Properties

 Label 2800.2.a.bb Level $2800$ Weight $2$ Character orbit 2800.a Self dual yes Analytic conductor $22.358$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1400) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + 2q^{3} - q^{7} + q^{9} + O(q^{10})$$ $$q + 2q^{3} - q^{7} + q^{9} - q^{11} - 4q^{13} - 6q^{19} - 2q^{21} + 3q^{23} - 4q^{27} - 3q^{29} - 2q^{33} - 9q^{37} - 8q^{39} + 2q^{41} + 9q^{43} - 6q^{47} + q^{49} - 6q^{53} - 12q^{57} - 8q^{59} - 10q^{61} - q^{63} - q^{67} + 6q^{69} + 7q^{71} + 2q^{73} + q^{77} + 9q^{79} - 11q^{81} + 12q^{83} - 6q^{87} - 4q^{89} + 4q^{91} - 16q^{97} - q^{99} + O(q^{100})$$

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 2.00000 0 0 0 −1.00000 0 1.00000 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.bb 1
4.b odd 2 1 1400.2.a.c 1
5.b even 2 1 2800.2.a.f 1
5.c odd 4 2 2800.2.g.f 2
20.d odd 2 1 1400.2.a.l yes 1
20.e even 4 2 1400.2.g.c 2
28.d even 2 1 9800.2.a.bk 1
140.c even 2 1 9800.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.a.c 1 4.b odd 2 1
1400.2.a.l yes 1 20.d odd 2 1
1400.2.g.c 2 20.e even 4 2
2800.2.a.f 1 5.b even 2 1
2800.2.a.bb 1 1.a even 1 1 trivial
2800.2.g.f 2 5.c odd 4 2
9800.2.a.h 1 140.c even 2 1
9800.2.a.bk 1 28.d even 2 1

Hecke kernels

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

 $$T_{3} - 2$$ $$T_{11} + 1$$ $$T_{13} + 4$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-2 + T$$
$5$ $$T$$
$7$ $$1 + T$$
$11$ $$1 + T$$
$13$ $$4 + T$$
$17$ $$T$$
$19$ $$6 + T$$
$23$ $$-3 + T$$
$29$ $$3 + T$$
$31$ $$T$$
$37$ $$9 + T$$
$41$ $$-2 + T$$
$43$ $$-9 + T$$
$47$ $$6 + T$$
$53$ $$6 + T$$
$59$ $$8 + T$$
$61$ $$10 + T$$
$67$ $$1 + T$$
$71$ $$-7 + T$$
$73$ $$-2 + T$$
$79$ $$-9 + T$$
$83$ $$-12 + T$$
$89$ $$4 + T$$
$97$ $$16 + T$$