Embedded newform 784.6.a.d.1.1

• Label: 784.6.a.d.1.1
• Level: $784$
• Weight: $6$
• Character: 784.1
• Self dual: yes
• Analytic conductor: $125.741$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(125.740914733\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 4)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\)

\(=\)

\(q-12.0000 q^{3} -54.0000 q^{5} -99.0000 q^{9} -540.000 q^{11} +418.000 q^{13} +648.000 q^{15} -594.000 q^{17} +836.000 q^{19} +4104.00 q^{23} -209.000 q^{25} +4104.00 q^{27} -594.000 q^{29} +4256.00 q^{31} +6480.00 q^{33} -298.000 q^{37} -5016.00 q^{39} -17226.0 q^{41} +12100.0 q^{43} +5346.00 q^{45} -1296.00 q^{47} +7128.00 q^{51} +19494.0 q^{53} +29160.0 q^{55} -10032.0 q^{57} -7668.00 q^{59} +34738.0 q^{61} -22572.0 q^{65} -21812.0 q^{67} -49248.0 q^{69} +46872.0 q^{71} -67562.0 q^{73} +2508.00 q^{75} +76912.0 q^{79} -25191.0 q^{81} +67716.0 q^{83} +32076.0 q^{85} +7128.00 q^{87} -29754.0 q^{89} -51072.0 q^{93} -45144.0 q^{95} +122398. q^{97} +53460.0 q^{99} +O(q^{100})\)

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
			\(3\)	−12.0000	−0.769800	−0.384900	−	0.922958i	\(-0.625764\pi\)
−0.384900	+	0.922958i				\(0.625764\pi\)
\(5\)	−54.0000	−0.965981	−0.482991	−	0.875625i	\(-0.660450\pi\)
			−0.482991	+	0.875625i	\(0.660450\pi\)
\(7\)	0	0
			\(11\)	−540.000	−1.34559	−0.672794	−	0.739830i	\(-0.734906\pi\)
−0.672794	+	0.739830i				\(0.734906\pi\)
\(13\)	418.000	0.685990	0.342995	−	0.939337i	\(-0.388559\pi\)
			0.342995	+	0.939337i	\(0.388559\pi\)
\(17\)	−594.000	−0.498499	−0.249249	−	0.968439i	\(-0.580184\pi\)
			−0.249249	+	0.968439i	\(0.580184\pi\)
\(19\)	836.000	0.531279	0.265639	−	0.964072i	\(-0.414417\pi\)
			0.265639	+	0.964072i	\(0.414417\pi\)
\(23\)	4104.00	1.61766	0.808831	−	0.588041i	\(-0.200101\pi\)
			0.808831	+	0.588041i	\(0.200101\pi\)
\(29\)	−594.000	−0.131157	−0.0655785	−	0.997847i	\(-0.520889\pi\)
			−0.0655785	+	0.997847i	\(0.520889\pi\)
\(31\)	4256.00	0.795422	0.397711	−	0.917511i	\(-0.369805\pi\)
			0.397711	+	0.917511i	\(0.369805\pi\)
\(37\)	−298.000	−0.0357859	−0.0178930	−	0.999840i	\(-0.505696\pi\)
			−0.0178930	+	0.999840i	\(0.505696\pi\)
\(41\)	−17226.0	−1.60039	−0.800193	−	0.599742i	\(-0.795270\pi\)
			−0.800193	+	0.599742i	\(0.795270\pi\)
\(43\)	12100.0	0.997963	0.498981	−	0.866613i	\(-0.333708\pi\)
			0.498981	+	0.866613i	\(0.333708\pi\)
\(47\)	−1296.00	−0.0855777	−0.0427888	−	0.999084i	\(-0.513624\pi\)
			−0.0427888	+	0.999084i	\(0.513624\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.d.1.1		1
4.3	odd	2		196.6.a.e.1.1		1
7.6	odd	2		16.6.a.b.1.1		1
21.20	even	2		144.6.a.c.1.1		1
28.3	even	6		196.6.e.g.177.1		2
28.11	odd	6		196.6.e.d.177.1		2
28.19	even	6		196.6.e.g.165.1		2
28.23	odd	6		196.6.e.d.165.1		2
28.27	even	2		4.6.a.a.1.1	✓	1
35.13	even	4		400.6.c.f.49.2		2
35.27	even	4		400.6.c.f.49.1		2
35.34	odd	2		400.6.a.d.1.1		1
56.13	odd	2		64.6.a.b.1.1		1
56.27	even	2		64.6.a.f.1.1		1
84.83	odd	2		36.6.a.a.1.1		1
112.13	odd	4		256.6.b.c.129.2		2
112.27	even	4		256.6.b.g.129.2		2
112.69	odd	4		256.6.b.c.129.1		2
112.83	even	4		256.6.b.g.129.1		2
140.27	odd	4		100.6.c.b.49.2		2
140.83	odd	4		100.6.c.b.49.1		2
140.139	even	2		100.6.a.b.1.1		1
168.83	odd	2		576.6.a.bc.1.1		1
168.125	even	2		576.6.a.bd.1.1		1
252.83	odd	6		324.6.e.d.109.1		2
252.139	even	6		324.6.e.a.217.1		2
252.167	odd	6		324.6.e.d.217.1		2
252.223	even	6		324.6.e.a.109.1		2
308.307	odd	2		484.6.a.a.1.1		1
364.83	odd	4		676.6.d.a.337.1		2
364.307	odd	4		676.6.d.a.337.2		2
364.363	even	2		676.6.a.a.1.1		1
420.83	even	4		900.6.d.a.649.2		2
420.167	even	4		900.6.d.a.649.1		2
420.419	odd	2		900.6.a.h.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.6.a.a.1.1	✓	1	28.27	even	2
16.6.a.b.1.1		1	7.6	odd	2
36.6.a.a.1.1		1	84.83	odd	2
64.6.a.b.1.1		1	56.13	odd	2
64.6.a.f.1.1		1	56.27	even	2
100.6.a.b.1.1		1	140.139	even	2
100.6.c.b.49.1		2	140.83	odd	4
100.6.c.b.49.2		2	140.27	odd	4
144.6.a.c.1.1		1	21.20	even	2
196.6.a.e.1.1		1	4.3	odd	2
196.6.e.d.165.1		2	28.23	odd	6
196.6.e.d.177.1		2	28.11	odd	6
196.6.e.g.165.1		2	28.19	even	6
196.6.e.g.177.1		2	28.3	even	6
256.6.b.c.129.1		2	112.69	odd	4
256.6.b.c.129.2		2	112.13	odd	4
256.6.b.g.129.1		2	112.83	even	4
256.6.b.g.129.2		2	112.27	even	4
324.6.e.a.109.1		2	252.223	even	6
324.6.e.a.217.1		2	252.139	even	6
324.6.e.d.109.1		2	252.83	odd	6
324.6.e.d.217.1		2	252.167	odd	6
400.6.a.d.1.1		1	35.34	odd	2
400.6.c.f.49.1		2	35.27	even	4
400.6.c.f.49.2		2	35.13	even	4
484.6.a.a.1.1		1	308.307	odd	2
576.6.a.bc.1.1		1	168.83	odd	2
576.6.a.bd.1.1		1	168.125	even	2
676.6.a.a.1.1		1	364.363	even	2
676.6.d.a.337.1		2	364.83	odd	4
676.6.d.a.337.2		2	364.307	odd	4
784.6.a.d.1.1		1	1.1	even	1	trivial
900.6.a.h.1.1		1	420.419	odd	2
900.6.d.a.649.1		2	420.167	even	4
900.6.d.a.649.2		2	420.83	even	4