Embedded newform 4900.2.a.bf.1.4

• Label: 4900.2.a.bf.1.4
• 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			1.4
Root			\(-1.31342\) of defining polynomial
Character	\(\chi\)	\(=\)	4900.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} +2.27492 q^{11} -6.09095 q^{13} +4.77753 q^{17} +4.27492 q^{19} -0.894797 q^{23} -5.19615 q^{27} -3.27492 q^{29} +4.27492 q^{31} +3.94027 q^{33} +5.61478 q^{37} -10.5498 q^{39} +11.2749 q^{41} +6.50958 q^{43} +2.15068 q^{47} +8.27492 q^{51} -7.40437 q^{53} +7.40437 q^{57} -4.27492 q^{59} -1.54983 q^{61} +13.9140 q^{67} -1.54983 q^{69} +10.5498 q^{71} +2.15068 q^{73} -0.274917 q^{79} -9.00000 q^{81} +5.67232 q^{83} -5.67232 q^{87} +7.00000 q^{89} +7.40437 q^{93} -6.92820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{11} + 2 q^{19} + 2 q^{29} + 2 q^{31} - 12 q^{39} + 30 q^{41} + 18 q^{51} - 2 q^{59} + 24 q^{61} + 24 q^{69} + 12 q^{71} + 14 q^{79} - 36 q^{81} + 28 q^{89}+O(q^{100}) \)

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
\(11\)	2.27492	0.685913				0.342957	−	0.939351i	\(-0.388572\pi\)
			0.342957	+	0.939351i	\(0.388572\pi\)
\(13\)	−6.09095	−1.68933	−0.844663	−	0.535299i	\(-0.820199\pi\)
			−0.844663	+	0.535299i	\(0.820199\pi\)
\(17\)	4.77753	1.15872	0.579360	−	0.815072i	\(-0.303303\pi\)
			0.579360	+	0.815072i	\(0.303303\pi\)
\(19\)	4.27492	0.980733	0.490367	−	0.871516i	\(-0.336863\pi\)
			0.490367	+	0.871516i	\(0.336863\pi\)
\(23\)	−0.894797	−0.186578	−0.0932891	−	0.995639i	\(-0.529738\pi\)
			−0.0932891	+	0.995639i	\(0.529738\pi\)
\(29\)	−3.27492	−0.608137	−0.304068	−	0.952650i	\(-0.598345\pi\)
			−0.304068	+	0.952650i	\(0.598345\pi\)
\(31\)	4.27492	0.767798	0.383899	−	0.923375i	\(-0.374581\pi\)
			0.383899	+	0.923375i	\(0.374581\pi\)
\(37\)	5.61478	0.923064	0.461532	−	0.887124i	\(-0.347300\pi\)
			0.461532	+	0.887124i	\(0.347300\pi\)
\(41\)	11.2749	1.76085	0.880423	−	0.474189i	\(-0.157259\pi\)
			0.880423	+	0.474189i	\(0.157259\pi\)
\(43\)	6.50958	0.992701	0.496351	−	0.868122i	\(-0.334673\pi\)
			0.496351	+	0.868122i	\(0.334673\pi\)
\(47\)	2.15068	0.313709	0.156854	−	0.987622i	\(-0.449865\pi\)
			0.156854	+	0.987622i	\(0.449865\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.a.bf.1.4		4
5.2	odd	4		980.2.e.c.589.1		4
5.3	odd	4		980.2.e.c.589.3		4
5.4	even	2	inner	4900.2.a.bf.1.2		4
7.3	odd	6		700.2.i.f.401.4		8
7.5	odd	6		700.2.i.f.501.4		8
7.6	odd	2		4900.2.a.be.1.2		4
35.2	odd	12		980.2.q.g.949.2		4
35.3	even	12		140.2.q.a.9.1	✓	4
35.12	even	12		140.2.q.a.109.1	yes	4
35.13	even	4		980.2.e.f.589.2		4
35.17	even	12		140.2.q.b.9.2	yes	4
35.18	odd	12		980.2.q.g.569.2		4
35.19	odd	6		700.2.i.f.501.1		8
35.23	odd	12		980.2.q.b.949.2		4
35.24	odd	6		700.2.i.f.401.1		8
35.27	even	4		980.2.e.f.589.4		4
35.32	odd	12		980.2.q.b.569.1		4
35.33	even	12		140.2.q.b.109.1	yes	4
35.34	odd	2		4900.2.a.be.1.4		4
105.17	odd	12		1260.2.bm.b.289.1		4
105.38	odd	12		1260.2.bm.a.289.2		4
105.47	odd	12		1260.2.bm.a.109.2		4
105.68	odd	12		1260.2.bm.b.109.2		4
140.3	odd	12		560.2.bw.e.289.1		4
140.47	odd	12		560.2.bw.e.529.1		4
140.87	odd	12		560.2.bw.a.289.2		4
140.103	odd	12		560.2.bw.a.529.1		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.q.a.9.1	✓	4	35.3	even	12
140.2.q.a.109.1	yes	4	35.12	even	12
140.2.q.b.9.2	yes	4	35.17	even	12
140.2.q.b.109.1	yes	4	35.33	even	12
560.2.bw.a.289.2		4	140.87	odd	12
560.2.bw.a.529.1		4	140.103	odd	12
560.2.bw.e.289.1		4	140.3	odd	12
560.2.bw.e.529.1		4	140.47	odd	12
700.2.i.f.401.1		8	35.24	odd	6
700.2.i.f.401.4		8	7.3	odd	6
700.2.i.f.501.1		8	35.19	odd	6
700.2.i.f.501.4		8	7.5	odd	6
980.2.e.c.589.1		4	5.2	odd	4
980.2.e.c.589.3		4	5.3	odd	4
980.2.e.f.589.2		4	35.13	even	4
980.2.e.f.589.4		4	35.27	even	4
980.2.q.b.569.1		4	35.32	odd	12
980.2.q.b.949.2		4	35.23	odd	12
980.2.q.g.569.2		4	35.18	odd	12
980.2.q.g.949.2		4	35.2	odd	12
1260.2.bm.a.109.2		4	105.47	odd	12
1260.2.bm.a.289.2		4	105.38	odd	12
1260.2.bm.b.109.2		4	105.68	odd	12
1260.2.bm.b.289.1		4	105.17	odd	12
4900.2.a.be.1.2		4	7.6	odd	2
4900.2.a.be.1.4		4	35.34	odd	2
4900.2.a.bf.1.2		4	5.4	even	2	inner
4900.2.a.bf.1.4		4	1.1	even	1	trivial