Embedded newform 2800.2.a.p.1.1

• Label: 2800.2.a.p.1.1
• Level: $2800$
• Weight: $2$
• Character: 2800.1
• Self dual: yes
• Analytic conductor: $22.358$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2800.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(22.3581125660\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	2800.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{7} -3.00000 q^{9} +4.00000 q^{11} -2.00000 q^{13} +6.00000 q^{17} -8.00000 q^{19} +6.00000 q^{29} -8.00000 q^{31} +2.00000 q^{37} +2.00000 q^{41} -4.00000 q^{43} -8.00000 q^{47} +1.00000 q^{49} -6.00000 q^{53} -6.00000 q^{61} +3.00000 q^{63} -4.00000 q^{67} +8.00000 q^{71} -10.0000 q^{73} -4.00000 q^{77} -16.0000 q^{79} +9.00000 q^{81} +8.00000 q^{83} -6.00000 q^{89} +2.00000 q^{91} +6.00000 q^{97} -12.0000 q^{99} +O(q^{100})\)

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
			\(7\)	−1.00000	−0.377964
\(11\)	4.00000	1.20605				0.603023	−	0.797724i	\(-0.293963\pi\)
			0.603023	+	0.797724i	\(0.293963\pi\)
\(13\)	−2.00000	−0.554700	−0.277350	−	0.960769i	\(-0.589456\pi\)
			−0.277350	+	0.960769i	\(0.589456\pi\)
\(17\)	6.00000	1.45521	0.727607	−	0.685994i	\(-0.240633\pi\)
			0.727607	+	0.685994i	\(0.240633\pi\)
\(19\)	−8.00000	−1.83533	−0.917663	−	0.397360i	\(-0.869927\pi\)
			−0.917663	+	0.397360i	\(0.869927\pi\)
\(23\)	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	\(0.311919\pi\)
\(31\)	−8.00000	−1.43684	−0.718421	−	0.695608i	\(-0.755135\pi\)
			−0.718421	+	0.695608i	\(0.755135\pi\)
\(37\)	2.00000	0.328798	0.164399	−	0.986394i	\(-0.447432\pi\)
			0.164399	+	0.986394i	\(0.447432\pi\)
\(41\)	2.00000	0.312348	0.156174	−	0.987730i	\(-0.450084\pi\)
			0.156174	+	0.987730i	\(0.450084\pi\)
\(43\)	−4.00000	−0.609994	−0.304997	−	0.952353i	\(-0.598656\pi\)
			−0.304997	+	0.952353i	\(0.598656\pi\)
\(47\)	−8.00000	−1.16692	−0.583460	−	0.812142i	\(-0.698301\pi\)
			−0.583460	+	0.812142i	\(0.698301\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2800.2.a.p.1.1		1
4.3	odd	2		1400.2.a.g.1.1		1
5.2	odd	4		2800.2.g.p.449.1		2
5.3	odd	4		2800.2.g.p.449.2		2
5.4	even	2		112.2.a.b.1.1		1
15.14	odd	2		1008.2.a.d.1.1		1
20.3	even	4		1400.2.g.g.449.1		2
20.7	even	4		1400.2.g.g.449.2		2
20.19	odd	2		56.2.a.a.1.1	✓	1
28.27	even	2		9800.2.a.u.1.1		1
35.4	even	6		784.2.i.e.177.1		2
35.9	even	6		784.2.i.e.753.1		2
35.19	odd	6		784.2.i.g.753.1		2
35.24	odd	6		784.2.i.g.177.1		2
35.34	odd	2		784.2.a.e.1.1		1
40.19	odd	2		448.2.a.d.1.1		1
40.29	even	2		448.2.a.e.1.1		1
60.59	even	2		504.2.a.c.1.1		1
80.19	odd	4		1792.2.b.i.897.1		2
80.29	even	4		1792.2.b.d.897.1		2
80.59	odd	4		1792.2.b.i.897.2		2
80.69	even	4		1792.2.b.d.897.2		2
105.104	even	2		7056.2.a.bo.1.1		1
120.29	odd	2		4032.2.a.bk.1.1		1
120.59	even	2		4032.2.a.bb.1.1		1
140.19	even	6		392.2.i.d.361.1		2
140.39	odd	6		392.2.i.c.177.1		2
140.59	even	6		392.2.i.d.177.1		2
140.79	odd	6		392.2.i.c.361.1		2
140.139	even	2		392.2.a.d.1.1		1
220.219	even	2		6776.2.a.g.1.1		1
260.259	odd	2		9464.2.a.c.1.1		1
280.69	odd	2		3136.2.a.p.1.1		1
280.139	even	2		3136.2.a.q.1.1		1
420.59	odd	6		3528.2.s.e.3313.1		2
420.179	even	6		3528.2.s.t.3313.1		2
420.299	odd	6		3528.2.s.e.361.1		2
420.359	even	6		3528.2.s.t.361.1		2
420.419	odd	2		3528.2.a.x.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.2.a.a.1.1	✓	1	20.19	odd	2
112.2.a.b.1.1		1	5.4	even	2
392.2.a.d.1.1		1	140.139	even	2
392.2.i.c.177.1		2	140.39	odd	6
392.2.i.c.361.1		2	140.79	odd	6
392.2.i.d.177.1		2	140.59	even	6
392.2.i.d.361.1		2	140.19	even	6
448.2.a.d.1.1		1	40.19	odd	2
448.2.a.e.1.1		1	40.29	even	2
504.2.a.c.1.1		1	60.59	even	2
784.2.a.e.1.1		1	35.34	odd	2
784.2.i.e.177.1		2	35.4	even	6
784.2.i.e.753.1		2	35.9	even	6
784.2.i.g.177.1		2	35.24	odd	6
784.2.i.g.753.1		2	35.19	odd	6
1008.2.a.d.1.1		1	15.14	odd	2
1400.2.a.g.1.1		1	4.3	odd	2
1400.2.g.g.449.1		2	20.3	even	4
1400.2.g.g.449.2		2	20.7	even	4
1792.2.b.d.897.1		2	80.29	even	4
1792.2.b.d.897.2		2	80.69	even	4
1792.2.b.i.897.1		2	80.19	odd	4
1792.2.b.i.897.2		2	80.59	odd	4
2800.2.a.p.1.1		1	1.1	even	1	trivial
2800.2.g.p.449.1		2	5.2	odd	4
2800.2.g.p.449.2		2	5.3	odd	4
3136.2.a.p.1.1		1	280.69	odd	2
3136.2.a.q.1.1		1	280.139	even	2
3528.2.a.x.1.1		1	420.419	odd	2
3528.2.s.e.361.1		2	420.299	odd	6
3528.2.s.e.3313.1		2	420.59	odd	6
3528.2.s.t.361.1		2	420.359	even	6
3528.2.s.t.3313.1		2	420.179	even	6
4032.2.a.bb.1.1		1	120.59	even	2
4032.2.a.bk.1.1		1	120.29	odd	2
6776.2.a.g.1.1		1	220.219	even	2
7056.2.a.bo.1.1		1	105.104	even	2
9464.2.a.c.1.1		1	260.259	odd	2
9800.2.a.u.1.1		1	28.27	even	2