Embedded newform 31.2.g.a.10.1

• Label: 31.2.g.a.10.1
• Level: $31$
• Weight: $2$
• Character: 31.10
• 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			10.1
Root			\(1.03739i\) of defining polynomial
Character	\(\chi\)	\(=\)	31.10
Dual form			31.2.g.a.28.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.02470 - 0.744490i) q^{2} +(-0.155153 - 1.47618i) q^{3} +(-0.122284 - 0.376353i) q^{4} +(1.90016 + 3.29117i) q^{5} +(-0.940018 + 1.62816i) q^{6} +(-2.14115 - 0.455117i) q^{7} +(-0.937688 + 2.88591i) q^{8} +(0.779397 - 0.165666i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{2} - 12 q^{3} - 14 q^{4} - 3 q^{5} + 11 q^{6} + 2 q^{7} + 17 q^{8} - 10 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{7}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.02470	−	0.744490i	−0.724574	−	0.526434i	0.163268	−	0.986582i	\(-0.447796\pi\)
							−0.887842	+	0.460148i	\(0.847796\pi\)
\(3\)	−0.155153	−	1.47618i	−0.0895777	−	0.852275i	−0.943388	−	0.331691i	\(-0.892381\pi\)
							0.853810	−	0.520584i	\(-0.174286\pi\)
\(5\)	1.90016	+	3.29117i	0.849778	+	1.47186i	0.881407	+	0.472358i	\(0.156597\pi\)
							−0.0316291	+	0.999500i	\(0.510070\pi\)
\(7\)	−2.14115	−	0.455117i	−0.809280	−	0.172018i	−0.215353	−	0.976536i	\(-0.569090\pi\)
							−0.593928	+	0.804518i	\(0.702423\pi\)
\(11\)	−0.636073	+	0.706430i	−0.191783	+	0.212997i	−0.831366	−	0.555726i	\(-0.812441\pi\)
							0.639583	+	0.768722i	\(0.279107\pi\)
\(13\)	0.153606	+	0.0683897i	0.0426026	+	0.0189679i	0.427927	−	0.903813i	\(-0.359244\pi\)
							−0.385325	+	0.922781i	\(0.625911\pi\)
\(17\)	−4.40064	−	4.88741i	−1.06731	−	1.18537i	−0.981974	−	0.189018i	\(-0.939469\pi\)
							−0.0853387	−	0.996352i	\(-0.527197\pi\)
\(19\)	−1.05317	+	0.468903i	−0.241615	+	0.107574i	−0.523972	−	0.851736i	\(-0.675550\pi\)
							0.282357	+	0.959309i	\(0.408884\pi\)
\(23\)	−1.43029	+	4.40197i	−0.298235	+	0.917873i	0.683880	+	0.729594i	\(0.260291\pi\)
							−0.982116	+	0.188279i	\(0.939709\pi\)
\(29\)	−1.08143	−	0.785701i	−0.200816	−	0.145901i	0.482833	−	0.875713i	\(-0.339608\pi\)
							−0.683649	+	0.729811i	\(0.739608\pi\)
\(31\)	5.56350	−	0.217815i	0.999234	−	0.0391208i
							\(37\)	1.93582	−	3.35295i	0.318248	−	0.551221i	−0.661875	−	0.749614i	\(-0.730239\pi\)
0.980122	+	0.198393i	\(0.0635723\pi\)
\(41\)	−0.0343065	+	0.326405i	−0.00535778	+	0.0509758i	−0.996874	−	0.0790062i	\(-0.974825\pi\)
							0.991516	+	0.129982i	\(0.0414920\pi\)
\(43\)	8.79797	−	3.91711i	1.34168	−	0.597353i	0.394746	−	0.918790i	\(-0.370833\pi\)
							0.946931	+	0.321437i	\(0.104166\pi\)
\(47\)	−4.56170	+	3.31427i	−0.665393	+	0.483436i	−0.868480	−	0.495725i	\(-0.834903\pi\)
							0.203087	+	0.979161i	\(0.434903\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	31.2.g.a.10.1	✓	16
3.2	odd	2		279.2.y.c.10.2		16
