Embedded newform 3136.2.a.e.1.1

• Label: 3136.2.a.e.1.1
• Level: $3136$
• Weight: $2$
• Character: 3136.1
• Self dual: yes
• Analytic conductor: $25.041$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3136 = 2^{6} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3136.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(25.0410860739\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	3136.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-2.00000 q^{3} +1.00000 q^{9} +O(q^{10})\)

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	−1.15470	−0.577350	−	0.816497i	\(-0.695913\pi\)
−0.577350	+	0.816497i				\(0.695913\pi\)
\(5\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(7\)	0	0
			\(11\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(13\)	−4.00000	−1.10940	−0.554700	−	0.832050i	\(-0.687167\pi\)
			−0.554700	+	0.832050i	\(0.687167\pi\)
\(17\)	−6.00000	−1.45521	−0.727607	−	0.685994i	\(-0.759367\pi\)
			−0.727607	+	0.685994i	\(0.759367\pi\)
\(19\)	2.00000	0.458831	0.229416	−	0.973329i	\(-0.426318\pi\)
			0.229416	+	0.973329i	\(0.426318\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	6.00000	1.11417	0.557086	−	0.830455i	\(-0.311919\pi\)
			0.557086	+	0.830455i	\(0.311919\pi\)
\(31\)	4.00000	0.718421	0.359211	−	0.933257i	\(-0.383046\pi\)
			0.359211	+	0.933257i	\(0.383046\pi\)
\(37\)	−2.00000	−0.328798	−0.164399	−	0.986394i	\(-0.552568\pi\)
			−0.164399	+	0.986394i	\(0.552568\pi\)
\(41\)	−6.00000	−0.937043	−0.468521	−	0.883452i	\(-0.655213\pi\)
			−0.468521	+	0.883452i	\(0.655213\pi\)
\(43\)	−8.00000	−1.21999	−0.609994	−	0.792406i	\(-0.708828\pi\)
			−0.609994	+	0.792406i	\(0.708828\pi\)
\(47\)	12.0000	1.75038	0.875190	−	0.483779i	\(-0.160736\pi\)
			0.875190	+	0.483779i	\(0.160736\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3136.2.a.e.1.1		1
4.3	odd	2		3136.2.a.z.1.1		1
7.6	odd	2		448.2.a.g.1.1		1
8.3	odd	2		784.2.a.b.1.1		1
8.5	even	2		98.2.a.a.1.1		1
21.20	even	2		4032.2.a.w.1.1		1
24.5	odd	2		882.2.a.i.1.1		1
24.11	even	2		7056.2.a.bd.1.1		1
28.27	even	2		448.2.a.a.1.1		1
40.13	odd	4		2450.2.c.c.99.2		2
40.29	even	2		2450.2.a.t.1.1		1
40.37	odd	4		2450.2.c.c.99.1		2
56.3	even	6		784.2.i.c.177.1		2
56.5	odd	6		98.2.c.b.67.1		2
56.11	odd	6		784.2.i.i.177.1		2
56.13	odd	2		14.2.a.a.1.1	✓	1
56.19	even	6		784.2.i.c.753.1		2
56.27	even	2		112.2.a.c.1.1		1
56.37	even	6		98.2.c.a.67.1		2
56.45	odd	6		98.2.c.b.79.1		2
56.51	odd	6		784.2.i.i.753.1		2
56.53	even	6		98.2.c.a.79.1		2
84.83	odd	2		4032.2.a.r.1.1		1
112.13	odd	4		1792.2.b.c.897.2		2
112.27	even	4		1792.2.b.g.897.2		2
112.69	odd	4		1792.2.b.c.897.1		2
112.83	even	4		1792.2.b.g.897.1		2
