# Properties

 Label 6400.2.a.cq Level $6400$ Weight $2$ Character orbit 6400.a Self dual yes Analytic conductor $51.104$ Analytic rank $1$ 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: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{24})^+$$ Defining polynomial: $$x^{4} - 4x^{2} + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + (\beta_{3} + 2) q^{9}+O(q^{10})$$ q + b1 * q^3 + (b3 + 2) * q^9 $$q + \beta_1 q^{3} + (\beta_{3} + 2) q^{9} + ( - \beta_{2} - 2 \beta_1) q^{11} + ( - \beta_{3} - 3) q^{17} + (2 \beta_{2} - \beta_1) q^{19} + ( - \beta_{2} + 4 \beta_1) q^{27} + ( - 2 \beta_{3} - 11) q^{33} + ( - 2 \beta_{3} - 3) q^{41} + (3 \beta_{2} - 3 \beta_1) q^{43} - 7 q^{49} + (\beta_{2} - 8 \beta_1) q^{51} + ( - \beta_{3} - 3) q^{57} + ( - 5 \beta_{2} + 5 \beta_1) q^{59} + (5 \beta_{2} + 2 \beta_1) q^{67} + (3 \beta_{3} - 1) q^{73} + (\beta_{3} + 13) q^{81} + ( - 5 \beta_{2} - 4 \beta_1) q^{83} + ( - \beta_{3} - 9) q^{89} - 10 q^{97} + (5 \beta_{2} - 15 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^3 + (b3 + 2) * q^9 + (-b2 - 2*b1) * q^11 + (-b3 - 3) * q^17 + (2*b2 - b1) * q^19 + (-b2 + 4*b1) * q^27 + (-2*b3 - 11) * q^33 + (-2*b3 - 3) * q^41 + (3*b2 - 3*b1) * q^43 - 7 * q^49 + (b2 - 8*b1) * q^51 + (-b3 - 3) * q^57 + (-5*b2 + 5*b1) * q^59 + (5*b2 + 2*b1) * q^67 + (3*b3 - 1) * q^73 + (b3 + 13) * q^81 + (-5*b2 - 4*b1) * q^83 + (-b3 - 9) * q^89 - 10 * q^97 + (5*b2 - 15*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{9}+O(q^{10})$$ 4 * q + 8 * q^9 $$4 q + 8 q^{9} - 12 q^{17} - 44 q^{33} - 12 q^{41} - 28 q^{49} - 12 q^{57} - 4 q^{73} + 52 q^{81} - 36 q^{89} - 40 q^{97}+O(q^{100})$$ 4 * q + 8 * q^9 - 12 * q^17 - 44 * q^33 - 12 * q^41 - 28 * q^49 - 12 * q^57 - 4 * q^73 + 52 * q^81 - 36 * q^89 - 40 * q^97

Basis of coefficient ring in terms of $$\nu = \zeta_{24} + \zeta_{24}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu^{3} + \nu^{2} - 3\nu - 2$$ v^3 + v^2 - 3*v - 2 $$\beta_{2}$$ $$=$$ $$-\nu^{3} + \nu^{2} + 3\nu - 2$$ -v^3 + v^2 + 3*v - 2 $$\beta_{3}$$ $$=$$ $$-2\nu^{3} + 10\nu$$ -2*v^3 + 10*v
 $$\nu$$ $$=$$ $$( \beta_{3} - \beta_{2} + \beta_1 ) / 4$$ (b3 - b2 + b1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{2} + \beta _1 + 4 ) / 2$$ (b2 + b1 + 4) / 2 $$\nu^{3}$$ $$=$$ $$( 3\beta_{3} - 5\beta_{2} + 5\beta_1 ) / 4$$ (3*b3 - 5*b2 + 5*b1) / 4

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

 $$T_{3}^{4} - 10T_{3}^{2} + 1$$ T3^4 - 10*T3^2 + 1 $$T_{7}$$ T7 $$T_{11}^{4} - 58T_{11}^{2} + 625$$ T11^4 - 58*T11^2 + 625 $$T_{13}$$ T13 $$T_{17}^{2} + 6T_{17} - 15$$ T17^2 + 6*T17 - 15 $$T_{29}$$ T29 $$T_{31}$$ T31

## Hecke characteristic polynomials

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