# Properties

 Label 6400.2.a.x Level $6400$ Weight $2$ Character orbit 6400.a Self dual yes Analytic conductor $51.104$ Analytic rank $1$ Dimension $1$ CM discriminant -8 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6400 = 2^{8} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6400.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$51.1042572936$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 64) Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2q^{3} + q^{9} + O(q^{10})$$ $$q + 2q^{3} + q^{9} - 6q^{11} + 6q^{17} - 2q^{19} - 4q^{27} - 12q^{33} + 6q^{41} - 10q^{43} - 7q^{49} + 12q^{51} - 4q^{57} - 6q^{59} - 14q^{67} + 2q^{73} - 11q^{81} + 18q^{83} - 18q^{89} - 10q^{97} - 6q^{99} + O(q^{100})$$

## 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 2.00000 0 0 0 0 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
$$2$$ $$1$$
$$5$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6400.2.a.x 1
4.b odd 2 1 6400.2.a.a 1
5.b even 2 1 256.2.a.a 1
8.b even 2 1 6400.2.a.a 1
8.d odd 2 1 CM 6400.2.a.x 1
15.d odd 2 1 2304.2.a.i 1
16.e even 4 2 1600.2.d.a 2
16.f odd 4 2 1600.2.d.a 2
20.d odd 2 1 256.2.a.d 1
40.e odd 2 1 256.2.a.a 1
40.f even 2 1 256.2.a.d 1
60.h even 2 1 2304.2.a.h 1
80.i odd 4 2 1600.2.f.a 2
80.j even 4 2 1600.2.f.a 2
80.k odd 4 2 64.2.b.a 2
80.q even 4 2 64.2.b.a 2
80.s even 4 2 1600.2.f.b 2
80.t odd 4 2 1600.2.f.b 2
120.i odd 2 1 2304.2.a.h 1
120.m even 2 1 2304.2.a.i 1
160.y odd 8 4 1024.2.e.l 4
160.z even 8 4 1024.2.e.l 4
240.t even 4 2 576.2.d.a 2
240.bm odd 4 2 576.2.d.a 2
560.be even 4 2 3136.2.b.b 2
560.bf odd 4 2 3136.2.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.2.b.a 2 80.k odd 4 2
64.2.b.a 2 80.q even 4 2
256.2.a.a 1 5.b even 2 1
256.2.a.a 1 40.e odd 2 1
256.2.a.d 1 20.d odd 2 1
256.2.a.d 1 40.f even 2 1
576.2.d.a 2 240.t even 4 2
576.2.d.a 2 240.bm odd 4 2
1024.2.e.l 4 160.y odd 8 4
1024.2.e.l 4 160.z even 8 4
1600.2.d.a 2 16.e even 4 2
1600.2.d.a 2 16.f odd 4 2
1600.2.f.a 2 80.i odd 4 2
1600.2.f.a 2 80.j even 4 2
1600.2.f.b 2 80.s even 4 2
1600.2.f.b 2 80.t odd 4 2
2304.2.a.h 1 60.h even 2 1
2304.2.a.h 1 120.i odd 2 1
2304.2.a.i 1 15.d odd 2 1
2304.2.a.i 1 120.m even 2 1
3136.2.b.b 2 560.be even 4 2
3136.2.b.b 2 560.bf odd 4 2
6400.2.a.a 1 4.b odd 2 1
6400.2.a.a 1 8.b even 2 1
6400.2.a.x 1 1.a even 1 1 trivial
6400.2.a.x 1 8.d odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6400))$$:

 $$T_{3} - 2$$ $$T_{7}$$ $$T_{11} + 6$$ $$T_{13}$$ $$T_{17} - 6$$ $$T_{29}$$ $$T_{31}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-2 + T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$6 + T$$
$13$ $$T$$
$17$ $$-6 + T$$
$19$ $$2 + T$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$-6 + T$$
$43$ $$10 + T$$
$47$ $$T$$
$53$ $$T$$
$59$ $$6 + T$$
$61$ $$T$$
$67$ $$14 + T$$
$71$ $$T$$
$73$ $$-2 + T$$
$79$ $$T$$
$83$ $$-18 + T$$
$89$ $$18 + T$$
$97$ $$10 + T$$