# Properties

 Label 800.2.a.g Level $800$ Weight $2$ Character orbit 800.a Self dual yes Analytic conductor $6.388$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 800.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.38803216170$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - 2q^{7} - 2q^{9} + O(q^{10})$$ $$q + q^{3} - 2q^{7} - 2q^{9} - 5q^{11} - 5q^{17} - 5q^{19} - 2q^{21} + 6q^{23} - 5q^{27} + 4q^{29} - 10q^{31} - 5q^{33} + 10q^{37} + 5q^{41} + 4q^{43} - 8q^{47} - 3q^{49} - 5q^{51} + 10q^{53} - 5q^{57} - 10q^{61} + 4q^{63} + 3q^{67} + 6q^{69} + 5q^{73} + 10q^{77} - 10q^{79} + q^{81} - q^{83} + 4q^{87} - 9q^{89} - 10q^{93} - 10q^{97} + 10q^{99} + O(q^{100})$$

## 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 −2.00000 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
$$2$$ $$1$$
$$5$$ $$1$$

## 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 800.2.a.g yes 1
3.b odd 2 1 7200.2.a.o 1
4.b odd 2 1 800.2.a.c yes 1
5.b even 2 1 800.2.a.b 1
5.c odd 4 2 800.2.c.c 2
8.b even 2 1 1600.2.a.h 1
8.d odd 2 1 1600.2.a.r 1
12.b even 2 1 7200.2.a.bm 1
15.d odd 2 1 7200.2.a.bq 1
15.e even 4 2 7200.2.f.bc 2
20.d odd 2 1 800.2.a.h yes 1
20.e even 4 2 800.2.c.d 2
40.e odd 2 1 1600.2.a.g 1
40.f even 2 1 1600.2.a.s 1
40.i odd 4 2 1600.2.c.j 2
40.k even 4 2 1600.2.c.g 2
60.h even 2 1 7200.2.a.k 1
60.l odd 4 2 7200.2.f.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.2.a.b 1 5.b even 2 1
800.2.a.c yes 1 4.b odd 2 1
800.2.a.g yes 1 1.a even 1 1 trivial
800.2.a.h yes 1 20.d odd 2 1
800.2.c.c 2 5.c odd 4 2
800.2.c.d 2 20.e even 4 2
1600.2.a.g 1 40.e odd 2 1
1600.2.a.h 1 8.b even 2 1
1600.2.a.r 1 8.d odd 2 1
1600.2.a.s 1 40.f even 2 1
1600.2.c.g 2 40.k even 4 2
1600.2.c.j 2 40.i odd 4 2
7200.2.a.k 1 60.h even 2 1
7200.2.a.o 1 3.b odd 2 1
7200.2.a.bm 1 12.b even 2 1
7200.2.a.bq 1 15.d odd 2 1
7200.2.f.a 2 60.l odd 4 2
7200.2.f.bc 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(800))$$:

 $$T_{3} - 1$$ $$T_{11} + 5$$ $$T_{13}$$

## Hecke characteristic polynomials

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