Embedded newform 2240.4.a.bg.1.1

• Label: 2240.4.a.bg.1.1
• Level: $2240$
• Weight: $4$
• Character: 2240.1
• Self dual: yes
• Analytic conductor: $132.164$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2240.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(132.164278413\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 280)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	2240.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+7.00000 q^{3} -5.00000 q^{5} -7.00000 q^{7} +22.0000 q^{9} +9.00000 q^{11} -23.0000 q^{13} -35.0000 q^{15} +41.0000 q^{17} +34.0000 q^{19} -49.0000 q^{21} +6.00000 q^{23} +25.0000 q^{25} -35.0000 q^{27} -131.000 q^{29} -4.00000 q^{31} +63.0000 q^{33} +35.0000 q^{35} -26.0000 q^{37} -161.000 q^{39} -260.000 q^{41} -190.000 q^{43} -110.000 q^{45} -167.000 q^{47} +49.0000 q^{49} +287.000 q^{51} +368.000 q^{53} -45.0000 q^{55} +238.000 q^{57} +324.000 q^{59} +164.000 q^{61} -154.000 q^{63} +115.000 q^{65} +200.000 q^{67} +42.0000 q^{69} -784.000 q^{71} -410.000 q^{73} +175.000 q^{75} -63.0000 q^{77} -1211.00 q^{79} -839.000 q^{81} -1132.00 q^{83} -205.000 q^{85} -917.000 q^{87} -72.0000 q^{89} +161.000 q^{91} -28.0000 q^{93} -170.000 q^{95} -707.000 q^{97} +198.000 q^{99} +O(q^{100})\)

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
			\(3\)	7.00000	1.34715	0.673575	−	0.739119i	\(-0.264758\pi\)
0.673575	+	0.739119i				\(0.264758\pi\)
\(5\)	−5.00000	−0.447214
			\(7\)	−7.00000	−0.377964
\(11\)	9.00000	0.246691				0.123346	−	0.992364i	\(-0.460638\pi\)
			0.123346	+	0.992364i	\(0.460638\pi\)
\(13\)	−23.0000	−0.490696	−0.245348	−	0.969435i	\(-0.578902\pi\)
			−0.245348	+	0.969435i	\(0.578902\pi\)
\(17\)	41.0000	0.584939	0.292469	−	0.956275i	\(-0.405523\pi\)
			0.292469	+	0.956275i	\(0.405523\pi\)
\(19\)	34.0000	0.410533	0.205267	−	0.978706i	\(-0.434194\pi\)
			0.205267	+	0.978706i	\(0.434194\pi\)
\(23\)	6.00000	0.0543951	0.0271975	−	0.999630i	\(-0.491342\pi\)
			0.0271975	+	0.999630i	\(0.491342\pi\)
\(29\)	−131.000	−0.838831	−0.419415	−	0.907794i	\(-0.637765\pi\)
			−0.419415	+	0.907794i	\(0.637765\pi\)
\(31\)	−4.00000	−0.0231749	−0.0115874	−	0.999933i	\(-0.503688\pi\)
			−0.0115874	+	0.999933i	\(0.503688\pi\)
\(37\)	−26.0000	−0.115524	−0.0577618	−	0.998330i	\(-0.518396\pi\)
			−0.0577618	+	0.998330i	\(0.518396\pi\)
\(41\)	−260.000	−0.990370	−0.495185	−	0.868787i	\(-0.664900\pi\)
			−0.495185	+	0.868787i	\(0.664900\pi\)
\(43\)	−190.000	−0.673831	−0.336915	−	0.941535i	\(-0.609384\pi\)
			−0.336915	+	0.941535i	\(0.609384\pi\)
\(47\)	−167.000	−0.518286	−0.259143	−	0.965839i	\(-0.583440\pi\)
			−0.259143	+	0.965839i	\(0.583440\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.4.a.bg.1.1		1
4.3	odd	2		2240.4.a.e.1.1		1
8.3	odd	2		280.4.a.d.1.1	✓	1
8.5	even	2		560.4.a.d.1.1		1
40.3	even	4		1400.4.g.a.449.2		2
40.19	odd	2		1400.4.a.a.1.1		1
40.27	even	4		1400.4.g.a.449.1		2
56.27	even	2		1960.4.a.b.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.4.a.d.1.1	✓	1	8.3	odd	2
560.4.a.d.1.1		1	8.5	even	2
1400.4.a.a.1.1		1	40.19	odd	2
1400.4.g.a.449.1		2	40.27	even	4
1400.4.g.a.449.2		2	40.3	even	4
1960.4.a.b.1.1		1	56.27	even	2
2240.4.a.e.1.1		1	4.3	odd	2
2240.4.a.bg.1.1		1	1.1	even	1	trivial