Embedded newform 3136.2.a.k.1.1

• Label: 3136.2.a.k.1.1
• Level: $3136$
• Weight: $2$
• Character: 3136.1
• Self dual: yes
• Analytic conductor: $25.041$
• Analytic rank: $1$
• 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:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 28)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{3} +3.00000 q^{5} -2.00000 q^{9} -3.00000 q^{11} +2.00000 q^{13} -3.00000 q^{15} -3.00000 q^{17} +1.00000 q^{19} -3.00000 q^{23} +4.00000 q^{25} +5.00000 q^{27} +6.00000 q^{29} -7.00000 q^{31} +3.00000 q^{33} +1.00000 q^{37} -2.00000 q^{39} -6.00000 q^{41} -4.00000 q^{43} -6.00000 q^{45} -9.00000 q^{47} +3.00000 q^{51} -3.00000 q^{53} -9.00000 q^{55} -1.00000 q^{57} -9.00000 q^{59} -1.00000 q^{61} +6.00000 q^{65} -7.00000 q^{67} +3.00000 q^{69} +1.00000 q^{73} -4.00000 q^{75} +13.0000 q^{79} +1.00000 q^{81} -12.0000 q^{83} -9.00000 q^{85} -6.00000 q^{87} -15.0000 q^{89} +7.00000 q^{93} +3.00000 q^{95} +10.0000 q^{97} +6.00000 q^{99} +O(q^{100})\)

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
			\(3\)	−1.00000	−0.577350	−0.288675	−	0.957427i	\(-0.593215\pi\)
−0.288675	+	0.957427i				\(0.593215\pi\)
\(5\)	3.00000	1.34164	0.670820	−	0.741620i	\(-0.265942\pi\)
			0.670820	+	0.741620i	\(0.265942\pi\)
\(7\)	0	0
			\(11\)	−3.00000	−0.904534	−0.452267	−	0.891883i	\(-0.649385\pi\)
−0.452267	+	0.891883i				\(0.649385\pi\)
\(13\)	2.00000	0.554700	0.277350	−	0.960769i	\(-0.410544\pi\)
			0.277350	+	0.960769i	\(0.410544\pi\)
\(17\)	−3.00000	−0.727607	−0.363803	−	0.931476i	\(-0.618522\pi\)
			−0.363803	+	0.931476i	\(0.618522\pi\)
\(19\)	1.00000	0.229416	0.114708	−	0.993399i	\(-0.463407\pi\)
			0.114708	+	0.993399i	\(0.463407\pi\)
\(23\)	−3.00000	−0.625543	−0.312772	−	0.949828i	\(-0.601257\pi\)
			−0.312772	+	0.949828i	\(0.601257\pi\)
\(29\)	6.00000	1.11417	0.557086	−	0.830455i	\(-0.311919\pi\)
			0.557086	+	0.830455i	\(0.311919\pi\)
\(31\)	−7.00000	−1.25724	−0.628619	−	0.777714i	\(-0.716379\pi\)
			−0.628619	+	0.777714i	\(0.716379\pi\)
\(37\)	1.00000	0.164399	0.0821995	−	0.996616i	\(-0.473806\pi\)
			0.0821995	+	0.996616i	\(0.473806\pi\)
\(41\)	−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	\(0.598656\pi\)
\(47\)	−9.00000	−1.31278	−0.656392	−	0.754420i	\(-0.727918\pi\)
			−0.656392	+	0.754420i	\(0.727918\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3136.2.a.k.1.1		1
4.3	odd	2		3136.2.a.v.1.1		1
7.3	odd	6		448.2.i.c.65.1		2
7.5	odd	6		448.2.i.c.193.1		2
7.6	odd	2		3136.2.a.s.1.1		1
8.3	odd	2		196.2.a.a.1.1		1
8.5	even	2		784.2.a.g.1.1		1
24.5	odd	2		7056.2.a.bw.1.1		1
24.11	even	2		1764.2.a.j.1.1		1
28.3	even	6		448.2.i.e.65.1		2
28.19	even	6		448.2.i.e.193.1		2
28.27	even	2		3136.2.a.h.1.1		1
40.3	even	4		4900.2.e.h.2549.1		2
40.19	odd	2		4900.2.a.n.1.1		1
40.27	even	4		4900.2.e.h.2549.2		2
56.3	even	6		28.2.e.a.9.1	✓	2
