# Properties

 Label 6400.2.a.cr Level $6400$ Weight $2$ Character orbit 6400.a Self dual yes Analytic conductor $51.104$ Analytic rank $0$ Dimension $4$ CM discriminant -8 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: $$\Q(\zeta_{24})^+$$ Defining polynomial: $$x^{4} - 4 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 3200) Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( 2 + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( 2 + \beta_{3} ) q^{9} + ( 2 \beta_{1} + \beta_{2} ) q^{11} + ( 3 + \beta_{3} ) q^{17} + ( \beta_{1} - 2 \beta_{2} ) q^{19} + ( 4 \beta_{1} - \beta_{2} ) q^{27} + ( 11 + 2 \beta_{3} ) q^{33} + ( -3 - 2 \beta_{3} ) q^{41} + ( -3 \beta_{1} + 3 \beta_{2} ) q^{43} -7 q^{49} + ( 8 \beta_{1} - \beta_{2} ) q^{51} + ( 3 + \beta_{3} ) q^{57} + ( -5 \beta_{1} + 5 \beta_{2} ) q^{59} + ( 2 \beta_{1} + 5 \beta_{2} ) q^{67} + ( 1 - 3 \beta_{3} ) q^{73} + ( 13 + \beta_{3} ) q^{81} + ( -4 \beta_{1} - 5 \beta_{2} ) q^{83} + ( -9 - \beta_{3} ) q^{89} + 10 q^{97} + ( 15 \beta_{1} - 5 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{9} + O(q^{10})$$ $$4q + 8q^{9} + 12q^{17} + 44q^{33} - 12q^{41} - 28q^{49} + 12q^{57} + 4q^{73} + 52q^{81} - 36q^{89} + 40q^{97} + O(q^{100})$$

## 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
 0.517638 −0.517638 −1.93185 1.93185
0 −3.14626 0 0 0 0 0 6.89898 0
1.2 0 −0.317837 0 0 0 0 0 −2.89898 0
1.3 0 0.317837 0 0 0 0 0 −2.89898 0
1.4 0 3.14626 0 0 0 0 0 6.89898 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 $$\Q(\sqrt{-2})$$
4.b odd 2 1 inner
8.b even 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.cr 4
4.b odd 2 1 inner 6400.2.a.cr 4
5.b even 2 1 6400.2.a.cq 4
8.b even 2 1 inner 6400.2.a.cr 4
8.d odd 2 1 CM 6400.2.a.cr 4
16.e even 4 2 3200.2.d.q yes 4
16.f odd 4 2 3200.2.d.q yes 4
20.d odd 2 1 6400.2.a.cq 4
40.e odd 2 1 6400.2.a.cq 4
40.f even 2 1 6400.2.a.cq 4
80.i odd 4 2 3200.2.f.s 8
80.j even 4 2 3200.2.f.s 8
80.k odd 4 2 3200.2.d.n 4
80.q even 4 2 3200.2.d.n 4
80.s even 4 2 3200.2.f.s 8
80.t odd 4 2 3200.2.f.s 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.2.d.n 4 80.k odd 4 2
3200.2.d.n 4 80.q even 4 2
3200.2.d.q yes 4 16.e even 4 2
3200.2.d.q yes 4 16.f odd 4 2
3200.2.f.s 8 80.i odd 4 2
3200.2.f.s 8 80.j even 4 2
3200.2.f.s 8 80.s even 4 2
3200.2.f.s 8 80.t odd 4 2
6400.2.a.cq 4 5.b even 2 1
6400.2.a.cq 4 20.d odd 2 1
6400.2.a.cq 4 40.e odd 2 1
6400.2.a.cq 4 40.f even 2 1
6400.2.a.cr 4 1.a even 1 1 trivial
6400.2.a.cr 4 4.b odd 2 1 inner
6400.2.a.cr 4 8.b even 2 1 inner
6400.2.a.cr 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6400))$$:

 $$T_{3}^{4} - 10 T_{3}^{2} + 1$$ $$T_{7}$$ $$T_{11}^{4} - 58 T_{11}^{2} + 625$$ $$T_{13}$$ $$T_{17}^{2} - 6 T_{17} - 15$$ $$T_{29}$$ $$T_{31}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1 - 10 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$625 - 58 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$( -15 - 6 T + T^{2} )^{2}$$
$19$ $$225 - 42 T^{2} + T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$( -87 + 6 T + T^{2} )^{2}$$
$43$ $$( -72 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$( -200 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$16641 - 330 T^{2} + T^{4}$$
$71$ $$T^{4}$$
$73$ $$( -215 - 2 T + T^{2} )^{2}$$
$79$ $$T^{4}$$
$83$ $$58081 - 490 T^{2} + T^{4}$$
$89$ $$( 57 + 18 T + T^{2} )^{2}$$
$97$ $$( -10 + T )^{4}$$