# Properties

 Label 2800.2.a.bf Level $2800$ Weight $2$ Character orbit 2800.a Self dual yes Analytic conductor $22.358$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3q^{3} + q^{7} + 6q^{9} + O(q^{10})$$ $$q + 3q^{3} + q^{7} + 6q^{9} - 3q^{11} - q^{13} + 5q^{17} + 8q^{19} + 3q^{21} + 2q^{23} + 9q^{27} - q^{29} + 2q^{31} - 9q^{33} - 10q^{37} - 3q^{39} - 6q^{41} - 4q^{43} + 11q^{47} + q^{49} + 15q^{51} - 6q^{53} + 24q^{57} + 10q^{59} + 6q^{63} - 10q^{67} + 6q^{69} + 10q^{73} - 3q^{77} + 7q^{79} + 9q^{81} + 12q^{83} - 3q^{87} + 8q^{89} - q^{91} + 6q^{93} - 3q^{97} - 18q^{99} + O(q^{100})$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 3.00000 0 0 0 1.00000 0 6.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2800.2.a.bf 1
4.b odd 2 1 700.2.a.a 1
5.b even 2 1 2800.2.a.a 1
5.c odd 4 2 560.2.g.a 2
12.b even 2 1 6300.2.a.c 1
15.e even 4 2 5040.2.t.s 2
20.d odd 2 1 700.2.a.j 1
20.e even 4 2 140.2.e.a 2
28.d even 2 1 4900.2.a.w 1
40.i odd 4 2 2240.2.g.f 2
40.k even 4 2 2240.2.g.e 2
60.h even 2 1 6300.2.a.t 1
60.l odd 4 2 1260.2.k.c 2
140.c even 2 1 4900.2.a.b 1
140.j odd 4 2 980.2.e.b 2
140.w even 12 4 980.2.q.f 4
140.x odd 12 4 980.2.q.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.e.a 2 20.e even 4 2
560.2.g.a 2 5.c odd 4 2
700.2.a.a 1 4.b odd 2 1
700.2.a.j 1 20.d odd 2 1
980.2.e.b 2 140.j odd 4 2
980.2.q.c 4 140.x odd 12 4
980.2.q.f 4 140.w even 12 4
1260.2.k.c 2 60.l odd 4 2
2240.2.g.e 2 40.k even 4 2
2240.2.g.f 2 40.i odd 4 2
2800.2.a.a 1 5.b even 2 1
2800.2.a.bf 1 1.a even 1 1 trivial
4900.2.a.b 1 140.c even 2 1
4900.2.a.w 1 28.d even 2 1
5040.2.t.s 2 15.e even 4 2
6300.2.a.c 1 12.b even 2 1
6300.2.a.t 1 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2800))$$:

 $$T_{3} - 3$$ $$T_{11} + 3$$ $$T_{13} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-3 + T$$
$5$ $$T$$
$7$ $$-1 + T$$
$11$ $$3 + T$$
$13$ $$1 + T$$
$17$ $$-5 + T$$
$19$ $$-8 + T$$
$23$ $$-2 + T$$
$29$ $$1 + T$$
$31$ $$-2 + T$$
$37$ $$10 + T$$
$41$ $$6 + T$$
$43$ $$4 + T$$
$47$ $$-11 + T$$
$53$ $$6 + T$$
$59$ $$-10 + T$$
$61$ $$T$$
$67$ $$10 + T$$
$71$ $$T$$
$73$ $$-10 + T$$
$79$ $$-7 + T$$
$83$ $$-12 + T$$
$89$ $$-8 + T$$
$97$ $$3 + T$$