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

$f(q)$

$=$

$q-2.00000 q^{3} +1.00000 q^{9} -4.00000 q^{13} -6.00000 q^{17} +2.00000 q^{19} -5.00000 q^{25} +4.00000 q^{27} +6.00000 q^{29} +4.00000 q^{31} -2.00000 q^{37} +8.00000 q^{39} -6.00000 q^{41} -8.00000 q^{43} +12.0000 q^{47} +12.0000 q^{51} -6.00000 q^{53} -4.00000 q^{57} -6.00000 q^{59} +8.00000 q^{61} +4.00000 q^{67} -2.00000 q^{73} +10.0000 q^{75} +8.00000 q^{79} -11.0000 q^{81} -6.00000 q^{83} -12.0000 q^{87} +6.00000 q^{89} -8.00000 q^{93} +10.0000 q^{97} +O(q^{100})$

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	−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