Embedded newform 4900.2.a.a.1.1

• Label: 4900.2.a.a.1.1
• Level: $4900$
• Weight: $2$
• Character: 4900.1
• Self dual: yes
• Analytic conductor: $39.127$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

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:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
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			1.1
Character	\(\chi\)	\(=\)	4900.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-3.00000 q^{3} +6.00000 q^{9} -2.00000 q^{11} -6.00000 q^{13} +2.00000 q^{17} +9.00000 q^{23} -9.00000 q^{27} +3.00000 q^{29} -2.00000 q^{31} +6.00000 q^{33} -8.00000 q^{37} +18.0000 q^{39} -5.00000 q^{41} -1.00000 q^{43} +8.00000 q^{47} -6.00000 q^{51} -4.00000 q^{53} +8.00000 q^{59} -7.00000 q^{61} +3.00000 q^{67} -27.0000 q^{69} +8.00000 q^{71} +14.0000 q^{73} +4.00000 q^{79} +9.00000 q^{81} -1.00000 q^{83} -9.00000 q^{87} -13.0000 q^{89} +6.00000 q^{93} -10.0000 q^{97} -12.0000 q^{99} +O(q^{100})\)

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
			\(3\)	−3.00000	−1.73205	−0.866025	−	0.500000i	\(-0.833333\pi\)
−0.866025	+	0.500000i				\(0.833333\pi\)
\(5\)	0	0
			\(7\)	0	0
\(11\)	−2.00000	−0.603023				−0.301511	−	0.953463i	\(-0.597491\pi\)
			−0.301511	+	0.953463i	\(0.597491\pi\)
\(13\)	−6.00000	−1.66410	−0.832050	−	0.554700i	\(-0.812833\pi\)
			−0.832050	+	0.554700i	\(0.812833\pi\)
\(17\)	2.00000	0.485071	0.242536	−	0.970143i	\(-0.422021\pi\)
			0.242536	+	0.970143i	\(0.422021\pi\)
\(19\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(23\)	9.00000	1.87663	0.938315	−	0.345782i	\(-0.112386\pi\)
			0.938315	+	0.345782i	\(0.112386\pi\)
\(29\)	3.00000	0.557086	0.278543	−	0.960424i	\(-0.410149\pi\)
			0.278543	+	0.960424i	\(0.410149\pi\)
\(31\)	−2.00000	−0.359211	−0.179605	−	0.983739i	\(-0.557482\pi\)
			−0.179605	+	0.983739i	\(0.557482\pi\)
\(37\)	−8.00000	−1.31519	−0.657596	−	0.753371i	\(-0.728427\pi\)
			−0.657596	+	0.753371i	\(0.728427\pi\)
\(41\)	−5.00000	−0.780869	−0.390434	−	0.920631i	\(-0.627675\pi\)
			−0.390434	+	0.920631i	\(0.627675\pi\)
\(43\)	−1.00000	−0.152499	−0.0762493	−	0.997089i	\(-0.524294\pi\)
			−0.0762493	+	0.997089i	\(0.524294\pi\)
\(47\)	8.00000	1.16692	0.583460	−	0.812142i	\(-0.301699\pi\)
			0.583460	+	0.812142i	\(0.301699\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.a.a.1.1		1
5.2	odd	4		4900.2.e.b.2549.2		2
5.3	odd	4		4900.2.e.b.2549.1		2
5.4	even	2		980.2.a.i.1.1		1
7.3	odd	6		700.2.i.a.401.1		2
7.5	odd	6		700.2.i.a.501.1		2
7.6	odd	2		4900.2.a.v.1.1		1
15.14	odd	2		8820.2.a.k.1.1		1
20.19	odd	2		3920.2.a.d.1.1		1
35.3	even	12		700.2.r.c.149.2		4
35.4	even	6		980.2.i.a.961.1		2
35.9	even	6		980.2.i.a.361.1		2
35.12	even	12		700.2.r.c.249.2		4
35.13	even	4		4900.2.e.c.2549.2		2
35.17	even	12		700.2.r.c.149.1		4
35.19	odd	6		140.2.i.b.81.1	✓	2
35.24	odd	6		140.2.i.b.121.1	yes	2
35.27	even	4		4900.2.e.c.2549.1		2
35.33	even	12		700.2.r.c.249.1		4
35.34	odd	2		980.2.a.a.1.1		1
105.59	even	6		1260.2.s.b.541.1		2
105.89	even	6		1260.2.s.b.361.1		2
105.104	even	2		8820.2.a.w.1.1		1
140.19	even	6		560.2.q.a.81.1		2
140.59	even	6		560.2.q.a.401.1		2
140.139	even	2		3920.2.a.bi.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.i.b.81.1	✓	2	35.19	odd	6
140.2.i.b.121.1	yes	2	35.24	odd	6
560.2.q.a.81.1		2	140.19	even	6
560.2.q.a.401.1		2	140.59	even	6
700.2.i.a.401.1		2	7.3	odd	6
700.2.i.a.501.1		2	7.5	odd	6
700.2.r.c.149.1		4	35.17	even	12
700.2.r.c.149.2		4	35.3	even	12
700.2.r.c.249.1		4	35.33	even	12
700.2.r.c.249.2		4	35.12	even	12
980.2.a.a.1.1		1	35.34	odd	2
980.2.a.i.1.1		1	5.4	even	2
980.2.i.a.361.1		2	35.9	even	6
980.2.i.a.961.1		2	35.4	even	6
1260.2.s.b.361.1		2	105.89	even	6
1260.2.s.b.541.1		2	105.59	even	6
3920.2.a.d.1.1		1	20.19	odd	2
3920.2.a.bi.1.1		1	140.139	even	2
4900.2.a.a.1.1		1	1.1	even	1	trivial
4900.2.a.v.1.1		1	7.6	odd	2
4900.2.e.b.2549.1		2	5.3	odd	4
4900.2.e.b.2549.2		2	5.2	odd	4
4900.2.e.c.2549.1		2	35.27	even	4
4900.2.e.c.2549.2		2	35.13	even	4
8820.2.a.k.1.1		1	15.14	odd	2
8820.2.a.w.1.1		1	105.104	even	2