Embedded newform 2366.2.a.n.1.1

• Label: 2366.2.a.n.1.1
• Level: $2366$
• Weight: $2$
• Character: 2366.1
• Self dual: yes
• Analytic conductor: $18.893$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2366.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(18.8926051182\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 182)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	2366.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\)

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	1.00000	0.707107
			\(3\)	1.00000	0.577350	0.288675	−	0.957427i	\(-0.406785\pi\)
0.288675	+	0.957427i				\(0.406785\pi\)
\(5\)	−3.00000	−1.34164	−0.670820	−	0.741620i	\(-0.734058\pi\)
			−0.670820	+	0.741620i	\(0.734058\pi\)
\(7\)	−1.00000	−0.377964
			\(11\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(13\)	0	0
			\(17\)	6.00000	1.45521	0.727607	−	0.685994i	\(-0.240633\pi\)
0.727607	+	0.685994i				\(0.240633\pi\)
\(19\)	4.00000	0.917663	0.458831	−	0.888523i	\(-0.348268\pi\)
			0.458831	+	0.888523i	\(0.348268\pi\)
\(23\)	3.00000	0.625543	0.312772	−	0.949828i	\(-0.398743\pi\)
			0.312772	+	0.949828i	\(0.398743\pi\)
\(29\)	6.00000	1.11417	0.557086	−	0.830455i	\(-0.311919\pi\)
			0.557086	+	0.830455i	\(0.311919\pi\)
\(31\)	10.0000	1.79605	0.898027	−	0.439941i	\(-0.145001\pi\)
			0.898027	+	0.439941i	\(0.145001\pi\)
\(37\)	−8.00000	−1.31519	−0.657596	−	0.753371i	\(-0.728427\pi\)
			−0.657596	+	0.753371i	\(0.728427\pi\)
\(41\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(43\)	8.00000	1.21999	0.609994	−	0.792406i	\(-0.291172\pi\)
			0.609994	+	0.792406i	\(0.291172\pi\)
\(47\)	−6.00000	−0.875190	−0.437595	−	0.899172i	\(-0.644170\pi\)
			−0.437595	+	0.899172i	\(0.644170\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2366.2.a.n.1.1		1
13.4	even	6		182.2.g.b.29.1	✓	2
13.5	odd	4		2366.2.d.f.337.1		2
13.8	odd	4		2366.2.d.f.337.2		2
13.10	even	6		182.2.g.b.113.1	yes	2
13.12	even	2		2366.2.a.f.1.1		1
39.17	odd	6		1638.2.r.c.757.1		2
39.23	odd	6		1638.2.r.c.1387.1		2
52.23	odd	6		1456.2.s.e.113.1		2
52.43	odd	6		1456.2.s.e.1121.1		2
91.4	even	6		1274.2.e.d.471.1		2
91.10	odd	6		1274.2.h.i.373.1		2
91.17	odd	6		1274.2.e.i.471.1		2
91.23	even	6		1274.2.e.d.165.1		2
91.30	even	6		1274.2.h.j.263.1		2
91.62	odd	6		1274.2.g.g.295.1		2
91.69	odd	6		1274.2.g.g.393.1		2
91.75	odd	6		1274.2.e.i.165.1		2
91.82	odd	6		1274.2.h.i.263.1		2
91.88	even	6		1274.2.h.j.373.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.2.g.b.29.1	✓	2	13.4	even	6
182.2.g.b.113.1	yes	2	13.10	even	6
1274.2.e.d.165.1		2	91.23	even	6
1274.2.e.d.471.1		2	91.4	even	6
1274.2.e.i.165.1		2	91.75	odd	6
1274.2.e.i.471.1		2	91.17	odd	6
1274.2.g.g.295.1		2	91.62	odd	6
1274.2.g.g.393.1		2	91.69	odd	6
1274.2.h.i.263.1		2	91.82	odd	6
1274.2.h.i.373.1		2	91.10	odd	6
1274.2.h.j.263.1		2	91.30	even	6
1274.2.h.j.373.1		2	91.88	even	6
1456.2.s.e.113.1		2	52.23	odd	6
1456.2.s.e.1121.1		2	52.43	odd	6
1638.2.r.c.757.1		2	39.17	odd	6
1638.2.r.c.1387.1		2	39.23	odd	6
2366.2.a.f.1.1		1	13.12	even	2
2366.2.a.n.1.1		1	1.1	even	1	trivial
2366.2.d.f.337.1		2	13.5	odd	4
2366.2.d.f.337.2		2	13.8	odd	4