# Properties

 Label 6400.2.a.n Level $6400$ Weight $2$ Character orbit 6400.a Self dual yes Analytic conductor $51.104$ Analytic rank $1$ Dimension $1$ CM discriminant -4 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 640) Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 3q^{9} + O(q^{10})$$ $$q - 3q^{9} + 6q^{13} - 8q^{17} + 4q^{29} + 2q^{37} + 10q^{41} - 7q^{49} - 14q^{53} - 12q^{61} + 16q^{73} + 9q^{81} - 10q^{89} + 8q^{97} + O(q^{100})$$

## 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 0 0 0 0 0 0 −3.00000 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6400.2.a.n 1
4.b odd 2 1 CM 6400.2.a.n 1
5.b even 2 1 6400.2.a.k 1
5.c odd 4 2 1280.2.c.d 2
8.b even 2 1 6400.2.a.j 1
8.d odd 2 1 6400.2.a.j 1
16.e even 4 2 3200.2.d.d 2
16.f odd 4 2 3200.2.d.d 2
20.d odd 2 1 6400.2.a.k 1
20.e even 4 2 1280.2.c.d 2
40.e odd 2 1 6400.2.a.o 1
40.f even 2 1 6400.2.a.o 1
40.i odd 4 2 1280.2.c.a 2
40.k even 4 2 1280.2.c.a 2
80.i odd 4 2 640.2.f.a 2
80.j even 4 2 640.2.f.b yes 2
80.k odd 4 2 3200.2.d.f 2
80.q even 4 2 3200.2.d.f 2
80.s even 4 2 640.2.f.a 2
80.t odd 4 2 640.2.f.b yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.f.a 2 80.i odd 4 2
640.2.f.a 2 80.s even 4 2
640.2.f.b yes 2 80.j even 4 2
640.2.f.b yes 2 80.t odd 4 2
1280.2.c.a 2 40.i odd 4 2
1280.2.c.a 2 40.k even 4 2
1280.2.c.d 2 5.c odd 4 2
1280.2.c.d 2 20.e even 4 2
3200.2.d.d 2 16.e even 4 2
3200.2.d.d 2 16.f odd 4 2
3200.2.d.f 2 80.k odd 4 2
3200.2.d.f 2 80.q even 4 2
6400.2.a.j 1 8.b even 2 1
6400.2.a.j 1 8.d odd 2 1
6400.2.a.k 1 5.b even 2 1
6400.2.a.k 1 20.d odd 2 1
6400.2.a.n 1 1.a even 1 1 trivial
6400.2.a.n 1 4.b odd 2 1 CM
6400.2.a.o 1 40.e odd 2 1
6400.2.a.o 1 40.f even 2 1

## Hecke kernels

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

 $$T_{3}$$ $$T_{7}$$ $$T_{11}$$ $$T_{13} - 6$$ $$T_{17} + 8$$ $$T_{29} - 4$$ $$T_{31}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$-6 + T$$
$17$ $$8 + T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$-4 + T$$
$31$ $$T$$
$37$ $$-2 + T$$
$41$ $$-10 + T$$
$43$ $$T$$
$47$ $$T$$
$53$ $$14 + T$$
$59$ $$T$$
$61$ $$12 + T$$
$67$ $$T$$
$71$ $$T$$
$73$ $$-16 + T$$
$79$ $$T$$
$83$ $$T$$
$89$ $$10 + T$$
$97$ $$-8 + T$$