Embedded newform 4900.2.a.o.1.1

• Label: 4900.2.a.o.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 980)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	4900.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{3} -2.00000 q^{9} -1.00000 q^{11} -5.00000 q^{13} +1.00000 q^{17} +6.00000 q^{19} +4.00000 q^{23} -5.00000 q^{27} +3.00000 q^{29} -2.00000 q^{31} -1.00000 q^{33} -8.00000 q^{37} -5.00000 q^{39} +10.0000 q^{41} +2.00000 q^{43} -7.00000 q^{47} +1.00000 q^{51} +2.00000 q^{53} +6.00000 q^{57} -14.0000 q^{59} +8.00000 q^{61} -14.0000 q^{67} +4.00000 q^{69} -10.0000 q^{73} -11.0000 q^{79} +1.00000 q^{81} -4.00000 q^{83} +3.00000 q^{87} -4.00000 q^{89} -2.00000 q^{93} -3.00000 q^{97} +2.00000 q^{99} +O(q^{100})\)

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
			\(3\)	1.00000	0.577350	0.288675	−	0.957427i	\(-0.406785\pi\)
0.288675	+	0.957427i				\(0.406785\pi\)
\(5\)	0	0
			\(7\)	0	0
\(11\)	−1.00000	−0.301511				−0.150756	−	0.988571i	\(-0.548171\pi\)
			−0.150756	+	0.988571i	\(0.548171\pi\)
\(13\)	−5.00000	−1.38675	−0.693375	−	0.720577i	\(-0.743877\pi\)
			−0.693375	+	0.720577i	\(0.743877\pi\)
\(17\)	1.00000	0.242536	0.121268	−	0.992620i	\(-0.461304\pi\)
			0.121268	+	0.992620i	\(0.461304\pi\)
\(19\)	6.00000	1.37649	0.688247	−	0.725476i	\(-0.258380\pi\)
			0.688247	+	0.725476i	\(0.258380\pi\)
\(23\)	4.00000	0.834058	0.417029	−	0.908893i	\(-0.363071\pi\)
			0.417029	+	0.908893i	\(0.363071\pi\)
\(29\)	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	\(0.728427\pi\)
\(41\)	10.0000	1.56174	0.780869	−	0.624695i	\(-0.214777\pi\)
			0.780869	+	0.624695i	\(0.214777\pi\)
\(43\)	2.00000	0.304997	0.152499	−	0.988304i	\(-0.451268\pi\)
			0.152499	+	0.988304i	\(0.451268\pi\)
\(47\)	−7.00000	−1.02105	−0.510527	−	0.859861i	\(-0.670550\pi\)
			−0.510527	+	0.859861i	\(0.670550\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.a.o.1.1		1
5.2	odd	4		4900.2.e.j.2549.1		2
5.3	odd	4		4900.2.e.j.2549.2		2
5.4	even	2		980.2.a.d.1.1	✓	1
7.6	odd	2		4900.2.a.h.1.1		1
15.14	odd	2		8820.2.a.i.1.1		1
20.19	odd	2		3920.2.a.z.1.1		1
35.4	even	6		980.2.i.g.961.1		2
35.9	even	6		980.2.i.g.361.1		2
35.13	even	4		4900.2.e.k.2549.1		2
35.19	odd	6		980.2.i.e.361.1		2
35.24	odd	6		980.2.i.e.961.1		2
35.27	even	4		4900.2.e.k.2549.2		2
35.34	odd	2		980.2.a.f.1.1	yes	1
105.104	even	2		8820.2.a.v.1.1		1
140.139	even	2		3920.2.a.n.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
980.2.a.d.1.1	✓	1	5.4	even	2
980.2.a.f.1.1	yes	1	35.34	odd	2
980.2.i.e.361.1		2	35.19	odd	6
980.2.i.e.961.1		2	35.24	odd	6
980.2.i.g.361.1		2	35.9	even	6
980.2.i.g.961.1		2	35.4	even	6
3920.2.a.n.1.1		1	140.139	even	2
3920.2.a.z.1.1		1	20.19	odd	2
4900.2.a.h.1.1		1	7.6	odd	2
4900.2.a.o.1.1		1	1.1	even	1	trivial
4900.2.e.j.2549.1		2	5.2	odd	4
4900.2.e.j.2549.2		2	5.3	odd	4
4900.2.e.k.2549.1		2	35.13	even	4
4900.2.e.k.2549.2		2	35.27	even	4
8820.2.a.i.1.1		1	15.14	odd	2
8820.2.a.v.1.1		1	105.104	even	2