Embedded newform 4900.2.a.bj.1.3

• Label: 4900.2.a.bj.1.3
• Level: $4900$
• Weight: $2$
• Character: 4900.1
• Self dual: yes
• Analytic conductor: $39.127$
• Analytic rank: $0$
• Dimension: $4$
• CM discriminant: -35
• Inner twists: $4$

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:	\(\Q(\sqrt{5}, \sqrt{21})\)
Defining polynomial:	\( x^{4} - 13x^{2} + 16 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{3} \)
Twist minimal:	no (minimal twist has level 980)
Fricke sign:	\(-1\)
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.3
Root			\(1.17325\) of defining polynomial
Character	\(\chi\)	\(=\)	4900.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.17325 q^{3} -1.62348 q^{9} +6.62348 q^{11} +5.64539 q^{13} +7.99190 q^{17} -5.42451 q^{27} -0.623475 q^{29} +7.77102 q^{33} +6.62348 q^{39} -12.4640 q^{47} +9.37652 q^{51} -12.0000 q^{71} -13.4164 q^{73} +15.8704 q^{79} -1.49390 q^{81} +8.94427 q^{83} -0.731495 q^{87} +12.6849 q^{97} -10.7530 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 14 q^{9} + 6 q^{11} + 18 q^{29} + 6 q^{39} + 58 q^{51} - 48 q^{71} + 2 q^{79} + 76 q^{81} - 84 q^{99}+O(q^{100}) \)

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
			\(3\)	1.17325	0.677378	0.338689	−	0.940898i	\(-0.390016\pi\)
0.338689	+	0.940898i				\(0.390016\pi\)
\(5\)	0	0
			\(7\)	0	0
\(11\)	6.62348	1.99705				0.998526	−	0.0542666i	\(-0.0172821\pi\)
			0.998526	+	0.0542666i	\(0.0172821\pi\)
\(13\)	5.64539	1.56575	0.782875	−	0.622179i	\(-0.213753\pi\)
			0.782875	+	0.622179i	\(0.213753\pi\)
\(17\)	7.99190	1.93832	0.969160	−	0.246433i	\(-0.0792584\pi\)
			0.969160	+	0.246433i	\(0.0792584\pi\)
\(19\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	−0.623475	−0.115776	−0.0578882	−	0.998323i	\(-0.518437\pi\)
			−0.0578882	+	0.998323i	\(0.518437\pi\)
\(31\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(37\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(41\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(43\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(47\)	−12.4640	−1.81807	−0.909033	−	0.416724i	\(-0.863178\pi\)
			−0.909033	+	0.416724i	\(0.863178\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.a.bj.1.3		4
5.2	odd	4		980.2.e.d.589.2	✓	4
5.3	odd	4		980.2.e.d.589.3	yes	4
5.4	even	2	inner	4900.2.a.bj.1.2		4
7.6	odd	2	inner	4900.2.a.bj.1.2		4
35.2	odd	12		980.2.q.i.949.3		8
35.3	even	12		980.2.q.i.569.2		8
35.12	even	12		980.2.q.i.949.2		8
35.13	even	4		980.2.e.d.589.2	✓	4
35.17	even	12		980.2.q.i.569.3		8
35.18	odd	12		980.2.q.i.569.3		8
35.23	odd	12		980.2.q.i.949.2		8
35.27	even	4		980.2.e.d.589.3	yes	4
35.32	odd	12		980.2.q.i.569.2		8
35.33	even	12		980.2.q.i.949.3		8
35.34	odd	2	CM	4900.2.a.bj.1.3		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
980.2.e.d.589.2	✓	4	5.2	odd	4
980.2.e.d.589.2	✓	4	35.13	even	4
980.2.e.d.589.3	yes	4	5.3	odd	4
980.2.e.d.589.3	yes	4	35.27	even	4
980.2.q.i.569.2		8	35.3	even	12
980.2.q.i.569.2		8	35.32	odd	12
980.2.q.i.569.3		8	35.17	even	12
980.2.q.i.569.3		8	35.18	odd	12
980.2.q.i.949.2		8	35.12	even	12
980.2.q.i.949.2		8	35.23	odd	12
980.2.q.i.949.3		8	35.2	odd	12
980.2.q.i.949.3		8	35.33	even	12
4900.2.a.bj.1.2		4	5.4	even	2	inner
4900.2.a.bj.1.2		4	7.6	odd	2	inner
4900.2.a.bj.1.3		4	1.1	even	1	trivial
4900.2.a.bj.1.3		4	35.34	odd	2	CM