
5.3	odd	4		775.2.ck.a.474.4		32
5.4	even	2		775.2.bl.a.226.2		16
31.2	even	5		961.2.c.j.439.2		16
31.3	odd	30		961.2.g.n.732.2		16
31.4	even	5		961.2.g.k.547.1		16
31.5	even	3		961.2.d.o.628.2		16
31.6	odd	6		961.2.g.j.448.1		16
31.7	even	15	inner	31.2.g.a.7.1	✓	16
31.8	even	5		961.2.g.t.235.2		16
31.9	even	15		961.2.d.p.388.3		16
31.10	even	15		961.2.a.i.1.2		8
31.11	odd	30		961.2.d.n.531.2		16
31.12	odd	30		961.2.c.i.521.2		16
31.13	odd	30		961.2.d.q.374.3		16
31.14	even	15		961.2.g.s.338.2		16
31.15	odd	10		961.2.g.m.816.2		16
31.16	even	5		961.2.g.s.816.2		16
31.17	odd	30		961.2.g.m.338.2		16
31.18	even	15		961.2.d.p.374.3		16
31.19	even	15		961.2.c.j.521.2		16
31.20	even	15		961.2.d.o.531.2		16
31.21	odd	30		961.2.a.j.1.2		8
31.22	odd	30		961.2.d.q.388.3		16
31.23	odd	10		961.2.g.n.235.2		16
31.24	odd	30		961.2.g.l.844.1		16
31.25	even	3		961.2.g.k.448.1		16
31.26	odd	6		961.2.d.n.628.2		16
31.27	odd	10		961.2.g.j.547.1		16
31.28	even	15		961.2.g.t.732.2		16
31.29	odd	10		961.2.c.i.439.2		16
31.30	odd	2		961.2.g.l.846.1		16
93.38	odd	30		279.2.y.c.100.2		16
93.41	odd	30		8649.2.a.bf.1.7		8
93.83	even	30		8649.2.a.be.1.7		8
124.7	odd	30		496.2.bg.c.193.1		16
155.7	odd	60		775.2.ck.a.224.4		32
155.38	odd	60		775.2.ck.a.224.1		32
155.69	even	30		775.2.bl.a.751.2		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.2.g.a.7.1	✓	16	31.7	even	15	inner
31.2.g.a.9.1	yes	16	1.1	even	1	trivial
279.2.y.c.100.2		16	93.38	odd	30
279.2.y.c.226.2		16	3.2	odd	2
496.2.bg.c.193.1		16	124.7	odd	30
496.2.bg.c.257.1		16	4.3	odd	2
775.2.bl.a.226.2		16	5.4	even	2
775.2.bl.a.751.2		16	155.69	even	30
775.2.ck.a.224.1		32	155.38	odd	60
775.2.ck.a.224.4		32	155.7	odd	60
775.2.ck.a.474.1		32	5.2	odd	4
775.2.ck.a.474.4		32	5.3	odd	4
961.2.a.i.1.2		8	31.10	even	15
961.2.a.j.1.2		8	31.21	odd	30
961.2.c.i.439.2		16	31.29	odd	10
961.2.c.i.521.2		16	31.12	odd	30
961.2.c.j.439.2		16	31.2	even	5
961.2.c.j.521.2		16	31.19	even	15
961.2.d.n.531.2		16	31.11	odd	30
961.2.d.n.628.2		16	31.26	odd	6
961.2.d.o.531.2		16	31.20	even	15
961.2.d.o.628.2		16	31.5	even	3
961.2.d.p.374.3		16	31.18	even	15
961.2.d.p.388.3		16	31.9	even	15
961.2.d.q.374.3		16	31.13	odd	30
961.2.d.q.388.3		16	31.22	odd	30
961.2.g.j.448.1		16	31.6	odd	6
961.2.g.j.547.1		16	31.27	odd	10
961.2.g.k.448.1		16	31.25	even	3
961.2.g.k.547.1		16	31.4	even	5
961.2.g.l.844.1		16	31.24	odd	30
961.2.g.l.846.1		16	31.30	odd	2
961.2.g.m.338.2		16	31.17	odd	30
961.2.g.m.816.2		16	31.15	odd	10
961.2.g.n.235.2		16	31.23	odd	10
961.2.g.n.732.2		16	31.3	odd	30
961.2.g.s.338.2		16	31.14	even	15
961.2.g.s.816.2		16	31.16	even	5
961.2.g.t.235.2		16	31.8	even	5
961.2.g.t.732.2		16	31.28	even	15
8649.2.a.be.1.7		8	93.83	even	30
8649.2.a.bf.1.7		8	93.41	odd	30