Properties

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

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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 350) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + 3q^{3} + q^{7} + 6q^{9} + O(q^{10})$$ $$q + 3q^{3} + q^{7} + 6q^{9} + 5q^{11} - 6q^{13} - q^{17} + 3q^{19} + 3q^{21} + 9q^{27} - 6q^{29} + 4q^{31} + 15q^{33} + 8q^{37} - 18q^{39} + 11q^{41} + 8q^{43} - 2q^{47} + q^{49} - 3q^{51} + 4q^{53} + 9q^{57} - 4q^{59} - 2q^{61} + 6q^{63} - 9q^{67} + 10q^{71} - 7q^{73} + 5q^{77} + 2q^{79} + 9q^{81} - 11q^{83} - 18q^{87} - 11q^{89} - 6q^{91} + 12q^{93} - 10q^{97} + 30q^{99} + O(q^{100})$$

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.bg 1
4.b odd 2 1 350.2.a.d yes 1
5.b even 2 1 2800.2.a.b 1
5.c odd 4 2 2800.2.g.a 2
12.b even 2 1 3150.2.a.j 1
20.d odd 2 1 350.2.a.c 1
20.e even 4 2 350.2.c.a 2
28.d even 2 1 2450.2.a.bg 1
60.h even 2 1 3150.2.a.bq 1
60.l odd 4 2 3150.2.g.v 2
140.c even 2 1 2450.2.a.a 1
140.j odd 4 2 2450.2.c.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.a.c 1 20.d odd 2 1
350.2.a.d yes 1 4.b odd 2 1
350.2.c.a 2 20.e even 4 2
2450.2.a.a 1 140.c even 2 1
2450.2.a.bg 1 28.d even 2 1
2450.2.c.r 2 140.j odd 4 2
2800.2.a.b 1 5.b even 2 1
2800.2.a.bg 1 1.a even 1 1 trivial
2800.2.g.a 2 5.c odd 4 2
3150.2.a.j 1 12.b even 2 1
3150.2.a.bq 1 60.h even 2 1
3150.2.g.v 2 60.l odd 4 2

Hecke kernels

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

 $$T_{3} - 3$$ $$T_{11} - 5$$ $$T_{13} + 6$$

Hecke characteristic polynomials

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