Embedded newform 2366.2.a.b.1.1

• Label: 2366.2.a.b.1.1
• Level: $2366$
• Weight: $2$
• Character: 2366.1
• Self dual: yes
• Analytic conductor: $18.893$
• Analytic rank: $1$
• 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:	\(1\)
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

\(f(q)\)

\(=\)

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

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
			\(3\)	−2.00000	−1.15470	−0.577350	−	0.816497i	\(-0.695913\pi\)
−0.577350	+	0.816497i				\(0.695913\pi\)
\(5\)	1.00000	0.447214	0.223607	−	0.974679i	\(-0.428217\pi\)
			0.223607	+	0.974679i	\(0.428217\pi\)
\(7\)	−1.00000	−0.377964
			\(11\)	2.00000	0.603023	0.301511	−	0.953463i	\(-0.402509\pi\)
0.301511	+	0.953463i				\(0.402509\pi\)
\(13\)	0	0
			\(17\)	1.00000	0.242536	0.121268	−	0.992620i	\(-0.461304\pi\)
0.121268	+	0.992620i				\(0.461304\pi\)
\(19\)	−4.00000	−0.917663	−0.458831	−	0.888523i	\(-0.651732\pi\)
			−0.458831	+	0.888523i	\(0.651732\pi\)
\(23\)	2.00000	0.417029	0.208514	−	0.978019i	\(-0.433137\pi\)
			0.208514	+	0.978019i	\(0.433137\pi\)
\(29\)	−5.00000	−0.928477	−0.464238	−	0.885710i	\(-0.653672\pi\)
			−0.464238	+	0.885710i	\(0.653672\pi\)
\(31\)	6.00000	1.07763	0.538816	−	0.842424i	\(-0.318872\pi\)
			0.538816	+	0.842424i	\(0.318872\pi\)
\(37\)	7.00000	1.15079	0.575396	−	0.817875i	\(-0.304848\pi\)
			0.575396	+	0.817875i	\(0.304848\pi\)
\(41\)	−7.00000	−1.09322	−0.546608	−	0.837389i	\(-0.684081\pi\)
			−0.546608	+	0.837389i	\(0.684081\pi\)
\(43\)	−2.00000	−0.304997	−0.152499	−	0.988304i	\(-0.548732\pi\)
			−0.152499	+	0.988304i	\(0.548732\pi\)
\(47\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2366.2.a.b.1.1		1
13.3	even	3		182.2.g.d.113.1	yes	2
13.5	odd	4		2366.2.d.c.337.2		2
13.8	odd	4		2366.2.d.c.337.1		2
13.9	even	3		182.2.g.d.29.1	✓	2
13.12	even	2		2366.2.a.i.1.1		1
39.29	odd	6		1638.2.r.e.1387.1		2
39.35	odd	6		1638.2.r.e.757.1		2
52.3	odd	6		1456.2.s.b.113.1		2
52.35	odd	6		1456.2.s.b.1121.1		2
91.3	odd	6		1274.2.h.m.373.1		2
91.9	even	3		1274.2.h.d.263.1		2
91.16	even	3		1274.2.e.j.165.1		2
91.48	odd	6		1274.2.g.b.393.1		2
91.55	odd	6		1274.2.g.b.295.1		2
91.61	odd	6		1274.2.h.m.263.1		2
91.68	odd	6		1274.2.e.c.165.1		2
91.74	even	3		1274.2.e.j.471.1		2
91.81	even	3		1274.2.h.d.373.1		2
91.87	odd	6		1274.2.e.c.471.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.2.g.d.29.1	✓	2	13.9	even	3
182.2.g.d.113.1	yes	2	13.3	even	3
1274.2.e.c.165.1		2	91.68	odd	6
1274.2.e.c.471.1		2	91.87	odd	6
1274.2.e.j.165.1		2	91.16	even	3
1274.2.e.j.471.1		2	91.74	even	3
1274.2.g.b.295.1		2	91.55	odd	6
1274.2.g.b.393.1		2	91.48	odd	6
1274.2.h.d.263.1		2	91.9	even	3
1274.2.h.d.373.1		2	91.81	even	3
1274.2.h.m.263.1		2	91.61	odd	6
1274.2.h.m.373.1		2	91.3	odd	6
1456.2.s.b.113.1		2	52.3	odd	6
1456.2.s.b.1121.1		2	52.35	odd	6
1638.2.r.e.757.1		2	39.35	odd	6
1638.2.r.e.1387.1		2	39.29	odd	6
2366.2.a.b.1.1		1	1.1	even	1	trivial
2366.2.a.i.1.1		1	13.12	even	2
2366.2.d.c.337.1		2	13.8	odd	4
2366.2.d.c.337.2		2	13.5	odd	4