Embedded newform 784.4.a.g.1.1

• Label: 784.4.a.g.1.1
• Level: $784$
• Weight: $4$
• Character: 784.1
• Self dual: yes
• Analytic conductor: $46.257$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(46.2574974445\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 7)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	784.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-2.00000 q^{3} -16.0000 q^{5} -23.0000 q^{9} +8.00000 q^{11} -28.0000 q^{13} +32.0000 q^{15} -54.0000 q^{17} -110.000 q^{19} -48.0000 q^{23} +131.000 q^{25} +100.000 q^{27} -110.000 q^{29} +12.0000 q^{31} -16.0000 q^{33} -246.000 q^{37} +56.0000 q^{39} -182.000 q^{41} -128.000 q^{43} +368.000 q^{45} +324.000 q^{47} +108.000 q^{51} -162.000 q^{53} -128.000 q^{55} +220.000 q^{57} +810.000 q^{59} +488.000 q^{61} +448.000 q^{65} -244.000 q^{67} +96.0000 q^{69} +768.000 q^{71} +702.000 q^{73} -262.000 q^{75} -440.000 q^{79} +421.000 q^{81} -1302.00 q^{83} +864.000 q^{85} +220.000 q^{87} -730.000 q^{89} -24.0000 q^{93} +1760.00 q^{95} -294.000 q^{97} -184.000 q^{99} +O(q^{100})\)

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
			\(3\)	−2.00000	−0.384900	−0.192450	−	0.981307i	\(-0.561643\pi\)
−0.192450	+	0.981307i				\(0.561643\pi\)
\(5\)	−16.0000	−1.43108	−0.715542	−	0.698570i	\(-0.753820\pi\)
			−0.715542	+	0.698570i	\(0.753820\pi\)
\(7\)	0	0
			\(11\)	8.00000	0.219281	0.109640	−	0.993971i	\(-0.465030\pi\)
0.109640	+	0.993971i				\(0.465030\pi\)
\(13\)	−28.0000	−0.597369	−0.298685	−	0.954352i	\(-0.596548\pi\)
			−0.298685	+	0.954352i	\(0.596548\pi\)
\(17\)	−54.0000	−0.770407	−0.385204	−	0.922832i	\(-0.625869\pi\)
			−0.385204	+	0.922832i	\(0.625869\pi\)
\(19\)	−110.000	−1.32820	−0.664098	−	0.747645i	\(-0.731184\pi\)
			−0.664098	+	0.747645i	\(0.731184\pi\)
\(23\)	−48.0000	−0.435161	−0.217580	−	0.976042i	\(-0.569816\pi\)
			−0.217580	+	0.976042i	\(0.569816\pi\)
\(29\)	−110.000	−0.704362	−0.352181	−	0.935932i	\(-0.614560\pi\)
			−0.352181	+	0.935932i	\(0.614560\pi\)
\(31\)	12.0000	0.0695246	0.0347623	−	0.999396i	\(-0.488933\pi\)
			0.0347623	+	0.999396i	\(0.488933\pi\)
\(37\)	−246.000	−1.09303	−0.546516	−	0.837449i	\(-0.684046\pi\)
			−0.546516	+	0.837449i	\(0.684046\pi\)
\(41\)	−182.000	−0.693259	−0.346630	−	0.938002i	\(-0.612674\pi\)
			−0.346630	+	0.938002i	\(0.612674\pi\)
\(43\)	−128.000	−0.453949	−0.226975	−	0.973901i	\(-0.572883\pi\)
			−0.226975	+	0.973901i	\(0.572883\pi\)
\(47\)	324.000	1.00554	0.502769	−	0.864421i	\(-0.332315\pi\)
			0.502769	+	0.864421i	\(0.332315\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.a.g.1.1		1
4.3	odd	2		49.4.a.b.1.1		1
7.6	odd	2		112.4.a.f.1.1		1
12.11	even	2		441.4.a.i.1.1		1
20.19	odd	2		1225.4.a.j.1.1		1
21.20	even	2		1008.4.a.c.1.1		1
28.3	even	6		49.4.c.c.30.1		2
28.11	odd	6		49.4.c.b.30.1		2
28.19	even	6		49.4.c.c.18.1		2
28.23	odd	6		49.4.c.b.18.1		2
28.27	even	2		7.4.a.a.1.1	✓	1
56.13	odd	2		448.4.a.e.1.1		1
56.27	even	2		448.4.a.i.1.1		1
84.11	even	6		441.4.e.e.226.1		2
84.23	even	6		441.4.e.e.361.1		2
84.47	odd	6		441.4.e.h.361.1		2
84.59	odd	6		441.4.e.h.226.1		2
84.83	odd	2		63.4.a.b.1.1		1
140.27	odd	4		175.4.b.b.99.1		2
140.83	odd	4		175.4.b.b.99.2		2
140.139	even	2		175.4.a.b.1.1		1
308.307	odd	2		847.4.a.b.1.1		1
364.363	even	2		1183.4.a.b.1.1		1
420.419	odd	2		1575.4.a.e.1.1		1
476.475	even	2		2023.4.a.a.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.4.a.a.1.1	✓	1	28.27	even	2
49.4.a.b.1.1		1	4.3	odd	2
49.4.c.b.18.1		2	28.23	odd	6
49.4.c.b.30.1		2	28.11	odd	6
49.4.c.c.18.1		2	28.19	even	6
49.4.c.c.30.1		2	28.3	even	6
63.4.a.b.1.1		1	84.83	odd	2
112.4.a.f.1.1		1	7.6	odd	2
175.4.a.b.1.1		1	140.139	even	2
175.4.b.b.99.1		2	140.27	odd	4
175.4.b.b.99.2		2	140.83	odd	4
441.4.a.i.1.1		1	12.11	even	2
441.4.e.e.226.1		2	84.11	even	6
441.4.e.e.361.1		2	84.23	even	6
441.4.e.h.226.1		2	84.59	odd	6
441.4.e.h.361.1		2	84.47	odd	6
448.4.a.e.1.1		1	56.13	odd	2
448.4.a.i.1.1		1	56.27	even	2
784.4.a.g.1.1		1	1.1	even	1	trivial
847.4.a.b.1.1		1	308.307	odd	2
1008.4.a.c.1.1		1	21.20	even	2
1183.4.a.b.1.1		1	364.363	even	2
1225.4.a.j.1.1		1	20.19	odd	2
1575.4.a.e.1.1		1	420.419	odd	2
2023.4.a.a.1.1		1	476.475	even	2