Embedded newform 100.6.a.b.1.1

• Label: 100.6.a.b.1.1
• Level: $100$
• Weight: $6$
• Character: 100.1
• Self dual: yes
• Analytic conductor: $16.038$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 100 = 2^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	100.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(16.0383819813\)
Analytic rank:	\(0\)
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	\(\chi\)	\(=\)	100.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+12.0000 q^{3} +88.0000 q^{7} -99.0000 q^{9} +540.000 q^{11} +418.000 q^{13} -594.000 q^{17} +836.000 q^{19} +1056.00 q^{21} +4104.00 q^{23} -4104.00 q^{27} -594.000 q^{29} +4256.00 q^{31} +6480.00 q^{33} +298.000 q^{37} +5016.00 q^{39} +17226.0 q^{41} +12100.0 q^{43} +1296.00 q^{47} -9063.00 q^{49} -7128.00 q^{51} -19494.0 q^{53} +10032.0 q^{57} -7668.00 q^{59} -34738.0 q^{61} -8712.00 q^{63} -21812.0 q^{67} +49248.0 q^{69} -46872.0 q^{71} -67562.0 q^{73} +47520.0 q^{77} -76912.0 q^{79} -25191.0 q^{81} -67716.0 q^{83} -7128.00 q^{87} +29754.0 q^{89} +36784.0 q^{91} +51072.0 q^{93} +122398. q^{97} -53460.0 q^{99} +O(q^{100})\)

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	\(-0.374236\pi\)
0.384900	+	0.922958i				\(0.374236\pi\)
\(5\)	0	0
			\(7\)	88.0000	0.678793	0.339397	−	0.940643i	\(-0.389777\pi\)
0.339397	+	0.940643i				\(0.389777\pi\)
\(11\)	540.000	1.34559	0.672794	−	0.739830i	\(-0.265094\pi\)
			0.672794	+	0.739830i	\(0.265094\pi\)
\(13\)	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	\(-0.494304\pi\)
			0.0178930	+	0.999840i	\(0.494304\pi\)
\(41\)	17226.0	1.60039	0.800193	−	0.599742i	\(-0.204730\pi\)
			0.800193	+	0.599742i	\(0.204730\pi\)
\(43\)	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	\(-0.486376\pi\)
			0.0427888	+	0.999084i	\(0.486376\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	100.6.a.b.1.1		1
3.2	odd	2		900.6.a.h.1.1		1
4.3	odd	2		400.6.a.d.1.1		1
5.2	odd	4		100.6.c.b.49.1		2
5.3	odd	4		100.6.c.b.49.2		2
5.4	even	2		4.6.a.a.1.1	✓	1
15.2	even	4		900.6.d.a.649.2		2
15.8	even	4		900.6.d.a.649.1		2
15.14	odd	2		36.6.a.a.1.1		1
20.3	even	4		400.6.c.f.49.1		2
20.7	even	4		400.6.c.f.49.2		2
20.19	odd	2		16.6.a.b.1.1		1
35.4	even	6		196.6.e.g.177.1		2
35.9	even	6		196.6.e.g.165.1		2
35.19	odd	6		196.6.e.d.165.1		2
35.24	odd	6		196.6.e.d.177.1		2
35.34	odd	2		196.6.a.e.1.1		1
40.19	odd	2		64.6.a.b.1.1		1
40.29	even	2		64.6.a.f.1.1		1
45.4	even	6		324.6.e.a.217.1		2
45.14	odd	6		324.6.e.d.217.1		2
45.29	odd	6		324.6.e.d.109.1		2
45.34	even	6		324.6.e.a.109.1		2
55.54	odd	2		484.6.a.a.1.1		1
60.59	even	2		144.6.a.c.1.1		1
65.34	odd	4		676.6.d.a.337.2		2
65.44	odd	4		676.6.d.a.337.1		2
65.64	even	2		676.6.a.a.1.1		1
80.19	odd	4		256.6.b.c.129.2		2
80.29	even	4		256.6.b.g.129.1		2
80.59	odd	4		256.6.b.c.129.1		2
80.69	even	4		256.6.b.g.129.2		2
120.29	odd	2		576.6.a.bc.1.1		1
120.59	even	2		576.6.a.bd.1.1		1
140.139	even	2		784.6.a.d.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.6.a.a.1.1	✓	1	5.4	even	2
16.6.a.b.1.1		1	20.19	odd	2
36.6.a.a.1.1		1	15.14	odd	2
64.6.a.b.1.1		1	40.19	odd	2
64.6.a.f.1.1		1	40.29	even	2
100.6.a.b.1.1		1	1.1	even	1	trivial
100.6.c.b.49.1		2	5.2	odd	4
100.6.c.b.49.2		2	5.3	odd	4
144.6.a.c.1.1		1	60.59	even	2
196.6.a.e.1.1		1	35.34	odd	2
196.6.e.d.165.1		2	35.19	odd	6
196.6.e.d.177.1		2	35.24	odd	6
196.6.e.g.165.1		2	35.9	even	6
196.6.e.g.177.1		2	35.4	even	6
256.6.b.c.129.1		2	80.59	odd	4
256.6.b.c.129.2		2	80.19	odd	4
256.6.b.g.129.1		2	80.29	even	4
256.6.b.g.129.2		2	80.69	even	4
324.6.e.a.109.1		2	45.34	even	6
324.6.e.a.217.1		2	45.4	even	6
324.6.e.d.109.1		2	45.29	odd	6
324.6.e.d.217.1		2	45.14	odd	6
400.6.a.d.1.1		1	4.3	odd	2
400.6.c.f.49.1		2	20.3	even	4
400.6.c.f.49.2		2	20.7	even	4
484.6.a.a.1.1		1	55.54	odd	2
576.6.a.bc.1.1		1	120.29	odd	2
576.6.a.bd.1.1		1	120.59	even	2
676.6.a.a.1.1		1	65.64	even	2
676.6.d.a.337.1		2	65.44	odd	4
676.6.d.a.337.2		2	65.34	odd	4
784.6.a.d.1.1		1	140.139	even	2
900.6.a.h.1.1		1	3.2	odd	2
900.6.d.a.649.1		2	15.8	even	4
900.6.d.a.649.2		2	15.2	even	4