Embedded newform 5184.2.d.e.2593.3

• Label: 5184.2.d.e.2593.3
• Level: $5184$
• Weight: $2$
• Character: 5184.2593
• Analytic conductor: $41.394$
• Analytic rank: $0$
• Dimension: $4$
• CM discriminant: -8
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 5184 = 2^{6} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5184.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(41.3944484078\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(i, \sqrt{6})\)
Defining polynomial:	\( x^{4} + 9 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2^{3} \)
Twist minimal:	no (minimal twist has level 576)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			2593.3
Root			\(-1.22474 - 1.22474i\) of defining polynomial
Character	\(\chi\)	\(=\)	5184.2593
Dual form			5184.2.d.e.2593.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+0.550510i q^{11} -7.89898 q^{17} -6.34847i q^{19} +5.00000 q^{25} +6.79796 q^{41} +12.3485i q^{43} -7.00000 q^{49} +15.2474i q^{59} +0.348469i q^{67} -15.6969 q^{73} +18.0000i q^{83} -18.0000 q^{89} +9.69694 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{17} + 20 q^{25} - 12 q^{41} - 28 q^{49} - 4 q^{73} - 72 q^{89} - 20 q^{97}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(1217\)	\(2431\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)

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
\(5\)	0	0				1.00000			\(0\)
			−1.00000			\(\pi\)
\(7\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(11\)	0.550510i	0.165985i	0.996550	+	0.0829925i	\(0.0264478\pi\)
			−0.996550	+	0.0829925i	\(0.973552\pi\)
\(13\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(17\)	−7.89898	−1.91578	−0.957892	−	0.287129i	\(-0.907299\pi\)
			−0.957892	+	0.287129i	\(0.907299\pi\)
\(19\)	− 6.34847i	− 1.45644i	−0.685344	−	0.728219i	\(-0.740348\pi\)
			0.685344	−	0.728219i	\(-0.259652\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(31\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(37\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(41\)	6.79796	1.06166	0.530831	−	0.847477i	\(-0.321880\pi\)
			0.530831	+	0.847477i	\(0.321880\pi\)
\(43\)	12.3485i	1.88312i	0.336840	+	0.941562i	\(0.390642\pi\)
			−0.336840	+	0.941562i	\(0.609358\pi\)
\(47\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(53\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(59\)	15.2474i	1.98505i	0.122047	+	0.992524i	\(0.461054\pi\)
			−0.122047	+	0.992524i	\(0.538946\pi\)
\(61\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(67\)	0.348469i	0.0425723i	0.999773	+	0.0212861i	\(0.00677610\pi\)
			−0.999773	+	0.0212861i	\(0.993224\pi\)
\(71\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(73\)	−15.6969	−1.83719	−0.918594	−	0.395203i	\(-0.870674\pi\)
			−0.918594	+	0.395203i	\(0.870674\pi\)
\(79\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(83\)	18.0000i	1.97576i	0.155230	+	0.987878i	\(0.450388\pi\)
			−0.155230	+	0.987878i	\(0.549612\pi\)
\(89\)	−18.0000	−1.90800	−0.953998	−	0.299813i	\(-0.903076\pi\)
			−0.953998	+	0.299813i	\(0.903076\pi\)
\(97\)	9.69694	0.984575	0.492287	−	0.870433i	\(-0.336161\pi\)
			0.492287	+	0.870433i	\(0.336161\pi\)

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	5184.2.d.e.2593.3		4
3.2	odd	2		5184.2.d.l.2593.2		4
4.3	odd	2	inner	5184.2.d.e.2593.2		4
8.3	odd	2	CM	5184.2.d.e.2593.3		4
8.5	even	2	inner	5184.2.d.e.2593.2		4
9.2	odd	6		576.2.r.c.481.1	yes	8
9.4	even	3		1728.2.r.c.289.2		8
9.5	odd	6		576.2.r.c.97.4	yes	8
9.7	even	3		1728.2.r.c.1441.3		8
12.11	even	2		5184.2.d.l.2593.3		4
24.5	odd	2		5184.2.d.l.2593.3		4
24.11	even	2		5184.2.d.l.2593.2		4
36.7	odd	6		1728.2.r.c.1441.2		8
36.11	even	6		576.2.r.c.481.4	yes	8
36.23	even	6		576.2.r.c.97.1	✓	8
36.31	odd	6		1728.2.r.c.289.3		8
72.5	odd	6		576.2.r.c.97.1	✓	8
72.11	even	6		576.2.r.c.481.1	yes	8
72.13	even	6		1728.2.r.c.289.3		8
72.29	odd	6		576.2.r.c.481.4	yes	8
72.43	odd	6		1728.2.r.c.1441.3		8
72.59	even	6		576.2.r.c.97.4	yes	8
72.61	even	6		1728.2.r.c.1441.2		8
72.67	odd	6		1728.2.r.c.289.2		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.r.c.97.1	✓	8	36.23	even	6
576.2.r.c.97.1	✓	8	72.5	odd	6
576.2.r.c.97.4	yes	8	9.5	odd	6
576.2.r.c.97.4	yes	8	72.59	even	6
576.2.r.c.481.1	yes	8	9.2	odd	6
576.2.r.c.481.1	yes	8	72.11	even	6
576.2.r.c.481.4	yes	8	36.11	even	6
576.2.r.c.481.4	yes	8	72.29	odd	6
1728.2.r.c.289.2		8	9.4	even	3
1728.2.r.c.289.2		8	72.67	odd	6
1728.2.r.c.289.3		8	36.31	odd	6
1728.2.r.c.289.3		8	72.13	even	6
1728.2.r.c.1441.2		8	36.7	odd	6
1728.2.r.c.1441.2		8	72.61	even	6
1728.2.r.c.1441.3		8	9.7	even	3
1728.2.r.c.1441.3		8	72.43	odd	6
5184.2.d.e.2593.2		4	4.3	odd	2	inner
5184.2.d.e.2593.2		4	8.5	even	2	inner
5184.2.d.e.2593.3		4	1.1	even	1	trivial
5184.2.d.e.2593.3		4	8.3	odd	2	CM
5184.2.d.l.2593.2		4	3.2	odd	2
5184.2.d.l.2593.2		4	24.11	even	2
5184.2.d.l.2593.3		4	12.11	even	2
5184.2.d.l.2593.3		4	24.5	odd	2