# Properties

 Label 6400.2.a.cp Level $6400$ Weight $2$ Character orbit 6400.a Self dual yes Analytic conductor $51.104$ Analytic rank $1$ 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: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{5})$$ Defining polynomial: $$x^{4} - 6x^{2} + 4$$ 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 + b3 * q^3 + b1 * q^7 + 2 * q^9 $$q + \beta_{3} q^{3} + \beta_1 q^{7} + 2 q^{9} - \beta_{3} q^{11} - \beta_{2} q^{13} - 5 q^{17} - \beta_{3} q^{19} + \beta_{2} q^{21} - 2 \beta_1 q^{23} - \beta_{3} q^{27} + \beta_{2} q^{29} - 5 q^{33} + \beta_{2} q^{37} - 5 \beta_1 q^{39} - 3 q^{41} + 4 \beta_{3} q^{43} - \beta_1 q^{47} + q^{49} - 5 \beta_{3} q^{51} + 2 \beta_{2} q^{53} - 5 q^{57} - 4 \beta_{3} q^{59} - \beta_{2} q^{61} + 2 \beta_1 q^{63} - 5 \beta_{3} q^{67} - 2 \beta_{2} q^{69} - 5 \beta_1 q^{71} - 15 q^{73} - \beta_{2} q^{77} + 5 \beta_1 q^{79} - 11 q^{81} + 3 \beta_{3} q^{83} + 5 \beta_1 q^{87} - q^{89} - 8 \beta_{3} q^{91} + 10 q^{97} - 2 \beta_{3} q^{99}+O(q^{100})$$ q + b3 * q^3 + b1 * q^7 + 2 * q^9 - b3 * q^11 - b2 * q^13 - 5 * q^17 - b3 * q^19 + b2 * q^21 - 2*b1 * q^23 - b3 * q^27 + b2 * q^29 - 5 * q^33 + b2 * q^37 - 5*b1 * q^39 - 3 * q^41 + 4*b3 * q^43 - b1 * q^47 + q^49 - 5*b3 * q^51 + 2*b2 * q^53 - 5 * q^57 - 4*b3 * q^59 - b2 * q^61 + 2*b1 * q^63 - 5*b3 * q^67 - 2*b2 * q^69 - 5*b1 * q^71 - 15 * q^73 - b2 * q^77 + 5*b1 * q^79 - 11 * q^81 + 3*b3 * q^83 + 5*b1 * q^87 - q^89 - 8*b3 * q^91 + 10 * q^97 - 2*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{9}+O(q^{10})$$ 4 * q + 8 * q^9 $$4 q + 8 q^{9} - 20 q^{17} - 20 q^{33} - 12 q^{41} + 4 q^{49} - 20 q^{57} - 60 q^{73} - 44 q^{81} - 4 q^{89} + 40 q^{97}+O(q^{100})$$ 4 * q + 8 * q^9 - 20 * q^17 - 20 * q^33 - 12 * q^41 + 4 * q^49 - 20 * q^57 - 60 * q^73 - 44 * q^81 - 4 * q^89 + 40 * q^97

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

 $$\beta_{1}$$ $$=$$ $$\nu^{3} - 4\nu$$ v^3 - 4*v $$\beta_{2}$$ $$=$$ $$-\nu^{3} + 8\nu$$ -v^3 + 8*v $$\beta_{3}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 4$$ (b2 + b1) / 4 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3$$ b3 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{2} + 2\beta_1$$ b2 + 2*b1

## 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.cp 4
4.b odd 2 1 inner 6400.2.a.cp 4
5.b even 2 1 6400.2.a.cs 4
8.b even 2 1 inner 6400.2.a.cp 4
8.d odd 2 1 inner 6400.2.a.cp 4
16.e even 4 2 3200.2.d.m 4
16.f odd 4 2 3200.2.d.m 4
20.d odd 2 1 6400.2.a.cs 4
40.e odd 2 1 6400.2.a.cs 4
40.f even 2 1 6400.2.a.cs 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.r yes 4
80.q even 4 2 3200.2.d.r yes 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 16.e even 4 2
3200.2.d.m 4 16.f odd 4 2
3200.2.d.r yes 4 80.k odd 4 2
3200.2.d.r yes 4 80.q even 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 1.a even 1 1 trivial
6400.2.a.cp 4 4.b odd 2 1 inner
6400.2.a.cp 4 8.b even 2 1 inner
6400.2.a.cp 4 8.d odd 2 1 inner
6400.2.a.cs 4 5.b even 2 1
6400.2.a.cs 4 20.d odd 2 1
6400.2.a.cs 4 40.e odd 2 1
6400.2.a.cs 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6400))$$:

 $$T_{3}^{2} - 5$$ T3^2 - 5 $$T_{7}^{2} - 8$$ T7^2 - 8 $$T_{11}^{2} - 5$$ T11^2 - 5 $$T_{13}^{2} - 40$$ T13^2 - 40 $$T_{17} + 5$$ T17 + 5 $$T_{29}^{2} - 40$$ T29^2 - 40 $$T_{31}$$ T31

## Hecke characteristic polynomials

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