# Properties

 Label 6400.2.a.bk Level $6400$ Weight $2$ Character orbit 6400.a Self dual yes Analytic conductor $51.104$ Analytic rank $0$ Dimension $2$ CM no 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 640) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + 3 \beta q^{7} - q^{9} +O(q^{10})$$ $$q + \beta q^{3} + 3 \beta q^{7} - q^{9} -2 \beta q^{11} -6 q^{13} + 6 q^{17} + 6 q^{21} + 3 \beta q^{23} -4 \beta q^{27} + 6 \beta q^{31} -4 q^{33} -6 q^{37} -6 \beta q^{39} + 3 \beta q^{43} + 3 \beta q^{47} + 11 q^{49} + 6 \beta q^{51} -6 q^{53} + 8 \beta q^{59} + 6 q^{61} -3 \beta q^{63} + 9 \beta q^{67} + 6 q^{69} + 6 \beta q^{71} -2 q^{73} -12 q^{77} -5 q^{81} + 7 \beta q^{83} + 6 q^{89} -18 \beta q^{91} + 12 q^{93} + 10 q^{97} + 2 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} - 12q^{13} + 12q^{17} + 12q^{21} - 8q^{33} - 12q^{37} + 22q^{49} - 12q^{53} + 12q^{61} + 12q^{69} - 4q^{73} - 24q^{77} - 10q^{81} + 12q^{89} + 24q^{93} + 20q^{97} + O(q^{100})$$

## 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
 −1.41421 1.41421
0 −1.41421 0 0 0 −4.24264 0 −1.00000 0
1.2 0 1.41421 0 0 0 4.24264 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
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6400.2.a.bk 2
4.b odd 2 1 inner 6400.2.a.bk 2
5.b even 2 1 1280.2.a.g 2
8.b even 2 1 6400.2.a.bp 2
8.d odd 2 1 6400.2.a.bp 2
16.e even 4 2 3200.2.d.v 4
16.f odd 4 2 3200.2.d.v 4
20.d odd 2 1 1280.2.a.g 2
40.e odd 2 1 1280.2.a.i 2
40.f even 2 1 1280.2.a.i 2
80.i odd 4 2 3200.2.f.l 4
80.j even 4 2 3200.2.f.g 4
80.k odd 4 2 640.2.d.b 4
80.q even 4 2 640.2.d.b 4
80.s even 4 2 3200.2.f.l 4
80.t odd 4 2 3200.2.f.g 4
240.t even 4 2 5760.2.k.v 4
240.bm odd 4 2 5760.2.k.v 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.d.b 4 80.k odd 4 2
640.2.d.b 4 80.q even 4 2
1280.2.a.g 2 5.b even 2 1
1280.2.a.g 2 20.d odd 2 1
1280.2.a.i 2 40.e odd 2 1
1280.2.a.i 2 40.f even 2 1
3200.2.d.v 4 16.e even 4 2
3200.2.d.v 4 16.f odd 4 2
3200.2.f.g 4 80.j even 4 2
3200.2.f.g 4 80.t odd 4 2
3200.2.f.l 4 80.i odd 4 2
3200.2.f.l 4 80.s even 4 2
5760.2.k.v 4 240.t even 4 2
5760.2.k.v 4 240.bm odd 4 2
6400.2.a.bk 2 1.a even 1 1 trivial
6400.2.a.bk 2 4.b odd 2 1 inner
6400.2.a.bp 2 8.b even 2 1
6400.2.a.bp 2 8.d odd 2 1

## Hecke kernels

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

 $$T_{3}^{2} - 2$$ $$T_{7}^{2} - 18$$ $$T_{11}^{2} - 8$$ $$T_{13} + 6$$ $$T_{17} - 6$$ $$T_{29}$$ $$T_{31}^{2} - 72$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-2 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$-18 + T^{2}$$
$11$ $$-8 + T^{2}$$
$13$ $$( 6 + T )^{2}$$
$17$ $$( -6 + T )^{2}$$
$19$ $$T^{2}$$
$23$ $$-18 + T^{2}$$
$29$ $$T^{2}$$
$31$ $$-72 + T^{2}$$
$37$ $$( 6 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$-18 + T^{2}$$
$47$ $$-18 + T^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$-128 + T^{2}$$
$61$ $$( -6 + T )^{2}$$
$67$ $$-162 + T^{2}$$
$71$ $$-72 + T^{2}$$
$73$ $$( 2 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$-98 + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$( -10 + T )^{2}$$