# Properties

 Label 6400.2.a.cs Level $6400$ Weight $2$ Character orbit 6400.a Self dual yes Analytic conductor $51.104$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{5})$$ Defining polynomial: $$x^{4} - 6 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 3200) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} + \beta_{1} q^{7} + 2 q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} + \beta_{1} q^{7} + 2 q^{9} + \beta_{3} q^{11} + \beta_{2} q^{13} + 5 q^{17} + \beta_{3} q^{19} + \beta_{2} q^{21} -2 \beta_{1} q^{23} -\beta_{3} q^{27} + \beta_{2} q^{29} + 5 q^{33} -\beta_{2} q^{37} + 5 \beta_{1} q^{39} -3 q^{41} + 4 \beta_{3} q^{43} -\beta_{1} q^{47} + q^{49} + 5 \beta_{3} q^{51} -2 \beta_{2} q^{53} + 5 q^{57} + 4 \beta_{3} q^{59} -\beta_{2} q^{61} + 2 \beta_{1} q^{63} -5 \beta_{3} q^{67} -2 \beta_{2} q^{69} + 5 \beta_{1} q^{71} + 15 q^{73} + \beta_{2} q^{77} -5 \beta_{1} q^{79} -11 q^{81} + 3 \beta_{3} q^{83} + 5 \beta_{1} q^{87} - q^{89} + 8 \beta_{3} q^{91} -10 q^{97} + 2 \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{9} + O(q^{10})$$ $$4q + 8q^{9} + 20q^{17} + 20q^{33} - 12q^{41} + 4q^{49} + 20q^{57} + 60q^{73} - 44q^{81} - 4q^{89} - 40q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 6 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} - 4 \nu$$ $$\beta_{2}$$ $$=$$ $$-\nu^{3} + 8 \nu$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{2} + 2 \beta_{1}$$

## 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
 0.874032 −0.874032 −2.28825 2.28825
0 −2.23607 0 0 0 −2.82843 0 2.00000 0
1.2 0 −2.23607 0 0 0 2.82843 0 2.00000 0
1.3 0 2.23607 0 0 0 −2.82843 0 2.00000 0
1.4 0 2.23607 0 0 0 2.82843 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d 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.cs 4
4.b odd 2 1 inner 6400.2.a.cs 4
5.b even 2 1 6400.2.a.cp 4
8.b even 2 1 inner 6400.2.a.cs 4
8.d odd 2 1 inner 6400.2.a.cs 4
16.e even 4 2 3200.2.d.r yes 4
16.f odd 4 2 3200.2.d.r yes 4
20.d odd 2 1 6400.2.a.cp 4
40.e odd 2 1 6400.2.a.cp 4
40.f even 2 1 6400.2.a.cp 4
80.i odd 4 2 3200.2.f.r 8
80.j even 4 2 3200.2.f.r 8
80.k odd 4 2 3200.2.d.m 4
80.q even 4 2 3200.2.d.m 4
80.s even 4 2 3200.2.f.r 8
80.t odd 4 2 3200.2.f.r 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.2.d.m 4 80.k odd 4 2
3200.2.d.m 4 80.q even 4 2
3200.2.d.r yes 4 16.e even 4 2
3200.2.d.r yes 4 16.f odd 4 2
3200.2.f.r 8 80.i odd 4 2
3200.2.f.r 8 80.j even 4 2
3200.2.f.r 8 80.s even 4 2
3200.2.f.r 8 80.t odd 4 2
6400.2.a.cp 4 5.b even 2 1
6400.2.a.cp 4 20.d odd 2 1
6400.2.a.cp 4 40.e odd 2 1
6400.2.a.cp 4 40.f even 2 1
6400.2.a.cs 4 1.a even 1 1 trivial
6400.2.a.cs 4 4.b odd 2 1 inner
6400.2.a.cs 4 8.b even 2 1 inner
6400.2.a.cs 4 8.d odd 2 1 inner

## Hecke kernels

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

 $$T_{3}^{2} - 5$$ $$T_{7}^{2} - 8$$ $$T_{11}^{2} - 5$$ $$T_{13}^{2} - 40$$ $$T_{17} - 5$$ $$T_{29}^{2} - 40$$ $$T_{31}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -5 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( -8 + T^{2} )^{2}$$
$11$ $$( -5 + T^{2} )^{2}$$
$13$ $$( -40 + T^{2} )^{2}$$
$17$ $$( -5 + T )^{4}$$
$19$ $$( -5 + T^{2} )^{2}$$
$23$ $$( -32 + T^{2} )^{2}$$
$29$ $$( -40 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$( -40 + T^{2} )^{2}$$
$41$ $$( 3 + T )^{4}$$
$43$ $$( -80 + T^{2} )^{2}$$
$47$ $$( -8 + T^{2} )^{2}$$
$53$ $$( -160 + T^{2} )^{2}$$
$59$ $$( -80 + T^{2} )^{2}$$
$61$ $$( -40 + T^{2} )^{2}$$
$67$ $$( -125 + T^{2} )^{2}$$
$71$ $$( -200 + T^{2} )^{2}$$
$73$ $$( -15 + T )^{4}$$
$79$ $$( -200 + T^{2} )^{2}$$
$83$ $$( -45 + T^{2} )^{2}$$
$89$ $$( 1 + T )^{4}$$
$97$ $$( 10 + T )^{4}$$