168.5	even	6		882.2.g.c.361.1		2
168.53	odd	6		882.2.g.d.667.1		2
168.83	odd	2		1008.2.a.h.1.1		1
168.101	even	6		882.2.g.c.667.1		2
168.125	even	2		126.2.a.b.1.1		1
168.149	odd	6		882.2.g.d.361.1		2
280.13	even	4		350.2.c.d.99.2		2
280.27	odd	4		2800.2.g.h.449.1		2
280.69	odd	2		350.2.a.f.1.1		1
280.83	odd	4		2800.2.g.h.449.2		2
280.139	even	2		2800.2.a.g.1.1		1
280.237	even	4		350.2.c.d.99.1		2
504.13	odd	6		1134.2.f.l.379.1		2
504.293	even	6		1134.2.f.f.379.1		2
504.349	odd	6		1134.2.f.l.757.1		2
504.461	even	6		1134.2.f.f.757.1		2
616.461	even	2		1694.2.a.e.1.1		1
728.125	even	4		2366.2.d.b.337.1		2
728.181	odd	2		2366.2.a.j.1.1		1
728.629	even	4		2366.2.d.b.337.2		2
840.293	odd	4		3150.2.g.j.2899.1		2
840.629	even	2		3150.2.a.i.1.1		1
840.797	odd	4		3150.2.g.j.2899.2		2
952.237	odd	2		4046.2.a.f.1.1		1
1064.797	even	2		5054.2.a.c.1.1		1
1288.965	even	2		7406.2.a.a.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.2.a.a.1.1	✓	1	56.13	odd	2
98.2.a.a.1.1		1	8.5	even	2
98.2.c.a.67.1		2	56.37	even	6
98.2.c.a.79.1		2	56.53	even	6
98.2.c.b.67.1		2	56.5	odd	6
98.2.c.b.79.1		2	56.45	odd	6
112.2.a.c.1.1		1	56.27	even	2
126.2.a.b.1.1		1	168.125	even	2
350.2.a.f.1.1		1	280.69	odd	2
350.2.c.d.99.1		2	280.237	even	4
350.2.c.d.99.2		2	280.13	even	4
448.2.a.a.1.1		1	28.27	even	2
448.2.a.g.1.1		1	7.6	odd	2
784.2.a.b.1.1		1	8.3	odd	2
784.2.i.c.177.1		2	56.3	even	6
784.2.i.c.753.1		2	56.19	even	6
784.2.i.i.177.1		2	56.11	odd	6
784.2.i.i.753.1		2	56.51	odd	6
882.2.a.i.1.1		1	24.5	odd	2
882.2.g.c.361.1		2	168.5	even	6
882.2.g.c.667.1		2	168.101	even	6
882.2.g.d.361.1		2	168.149	odd	6
882.2.g.d.667.1		2	168.53	odd	6
1008.2.a.h.1.1		1	168.83	odd	2
1134.2.f.f.379.1		2	504.293	even	6
1134.2.f.f.757.1		2	504.461	even	6
1134.2.f.l.379.1		2	504.13	odd	6
1134.2.f.l.757.1		2	504.349	odd	6
1694.2.a.e.1.1		1	616.461	even	2
1792.2.b.c.897.1		2	112.69	odd	4
1792.2.b.c.897.2		2	112.13	odd	4
1792.2.b.g.897.1		2	112.83	even	4
1792.2.b.g.897.2		2	112.27	even	4
2366.2.a.j.1.1		1	728.181	odd	2
2366.2.d.b.337.1		2	728.125	even	4
2366.2.d.b.337.2		2	728.629	even	4
2450.2.a.t.1.1		1	40.29	even	2
2450.2.c.c.99.1		2	40.37	odd	4
2450.2.c.c.99.2		2	40.13	odd	4
2800.2.a.g.1.1		1	280.139	even	2
2800.2.g.h.449.1		2	280.27	odd	4
2800.2.g.h.449.2		2	280.83	odd	4
3136.2.a.e.1.1		1	1.1	even	1	trivial
3136.2.a.z.1.1		1	4.3	odd	2
3150.2.a.i.1.1		1	840.629	even	2
3150.2.g.j.2899.1		2	840.293	odd	4
3150.2.g.j.2899.2		2	840.797	odd	4
4032.2.a.r.1.1		1	84.83	odd	2
4032.2.a.w.1.1		1	21.20	even	2
4046.2.a.f.1.1		1	952.237	odd	2
5054.2.a.c.1.1		1	1064.797	even	2
7056.2.a.bd.1.1		1	24.11	even	2
7406.2.a.a.1.1		1	1288.965	even	2