Embedded newform 1960.4.a.g.1.1

• Label: 1960.4.a.g.1.1
• Level: $1960$
• Weight: $4$
• Character: 1960.1
• Self dual: yes
• Analytic conductor: $115.644$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1960.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(115.643743611\)
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	\(\chi\)	\(=\)	1960.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+4.00000 q^{3} -5.00000 q^{5} -11.0000 q^{9} +20.0000 q^{11} +10.0000 q^{13} -20.0000 q^{15} +14.0000 q^{17} -12.0000 q^{19} +104.000 q^{23} +25.0000 q^{25} -152.000 q^{27} -122.000 q^{29} -224.000 q^{31} +80.0000 q^{33} +158.000 q^{37} +40.0000 q^{39} -378.000 q^{41} +404.000 q^{43} +55.0000 q^{45} -112.000 q^{47} +56.0000 q^{51} +270.000 q^{53} -100.000 q^{55} -48.0000 q^{57} -324.000 q^{59} +186.000 q^{61} -50.0000 q^{65} +156.000 q^{67} +416.000 q^{69} -360.000 q^{71} +102.000 q^{73} +100.000 q^{75} -912.000 q^{79} -311.000 q^{81} -1068.00 q^{83} -70.0000 q^{85} -488.000 q^{87} +1590.00 q^{89} -896.000 q^{93} +60.0000 q^{95} -866.000 q^{97} -220.000 q^{99} +O(q^{100})\)

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
			\(3\)	4.00000	0.769800	0.384900	−	0.922958i	\(-0.374236\pi\)
0.384900	+	0.922958i				\(0.374236\pi\)
\(5\)	−5.00000	−0.447214
			\(7\)	0	0
\(11\)	20.0000	0.548202				0.274101	−	0.961701i	\(-0.411620\pi\)
			0.274101	+	0.961701i	\(0.411620\pi\)
\(13\)	10.0000	0.213346	0.106673	−	0.994294i	\(-0.465980\pi\)
			0.106673	+	0.994294i	\(0.465980\pi\)
\(17\)	14.0000	0.199735	0.0998676	−	0.995001i	\(-0.468158\pi\)
			0.0998676	+	0.995001i	\(0.468158\pi\)
\(19\)	−12.0000	−0.144894	−0.0724471	−	0.997372i	\(-0.523081\pi\)
			−0.0724471	+	0.997372i	\(0.523081\pi\)
\(23\)	104.000	0.942848	0.471424	−	0.881907i	\(-0.343740\pi\)
			0.471424	+	0.881907i	\(0.343740\pi\)
\(29\)	−122.000	−0.781201	−0.390601	−	0.920560i	\(-0.627733\pi\)
			−0.390601	+	0.920560i	\(0.627733\pi\)
\(31\)	−224.000	−1.29779	−0.648897	−	0.760877i	\(-0.724769\pi\)
			−0.648897	+	0.760877i	\(0.724769\pi\)
\(37\)	158.000	0.702028	0.351014	−	0.936370i	\(-0.385837\pi\)
			0.351014	+	0.936370i	\(0.385837\pi\)
\(41\)	−378.000	−1.43985	−0.719923	−	0.694054i	\(-0.755823\pi\)
			−0.719923	+	0.694054i	\(0.755823\pi\)
\(43\)	404.000	1.43278	0.716389	−	0.697701i	\(-0.245794\pi\)
			0.716389	+	0.697701i	\(0.245794\pi\)
\(47\)	−112.000	−0.347593	−0.173797	−	0.984782i	\(-0.555604\pi\)
			−0.173797	+	0.984782i	\(0.555604\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1960.4.a.g.1.1		1
7.6	odd	2		280.4.a.a.1.1	✓	1
28.27	even	2		560.4.a.l.1.1		1
35.13	even	4		1400.4.g.e.449.1		2
35.27	even	4		1400.4.g.e.449.2		2
35.34	odd	2		1400.4.a.g.1.1		1
56.13	odd	2		2240.4.a.z.1.1		1
56.27	even	2		2240.4.a.l.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.4.a.a.1.1	✓	1	7.6	odd	2
560.4.a.l.1.1		1	28.27	even	2
1400.4.a.g.1.1		1	35.34	odd	2
1400.4.g.e.449.1		2	35.13	even	4
1400.4.g.e.449.2		2	35.27	even	4
1960.4.a.g.1.1		1	1.1	even	1	trivial
2240.4.a.l.1.1		1	56.27	even	2
2240.4.a.z.1.1		1	56.13	odd	2