Embedded newform 1300.2.a.d.1.1

• Label: 1300.2.a.d.1.1
• Level: $1300$
• Weight: $2$
• Character: 1300.1
• Self dual: yes
• Analytic conductor: $10.381$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1300.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(10.3805522628\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 52)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	1300.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+2.00000 q^{7} -3.00000 q^{9} -2.00000 q^{11} +1.00000 q^{13} -6.00000 q^{17} -6.00000 q^{19} -8.00000 q^{23} +2.00000 q^{29} +10.0000 q^{31} +6.00000 q^{37} -6.00000 q^{41} -4.00000 q^{43} +2.00000 q^{47} -3.00000 q^{49} -6.00000 q^{53} -10.0000 q^{59} -2.00000 q^{61} -6.00000 q^{63} -10.0000 q^{67} +10.0000 q^{71} -2.00000 q^{73} -4.00000 q^{77} -4.00000 q^{79} +9.00000 q^{81} +6.00000 q^{83} -6.00000 q^{89} +2.00000 q^{91} -2.00000 q^{97} +6.00000 q^{99} +O(q^{100})\)

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
			\(3\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(5\)	0	0
			\(7\)	2.00000	0.755929	0.377964	−	0.925820i	\(-0.376624\pi\)
0.377964	+	0.925820i				\(0.376624\pi\)
\(11\)	−2.00000	−0.603023	−0.301511	−	0.953463i	\(-0.597491\pi\)
			−0.301511	+	0.953463i	\(0.597491\pi\)
\(13\)	1.00000	0.277350
			\(17\)	−6.00000	−1.45521	−0.727607	−	0.685994i	\(-0.759367\pi\)
−0.727607	+	0.685994i				\(0.759367\pi\)
\(19\)	−6.00000	−1.37649	−0.688247	−	0.725476i	\(-0.741620\pi\)
			−0.688247	+	0.725476i	\(0.741620\pi\)
\(23\)	−8.00000	−1.66812	−0.834058	−	0.551677i	\(-0.813988\pi\)
			−0.834058	+	0.551677i	\(0.813988\pi\)
\(29\)	2.00000	0.371391	0.185695	−	0.982607i	\(-0.440546\pi\)
			0.185695	+	0.982607i	\(0.440546\pi\)
\(31\)	10.0000	1.79605	0.898027	−	0.439941i	\(-0.145001\pi\)
			0.898027	+	0.439941i	\(0.145001\pi\)
\(37\)	6.00000	0.986394	0.493197	−	0.869918i	\(-0.335828\pi\)
			0.493197	+	0.869918i	\(0.335828\pi\)
\(41\)	−6.00000	−0.937043	−0.468521	−	0.883452i	\(-0.655213\pi\)
			−0.468521	+	0.883452i	\(0.655213\pi\)
\(43\)	−4.00000	−0.609994	−0.304997	−	0.952353i	\(-0.598656\pi\)
			−0.304997	+	0.952353i	\(0.598656\pi\)
\(47\)	2.00000	0.291730	0.145865	−	0.989305i	\(-0.453403\pi\)
			0.145865	+	0.989305i	\(0.453403\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1300.2.a.d.1.1		1
4.3	odd	2		5200.2.a.q.1.1		1
5.2	odd	4		1300.2.c.c.1249.2		2
5.3	odd	4		1300.2.c.c.1249.1		2
5.4	even	2		52.2.a.a.1.1	✓	1
15.14	odd	2		468.2.a.b.1.1		1
20.19	odd	2		208.2.a.c.1.1		1
35.4	even	6		2548.2.j.e.1353.1		2
35.9	even	6		2548.2.j.e.1145.1		2
