Embedded newform 1584.2.o.a.703.2

• Label: 1584.2.o.a.703.2
• Level: $1584$
• Weight: $2$
• Character: 1584.703
• Analytic conductor: $12.648$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -11
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1584.o (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.6483036802\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-11}) \)
Defining polynomial:	\( x^{2} - x + 3 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 176)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			703.2
Root			\(0.500000 - 1.65831i\) of defining polynomial
Character	\(\chi\)	\(=\)	1584.703
Dual form			1584.2.o.a.703.1

$q$-expansion

\(f(q)\)	\(=\)	\(q-3.00000 q^{5} +3.31662i q^{11} -3.31662i q^{23} +4.00000 q^{25} -9.94987i q^{31} +7.00000 q^{37} -6.63325i q^{47} -7.00000 q^{49} -6.00000 q^{53} -9.94987i q^{55} -3.31662i q^{59} -9.94987i q^{67} -16.5831i q^{71} +9.00000 q^{89} -17.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{5} + 8 q^{25} + 14 q^{37} - 14 q^{49} - 12 q^{53} + 18 q^{89} - 34 q^{97}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1584\mathbb{Z}\right)^\times\).

\(n\)	\(145\)	\(353\)	\(991\)	\(1189\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1\)	\(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	0	0
\(5\)	−3.00000	−1.34164				−0.670820	−	0.741620i	\(-0.734058\pi\)
			−0.670820	+	0.741620i	\(0.734058\pi\)
\(7\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(11\)	3.31662i	1.00000i
			\(13\)	0	0	1.00000			\(0\)
−1.00000						\(\pi\)
\(17\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(19\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(23\)	− 3.31662i	− 0.691564i	−0.938315	−	0.345782i	\(-0.887614\pi\)
			0.938315	−	0.345782i	\(-0.112386\pi\)
\(29\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(31\)	− 9.94987i	− 1.78705i	−0.449013	−	0.893525i	\(-0.648224\pi\)
			0.449013	−	0.893525i	\(-0.351776\pi\)
\(37\)	7.00000	1.15079	0.575396	−	0.817875i	\(-0.304848\pi\)
			0.575396	+	0.817875i	\(0.304848\pi\)
\(41\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(43\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(47\)	− 6.63325i	− 0.967559i	−0.875190	−	0.483779i	\(-0.839264\pi\)
			0.875190	−	0.483779i	\(-0.160736\pi\)
\(53\)	−6.00000	−0.824163	−0.412082	−	0.911147i	\(-0.635198\pi\)
			−0.412082	+	0.911147i	\(0.635198\pi\)
\(59\)	− 3.31662i	− 0.431788i	−0.976417	−	0.215894i	\(-0.930733\pi\)
			0.976417	−	0.215894i	\(-0.0692665\pi\)
\(61\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(67\)	− 9.94987i	− 1.21557i	−0.794101	−	0.607785i	\(-0.792058\pi\)
			0.794101	−	0.607785i	\(-0.207942\pi\)
\(71\)	− 16.5831i	− 1.96805i	−0.178017	−	0.984027i	\(-0.556968\pi\)
			0.178017	−	0.984027i	\(-0.443032\pi\)
\(73\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(79\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(83\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(89\)	9.00000	0.953998	0.476999	−	0.878904i	\(-0.341725\pi\)
			0.476999	+	0.878904i	\(0.341725\pi\)
\(97\)	−17.0000	−1.72609	−0.863044	−	0.505128i	\(-0.831445\pi\)
			−0.863044	+	0.505128i	\(0.831445\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1584.2.o.a.703.2		2
3.2	odd	2		176.2.e.a.175.2	yes	2
4.3	odd	2	inner	1584.2.o.a.703.1		2
11.10	odd	2	CM	1584.2.o.a.703.2		2
12.11	even	2		176.2.e.a.175.1	✓	2
24.5	odd	2		704.2.e.a.703.1		2
24.11	even	2		704.2.e.a.703.2		2
33.32	even	2		176.2.e.a.175.2	yes	2
44.43	even	2	inner	1584.2.o.a.703.1		2
48.5	odd	4		2816.2.g.a.1407.4		4
48.11	even	4		2816.2.g.a.1407.2		4
48.29	odd	4		2816.2.g.a.1407.1		4
48.35	even	4		2816.2.g.a.1407.3		4
132.131	odd	2		176.2.e.a.175.1	✓	2
264.131	odd	2		704.2.e.a.703.2		2
264.197	even	2		704.2.e.a.703.1		2
528.131	odd	4		2816.2.g.a.1407.3		4
528.197	even	4		2816.2.g.a.1407.4		4
528.395	odd	4		2816.2.g.a.1407.2		4
528.461	even	4		2816.2.g.a.1407.1		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
176.2.e.a.175.1	✓	2	12.11	even	2
176.2.e.a.175.1	✓	2	132.131	odd	2
176.2.e.a.175.2	yes	2	3.2	odd	2
176.2.e.a.175.2	yes	2	33.32	even	2
704.2.e.a.703.1		2	24.5	odd	2
704.2.e.a.703.1		2	264.197	even	2
704.2.e.a.703.2		2	24.11	even	2
704.2.e.a.703.2		2	264.131	odd	2
1584.2.o.a.703.1		2	4.3	odd	2	inner
1584.2.o.a.703.1		2	44.43	even	2	inner
1584.2.o.a.703.2		2	1.1	even	1	trivial
1584.2.o.a.703.2		2	11.10	odd	2	CM
2816.2.g.a.1407.1		4	48.29	odd	4
2816.2.g.a.1407.1		4	528.461	even	4
2816.2.g.a.1407.2		4	48.11	even	4
2816.2.g.a.1407.2		4	528.395	odd	4
2816.2.g.a.1407.3		4	48.35	even	4
2816.2.g.a.1407.3		4	528.131	odd	4
2816.2.g.a.1407.4		4	48.5	odd	4
2816.2.g.a.1407.4		4	528.197	even	4