4.3	odd	2		496.2.bg.c.289.2		16
5.2	odd	4		775.2.ck.a.599.4		32
5.3	odd	4		775.2.ck.a.599.1		32
5.4	even	2		775.2.bl.a.351.2		16
31.2	even	5		961.2.g.t.846.2		16
31.3	odd	30		961.2.g.l.338.1		16
31.4	even	5		961.2.c.j.439.6		16
31.5	even	3		961.2.d.o.374.2		16
31.6	odd	6		961.2.g.j.732.1		16
31.7	even	15		961.2.c.j.521.6		16
31.8	even	5		961.2.g.s.547.2		16
31.9	even	15		961.2.d.p.531.3		16
31.10	even	15		961.2.d.p.628.3		16
31.11	odd	30		961.2.a.j.1.6		8
31.12	odd	30		961.2.g.m.448.2		16
31.13	odd	30		961.2.d.n.388.2		16
31.14	even	15		961.2.g.t.844.2		16
31.15	odd	10		961.2.g.j.235.1		16
31.16	even	5		961.2.g.k.235.1		16
31.17	odd	30		961.2.g.n.844.2		16
31.18	even	15		961.2.d.o.388.2		16
31.19	even	15		961.2.g.s.448.2		16
31.20	even	15		961.2.a.i.1.6		8
31.21	odd	30		961.2.d.q.628.3		16
31.22	odd	30		961.2.d.q.531.3		16
31.23	odd	10		961.2.g.m.547.2		16
31.24	odd	30		961.2.c.i.521.6		16
31.25	even	3		961.2.g.k.732.1		16
31.26	odd	6		961.2.d.n.374.2		16
31.27	odd	10		961.2.c.i.439.6		16
31.28	even	15	inner	31.2.g.a.28.1	yes	16
31.29	odd	10		961.2.g.n.846.2		16
31.30	odd	2		961.2.g.l.816.1		16
93.11	even	30		8649.2.a.be.1.3		8
93.20	odd	30		8649.2.a.bf.1.3		8
93.59	odd	30		279.2.y.c.28.2		16
124.59	odd	30		496.2.bg.c.369.2		16
155.28	odd	60		775.2.ck.a.524.4		32
155.59	even	30		775.2.bl.a.276.2		16
155.152	odd	60		775.2.ck.a.524.1		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.2.g.a.10.1	✓	16	1.1	even	1	trivial
31.2.g.a.28.1	yes	16	31.28	even	15	inner
279.2.y.c.10.2		16	3.2	odd	2
279.2.y.c.28.2		16	93.59	odd	30
496.2.bg.c.289.2		16	4.3	odd	2
496.2.bg.c.369.2		16	124.59	odd	30
775.2.bl.a.276.2		16	155.59	even	30
775.2.bl.a.351.2		16	5.4	even	2
775.2.ck.a.524.1		32	155.152	odd	60
775.2.ck.a.524.4		32	155.28	odd	60
775.2.ck.a.599.1		32	5.3	odd	4
775.2.ck.a.599.4		32	5.2	odd	4
961.2.a.i.1.6		8	31.20	even	15
961.2.a.j.1.6		8	31.11	odd	30
961.2.c.i.439.6		16	31.27	odd	10
961.2.c.i.521.6		16	31.24	odd	30
961.2.c.j.439.6		16	31.4	even	5
961.2.c.j.521.6		16	31.7	even	15
961.2.d.n.374.2		16	31.26	odd	6
961.2.d.n.388.2		16	31.13	odd	30
961.2.d.o.374.2		16	31.5	even	3
961.2.d.o.388.2		16	31.18	even	15
961.2.d.p.531.3		16	31.9	even	15
961.2.d.p.628.3		16	31.10	even	15
961.2.d.q.531.3		16	31.22	odd	30
961.2.d.q.628.3		16	31.21	odd	30
961.2.g.j.235.1		16	31.15	odd	10
961.2.g.j.732.1		16	31.6	odd	6
961.2.g.k.235.1		16	31.16	even	5
961.2.g.k.732.1		16	31.25	even	3
961.2.g.l.338.1		16	31.3	odd	30
961.2.g.l.816.1		16	31.30	odd	2
961.2.g.m.448.2		16	31.12	odd	30
961.2.g.m.547.2		16	31.23	odd	10
961.2.g.n.844.2		16	31.17	odd	30
961.2.g.n.846.2		16	31.29	odd	10
961.2.g.s.448.2		16	31.19	even	15
961.2.g.s.547.2		16	31.8	even	5
961.2.g.t.844.2		16	31.14	even	15
961.2.g.t.846.2		16	31.2	even	5
8649.2.a.be.1.3		8	93.11	even	30
8649.2.a.bf.1.3		8	93.20	odd	30