35.19	odd	6		2548.2.j.f.1145.1		2
35.24	odd	6		2548.2.j.f.1353.1		2
35.34	odd	2		2548.2.a.e.1.1		1
40.19	odd	2		832.2.a.f.1.1		1
40.29	even	2		832.2.a.e.1.1		1
45.4	even	6		4212.2.i.d.2809.1		2
45.14	odd	6		4212.2.i.i.2809.1		2
45.29	odd	6		4212.2.i.i.1405.1		2
45.34	even	6		4212.2.i.d.1405.1		2
55.54	odd	2		6292.2.a.g.1.1		1
60.59	even	2		1872.2.a.f.1.1		1
65.4	even	6		676.2.e.b.653.1		2
65.9	even	6		676.2.e.c.653.1		2
65.19	odd	12		676.2.h.c.361.1		4
65.24	odd	12		676.2.h.c.485.2		4
65.29	even	6		676.2.e.c.529.1		2
65.34	odd	4		676.2.d.c.337.2		2
65.44	odd	4		676.2.d.c.337.1		2
65.49	even	6		676.2.e.b.529.1		2
65.54	odd	12		676.2.h.c.485.1		4
65.59	odd	12		676.2.h.c.361.2		4
65.64	even	2		676.2.a.c.1.1		1
80.19	odd	4		3328.2.b.e.1665.1		2
80.29	even	4		3328.2.b.q.1665.1		2
80.59	odd	4		3328.2.b.e.1665.2		2
80.69	even	4		3328.2.b.q.1665.2		2
120.29	odd	2		7488.2.a.bn.1.1		1
120.59	even	2		7488.2.a.bw.1.1		1
195.44	even	4		6084.2.b.m.4393.2		2
195.164	even	4		6084.2.b.m.4393.1		2
195.194	odd	2		6084.2.a.m.1.1		1
260.99	even	4		2704.2.f.f.337.2		2
260.239	even	4		2704.2.f.f.337.1		2
260.259	odd	2		2704.2.a.g.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.2.a.a.1.1	✓	1	5.4	even	2
208.2.a.c.1.1		1	20.19	odd	2
468.2.a.b.1.1		1	15.14	odd	2
676.2.a.c.1.1		1	65.64	even	2
676.2.d.c.337.1		2	65.44	odd	4
676.2.d.c.337.2		2	65.34	odd	4
676.2.e.b.529.1		2	65.49	even	6
676.2.e.b.653.1		2	65.4	even	6
676.2.e.c.529.1		2	65.29	even	6
676.2.e.c.653.1		2	65.9	even	6
676.2.h.c.361.1		4	65.19	odd	12
676.2.h.c.361.2		4	65.59	odd	12
676.2.h.c.485.1		4	65.54	odd	12
676.2.h.c.485.2		4	65.24	odd	12
832.2.a.e.1.1		1	40.29	even	2
832.2.a.f.1.1		1	40.19	odd	2
1300.2.a.d.1.1		1	1.1	even	1	trivial
1300.2.c.c.1249.1		2	5.3	odd	4
1300.2.c.c.1249.2		2	5.2	odd	4
1872.2.a.f.1.1		1	60.59	even	2
2548.2.a.e.1.1		1	35.34	odd	2
2548.2.j.e.1145.1		2	35.9	even	6
2548.2.j.e.1353.1		2	35.4	even	6
2548.2.j.f.1145.1		2	35.19	odd	6
2548.2.j.f.1353.1		2	35.24	odd	6
2704.2.a.g.1.1		1	260.259	odd	2
2704.2.f.f.337.1		2	260.239	even	4
2704.2.f.f.337.2		2	260.99	even	4
3328.2.b.e.1665.1		2	80.19	odd	4
3328.2.b.e.1665.2		2	80.59	odd	4
3328.2.b.q.1665.1		2	80.29	even	4
3328.2.b.q.1665.2		2	80.69	even	4
4212.2.i.d.1405.1		2	45.34	even	6
4212.2.i.d.2809.1		2	45.4	even	6
4212.2.i.i.1405.1		2	45.29	odd	6
4212.2.i.i.2809.1		2	45.14	odd	6
5200.2.a.q.1.1		1	4.3	odd	2
6084.2.a.m.1.1		1	195.194	odd	2
6084.2.b.m.4393.1		2	195.164	even	4
6084.2.b.m.4393.2		2	195.44	even	4
6292.2.a.g.1.1		1	55.54	odd	2
7488.2.a.bn.1.1		1	120.29	odd	2
7488.2.a.bw.1.1		1	120.59	even	2