Embedded newform 784.2.a.d.1.1

• Label: 784.2.a.d.1.1
• Level: $784$
• Weight: $2$
• Character: 784.1
• Self dual: yes
• Analytic conductor: $6.260$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(6.26027151847\)
Analytic rank:	\(0\)
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	\(\chi\)	\(=\)	784.1

$q$-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	\(-0.350615\pi\)
0.452267	+	0.891883i				\(0.350615\pi\)
\(13\)	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	\(-0.381478\pi\)
			0.363803	+	0.931476i	\(0.381478\pi\)
\(19\)	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	\(-0.688081\pi\)
			−0.557086	+	0.830455i	\(0.688081\pi\)
\(31\)	7.00000	1.25724	0.628619	−	0.777714i	\(-0.283621\pi\)
			0.628619	+	0.777714i	\(0.283621\pi\)
\(37\)	−1.00000	−0.164399	−0.0821995	−	0.996616i	\(-0.526194\pi\)
			−0.0821995	+	0.996616i	\(0.526194\pi\)
\(41\)	6.00000	0.937043	0.468521	−	0.883452i	\(-0.344787\pi\)
			0.468521	+	0.883452i	\(0.344787\pi\)
\(43\)	4.00000	0.609994	0.304997	−	0.952353i	\(-0.401344\pi\)
			0.304997	+	0.952353i	\(0.401344\pi\)
\(47\)	9.00000	1.31278	0.656392	−	0.754420i	\(-0.272082\pi\)
			0.656392	+	0.754420i	\(0.272082\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.2.a.d.1.1		1
3.2	odd	2		7056.2.a.f.1.1		1
4.3	odd	2		196.2.a.b.1.1		1
7.2	even	3		112.2.i.b.81.1		2
7.3	odd	6		784.2.i.d.177.1		2
7.4	even	3		112.2.i.b.65.1		2
7.5	odd	6		784.2.i.d.753.1		2
7.6	odd	2		784.2.a.g.1.1		1
8.3	odd	2		3136.2.a.h.1.1		1
8.5	even	2		3136.2.a.s.1.1		1
12.11	even	2		1764.2.a.a.1.1		1
20.3	even	4		4900.2.e.i.2549.2		2
20.7	even	4		4900.2.e.i.2549.1		2
20.19	odd	2		4900.2.a.g.1.1		1
21.2	odd	6		1008.2.s.p.865.1		2
21.11	odd	6		1008.2.s.p.289.1		2
21.20	even	2		7056.2.a.bw.1.1		1
28.3	even	6		196.2.e.a.177.1		2
28.11	odd	6		28.2.e.a.9.1	✓	2
28.19	even	6		196.2.e.a.165.1		2
28.23	odd	6		28.2.e.a.25.1	yes	2
28.27	even	2		196.2.a.a.1.1		1
56.11	odd	6		448.2.i.e.65.1		2
56.13	odd	2		3136.2.a.k.1.1		1
56.27	even	2		3136.2.a.v.1.1		1
56.37	even	6		448.2.i.c.193.1		2
56.51	odd	6		448.2.i.e.193.1		2
56.53	even	6		448.2.i.c.65.1		2
84.11	even	6		252.2.k.c.37.1		2
84.23	even	6		252.2.k.c.109.1		2
84.47	odd	6		1764.2.k.b.361.1		2
84.59	odd	6		1764.2.k.b.1549.1		2
84.83	odd	2		1764.2.a.j.1.1		1
140.23	even	12		700.2.r.b.249.1		4
140.27	odd	4		4900.2.e.h.2549.2		2
140.39	odd	6		700.2.i.c.401.1		2
140.67	even	12		700.2.r.b.149.1		4
140.79	odd	6		700.2.i.c.501.1		2
140.83	odd	4		4900.2.e.h.2549.1		2
140.107	even	12		700.2.r.b.249.2		4
140.123	even	12		700.2.r.b.149.2		4
140.139	even	2		4900.2.a.n.1.1		1
252.11	even	6		2268.2.i.h.2053.1		2
252.23	even	6		2268.2.i.h.865.1		2
252.67	odd	6		2268.2.l.h.541.1		2
252.79	odd	6		2268.2.l.h.109.1		2
252.95	even	6		2268.2.l.a.541.1		2
252.151	odd	6		2268.2.i.a.2053.1		2
252.191	even	6		2268.2.l.a.109.1		2
252.247	odd	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	28.11	odd	6
28.2.e.a.25.1	yes	2	28.23	odd	6
112.2.i.b.65.1		2	7.4	even	3
112.2.i.b.81.1		2	7.2	even	3
196.2.a.a.1.1		1	28.27	even	2
196.2.a.b.1.1		1	4.3	odd	2
196.2.e.a.165.1		2	28.19	even	6
196.2.e.a.177.1		2	28.3	even	6
252.2.k.c.37.1		2	84.11	even	6
252.2.k.c.109.1		2	84.23	even	6
448.2.i.c.65.1		2	56.53	even	6
448.2.i.c.193.1		2	56.37	even	6
448.2.i.e.65.1		2	56.11	odd	6
448.2.i.e.193.1		2	56.51	odd	6
700.2.i.c.401.1		2	140.39	odd	6
700.2.i.c.501.1		2	140.79	odd	6
700.2.r.b.149.1		4	140.67	even	12
700.2.r.b.149.2		4	140.123	even	12
700.2.r.b.249.1		4	140.23	even	12
700.2.r.b.249.2		4	140.107	even	12
784.2.a.d.1.1		1	1.1	even	1	trivial
784.2.a.g.1.1		1	7.6	odd	2
784.2.i.d.177.1		2	7.3	odd	6
784.2.i.d.753.1		2	7.5	odd	6
1008.2.s.p.289.1		2	21.11	odd	6
1008.2.s.p.865.1		2	21.2	odd	6
1764.2.a.a.1.1		1	12.11	even	2
1764.2.a.j.1.1		1	84.83	odd	2
1764.2.k.b.361.1		2	84.47	odd	6
1764.2.k.b.1549.1		2	84.59	odd	6
2268.2.i.a.865.1		2	252.247	odd	6
2268.2.i.a.2053.1		2	252.151	odd	6
2268.2.i.h.865.1		2	252.23	even	6
2268.2.i.h.2053.1		2	252.11	even	6
2268.2.l.a.109.1		2	252.191	even	6
2268.2.l.a.541.1		2	252.95	even	6
2268.2.l.h.109.1		2	252.79	odd	6
2268.2.l.h.541.1		2	252.67	odd	6
3136.2.a.h.1.1		1	8.3	odd	2
3136.2.a.k.1.1		1	56.13	odd	2
3136.2.a.s.1.1		1	8.5	even	2
3136.2.a.v.1.1		1	56.27	even	2
4900.2.a.g.1.1		1	20.19	odd	2
4900.2.a.n.1.1		1	140.139	even	2
4900.2.e.h.2549.1		2	140.83	odd	4
4900.2.e.h.2549.2		2	140.27	odd	4
4900.2.e.i.2549.1		2	20.7	even	4
4900.2.e.i.2549.2		2	20.3	even	4
7056.2.a.f.1.1		1	3.2	odd	2
7056.2.a.bw.1.1		1	21.20	even	2