56.5	odd	6		112.2.i.b.81.1		2
56.11	odd	6		196.2.e.a.177.1		2
56.13	odd	2		784.2.a.d.1.1		1
56.19	even	6		28.2.e.a.25.1	yes	2
56.27	even	2		196.2.a.b.1.1		1
56.37	even	6		784.2.i.d.753.1		2
56.45	odd	6		112.2.i.b.65.1		2
56.51	odd	6		196.2.e.a.165.1		2
56.53	even	6		784.2.i.d.177.1		2
168.5	even	6		1008.2.s.p.865.1		2
168.11	even	6		1764.2.k.b.1549.1		2
168.59	odd	6		252.2.k.c.37.1		2
168.83	odd	2		1764.2.a.a.1.1		1
168.101	even	6		1008.2.s.p.289.1		2
168.107	even	6		1764.2.k.b.361.1		2
168.125	even	2		7056.2.a.f.1.1		1
168.131	odd	6		252.2.k.c.109.1		2
280.3	odd	12		700.2.r.b.149.2		4
280.19	even	6		700.2.i.c.501.1		2
280.27	odd	4		4900.2.e.i.2549.1		2
280.59	even	6		700.2.i.c.401.1		2
280.83	odd	4		4900.2.e.i.2549.2		2
280.139	even	2		4900.2.a.g.1.1		1
280.187	odd	12		700.2.r.b.249.2		4
280.227	odd	12		700.2.r.b.149.1		4
280.243	odd	12		700.2.r.b.249.1		4
504.59	odd	6		2268.2.l.a.541.1		2
504.115	even	6		2268.2.i.a.2053.1		2
504.131	odd	6		2268.2.i.h.865.1		2
504.187	even	6		2268.2.l.h.109.1		2
504.227	odd	6		2268.2.i.h.2053.1		2
504.283	even	6		2268.2.l.h.541.1		2
504.299	odd	6		2268.2.l.a.109.1		2
504.355	even	6		2268.2.i.a.865.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.2.e.a.9.1	✓	2	56.3	even	6
28.2.e.a.25.1	yes	2	56.19	even	6
112.2.i.b.65.1		2	56.45	odd	6
112.2.i.b.81.1		2	56.5	odd	6
196.2.a.a.1.1		1	8.3	odd	2
196.2.a.b.1.1		1	56.27	even	2
196.2.e.a.165.1		2	56.51	odd	6
196.2.e.a.177.1		2	56.11	odd	6
252.2.k.c.37.1		2	168.59	odd	6
252.2.k.c.109.1		2	168.131	odd	6
448.2.i.c.65.1		2	7.3	odd	6
448.2.i.c.193.1		2	7.5	odd	6
448.2.i.e.65.1		2	28.3	even	6
448.2.i.e.193.1		2	28.19	even	6
700.2.i.c.401.1		2	280.59	even	6
700.2.i.c.501.1		2	280.19	even	6
700.2.r.b.149.1		4	280.227	odd	12
700.2.r.b.149.2		4	280.3	odd	12
700.2.r.b.249.1		4	280.243	odd	12
700.2.r.b.249.2		4	280.187	odd	12
784.2.a.d.1.1		1	56.13	odd	2
784.2.a.g.1.1		1	8.5	even	2
784.2.i.d.177.1		2	56.53	even	6
784.2.i.d.753.1		2	56.37	even	6
1008.2.s.p.289.1		2	168.101	even	6
1008.2.s.p.865.1		2	168.5	even	6
1764.2.a.a.1.1		1	168.83	odd	2
1764.2.a.j.1.1		1	24.11	even	2
1764.2.k.b.361.1		2	168.107	even	6
1764.2.k.b.1549.1		2	168.11	even	6
2268.2.i.a.865.1		2	504.355	even	6
2268.2.i.a.2053.1		2	504.115	even	6
2268.2.i.h.865.1		2	504.131	odd	6
2268.2.i.h.2053.1		2	504.227	odd	6
2268.2.l.a.109.1		2	504.299	odd	6
2268.2.l.a.541.1		2	504.59	odd	6
2268.2.l.h.109.1		2	504.187	even	6
2268.2.l.h.541.1		2	504.283	even	6
3136.2.a.h.1.1		1	28.27	even	2
3136.2.a.k.1.1		1	1.1	even	1	trivial
3136.2.a.s.1.1		1	7.6	odd	2
3136.2.a.v.1.1		1	4.3	odd	2
4900.2.a.g.1.1		1	280.139	even	2
4900.2.a.n.1.1		1	40.19	odd	2
4900.2.e.h.2549.1		2	40.3	even	4
4900.2.e.h.2549.2		2	40.27	even	4
4900.2.e.i.2549.1		2	280.27	odd	4
4900.2.e.i.2549.2		2	280.83	odd	4
7056.2.a.f.1.1		1	168.125	even	2
7056.2.a.bw.1.1		1	24.5	odd	2