# Properties

 Label 2800.2.a.l 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 35) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{7} - 2q^{9} + O(q^{10})$$ $$q - q^{3} + q^{7} - 2q^{9} + 3q^{11} + q^{13} + 7q^{17} - q^{21} - 6q^{23} + 5q^{27} - 5q^{29} - 2q^{31} - 3q^{33} + 2q^{37} - q^{39} + 2q^{41} + 4q^{43} + 3q^{47} + q^{49} - 7q^{51} + 6q^{53} - 10q^{59} - 8q^{61} - 2q^{63} - 2q^{67} + 6q^{69} + 8q^{71} + 6q^{73} + 3q^{77} + 5q^{79} + q^{81} + 4q^{83} + 5q^{87} + q^{91} + 2q^{93} + 7q^{97} - 6q^{99} + O(q^{100})$$

## 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 −1.00000 0 0 0 1.00000 0 −2.00000 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.l 1
4.b odd 2 1 175.2.a.c 1
5.b even 2 1 2800.2.a.w 1
5.c odd 4 2 560.2.g.b 2
12.b even 2 1 1575.2.a.a 1
15.e even 4 2 5040.2.t.p 2
20.d odd 2 1 175.2.a.a 1
20.e even 4 2 35.2.b.a 2
28.d even 2 1 1225.2.a.i 1
40.i odd 4 2 2240.2.g.g 2
40.k even 4 2 2240.2.g.h 2
60.h even 2 1 1575.2.a.k 1
60.l odd 4 2 315.2.d.a 2
140.c even 2 1 1225.2.a.a 1
140.j odd 4 2 245.2.b.a 2
140.w even 12 4 245.2.j.e 4
140.x odd 12 4 245.2.j.d 4
420.w even 4 2 2205.2.d.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.b.a 2 20.e even 4 2
175.2.a.a 1 20.d odd 2 1
175.2.a.c 1 4.b odd 2 1
245.2.b.a 2 140.j odd 4 2
245.2.j.d 4 140.x odd 12 4
245.2.j.e 4 140.w even 12 4
315.2.d.a 2 60.l odd 4 2
560.2.g.b 2 5.c odd 4 2
1225.2.a.a 1 140.c even 2 1
1225.2.a.i 1 28.d even 2 1
1575.2.a.a 1 12.b even 2 1
1575.2.a.k 1 60.h even 2 1
2205.2.d.b 2 420.w even 4 2
2240.2.g.g 2 40.i odd 4 2
2240.2.g.h 2 40.k even 4 2
2800.2.a.l 1 1.a even 1 1 trivial
2800.2.a.w 1 5.b even 2 1
5040.2.t.p 2 15.e even 4 2

## Hecke kernels

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

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

## Hecke characteristic polynomials

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