Embedded newform 4900.2.a.bf.1.3

• Label: 4900.2.a.bf.1.3
• Level: $4900$
• Weight: $2$
• Character: 4900.1
• Self dual: yes
• Analytic conductor: $39.127$
• Analytic rank: $0$
• Dimension: $4$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$4900 = 2^{2} \cdot 5^{2} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4900.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$39.1266969904$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{19})$
Defining polynomial:	$x^{4} - 11x^{2} + 16$
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 140)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.04547$ of defining polynomial
Character	$\chi$	$=$	4900.1

$q$-expansion

$f(q)$	$=$	$q+1.73205 q^{3} -5.27492 q^{11} +2.62685 q^{13} +0.418627 q^{17} -3.27492 q^{19} +7.82300 q^{23} -5.19615 q^{27} +4.27492 q^{29} -3.27492 q^{31} -9.13642 q^{33} +9.97368 q^{37} +4.54983 q^{39} +3.72508 q^{41} +2.15068 q^{43} +6.50958 q^{47} +0.725083 q^{51} +5.67232 q^{53} -5.67232 q^{57} +3.27492 q^{59} +13.5498 q^{61} -3.52165 q^{67} +13.5498 q^{69} -4.54983 q^{71} +6.50958 q^{73} +7.27492 q^{79} -9.00000 q^{81} -7.40437 q^{83} +7.40437 q^{87} +7.00000 q^{89} -5.67232 q^{93} -6.92820 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 6 q^{11} + 2 q^{19} + 2 q^{29} + 2 q^{31} - 12 q^{39} + 30 q^{41} + 18 q^{51} - 2 q^{59} + 24 q^{61} + 24 q^{69} + 12 q^{71} + 14 q^{79} - 36 q^{81} + 28 q^{89}+O(q^{100})$

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)

$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$
$2$	0	0
			$3$	1.73205	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
0.500000	+	0.866025i				$0.333333\pi$
$5$	0	0
			$7$	0	0
$11$	−5.27492	−1.59045				−0.795224	−	0.606316i	$-0.792647\pi$
			−0.795224	+	0.606316i	$0.792647\pi$
$13$	2.62685	0.728557	0.364278	−	0.931290i	$-0.381316\pi$
			0.364278	+	0.931290i	$0.381316\pi$
$17$	0.418627	0.101532	0.0507659	−	0.998711i	$-0.483834\pi$
			0.0507659	+	0.998711i	$0.483834\pi$
$19$	−3.27492	−0.751318	−0.375659	−	0.926758i	$-0.622584\pi$
			−0.375659	+	0.926758i	$0.622584\pi$
$23$	7.82300	1.63121	0.815604	−	0.578610i	$-0.196405\pi$
			0.815604	+	0.578610i	$0.196405\pi$
$29$	4.27492	0.793832	0.396916	−	0.917855i	$-0.370080\pi$
			0.396916	+	0.917855i	$0.370080\pi$
$31$	−3.27492	−0.588192	−0.294096	−	0.955776i	$-0.595019\pi$
			−0.294096	+	0.955776i	$0.595019\pi$
$37$	9.97368	1.63966	0.819831	−	0.572605i	$-0.194067\pi$
			0.819831	+	0.572605i	$0.194067\pi$
$41$	3.72508	0.581760	0.290880	−	0.956760i	$-0.406052\pi$
			0.290880	+	0.956760i	$0.406052\pi$
$43$	2.15068	0.327975	0.163988	−	0.986462i	$-0.447564\pi$
			0.163988	+	0.986462i	$0.447564\pi$
$47$	6.50958	0.949519	0.474760	−	0.880115i	$-0.342535\pi$
			0.474760	+	0.880115i	$0.342535\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.a.bf.1.3		4
5.2	odd	4		980.2.e.c.589.2		4
5.3	odd	4		980.2.e.c.589.4		4
5.4	even	2	inner	4900.2.a.bf.1.1		4
7.3	odd	6		700.2.i.f.401.3		8
7.5	odd	6		700.2.i.f.501.3		8
7.6	odd	2		4900.2.a.be.1.1		4
35.2	odd	12		980.2.q.g.949.1		4
35.3	even	12		140.2.q.a.9.2	✓	4
35.12	even	12		140.2.q.a.109.2	yes	4
35.13	even	4		980.2.e.f.589.1		4
35.17	even	12		140.2.q.b.9.1	yes	4
35.18	odd	12		980.2.q.g.569.1		4
35.19	odd	6		700.2.i.f.501.2		8
35.23	odd	12		980.2.q.b.949.1		4
35.24	odd	6		700.2.i.f.401.2		8
35.27	even	4		980.2.e.f.589.3		4
35.32	odd	12		980.2.q.b.569.2		4
35.33	even	12		140.2.q.b.109.2	yes	4
35.34	odd	2		4900.2.a.be.1.3		4
105.17	odd	12		1260.2.bm.b.289.2		4
105.38	odd	12		1260.2.bm.a.289.1		4
105.47	odd	12		1260.2.bm.a.109.1		4
105.68	odd	12		1260.2.bm.b.109.1		4
140.3	odd	12		560.2.bw.e.289.2		4
140.47	odd	12		560.2.bw.e.529.2		4
140.87	odd	12		560.2.bw.a.289.1		4
140.103	odd	12		560.2.bw.a.529.2		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.q.a.9.2	✓	4	35.3	even	12
140.2.q.a.109.2	yes	4	35.12	even	12
140.2.q.b.9.1	yes	4	35.17	even	12
140.2.q.b.109.2	yes	4	35.33	even	12
560.2.bw.a.289.1		4	140.87	odd	12
560.2.bw.a.529.2		4	140.103	odd	12
560.2.bw.e.289.2		4	140.3	odd	12
560.2.bw.e.529.2		4	140.47	odd	12
700.2.i.f.401.2		8	35.24	odd	6
700.2.i.f.401.3		8	7.3	odd	6
700.2.i.f.501.2		8	35.19	odd	6
700.2.i.f.501.3		8	7.5	odd	6
980.2.e.c.589.2		4	5.2	odd	4
980.2.e.c.589.4		4	5.3	odd	4
980.2.e.f.589.1		4	35.13	even	4
980.2.e.f.589.3		4	35.27	even	4
980.2.q.b.569.2		4	35.32	odd	12
980.2.q.b.949.1		4	35.23	odd	12
980.2.q.g.569.1		4	35.18	odd	12
980.2.q.g.949.1		4	35.2	odd	12
1260.2.bm.a.109.1		4	105.47	odd	12
1260.2.bm.a.289.1		4	105.38	odd	12
1260.2.bm.b.109.1		4	105.68	odd	12
1260.2.bm.b.289.2		4	105.17	odd	12
4900.2.a.be.1.1		4	7.6	odd	2
4900.2.a.be.1.3		4	35.34	odd	2
4900.2.a.bf.1.1		4	5.4	even	2	inner
4900.2.a.bf.1.3		4	1.1	even	1	trivial