# Properties

 Label 6400.2.a.ci Level $6400$ Weight $2$ Character orbit 6400.a Self dual yes Analytic conductor $51.104$ Analytic rank $0$ Dimension $2$ CM discriminant -8 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: $$\Q(\sqrt{6})$$ Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1600) Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{3} + (2 \beta + 4) q^{9}+O(q^{10})$$ q + (b + 1) * q^3 + (2*b + 4) * q^9 $$q + (\beta + 1) q^{3} + (2 \beta + 4) q^{9} + (\beta + 3) q^{11} + ( - 2 \beta + 3) q^{17} + ( - 3 \beta + 1) q^{19} + (3 \beta + 13) q^{27} + (4 \beta + 9) q^{33} + (4 \beta - 3) q^{41} + 10 q^{43} - 7 q^{49} + (\beta - 9) q^{51} + ( - 2 \beta - 17) q^{57} - 6 q^{59} + (3 \beta - 7) q^{67} + (6 \beta + 1) q^{73} + (10 \beta + 19) q^{81} + ( - \beta + 9) q^{83} + ( - 2 \beta + 9) q^{89} + 10 q^{97} + (10 \beta + 24) q^{99}+O(q^{100})$$ q + (b + 1) * q^3 + (2*b + 4) * q^9 + (b + 3) * q^11 + (-2*b + 3) * q^17 + (-3*b + 1) * q^19 + (3*b + 13) * q^27 + (4*b + 9) * q^33 + (4*b - 3) * q^41 + 10 * q^43 - 7 * q^49 + (b - 9) * q^51 + (-2*b - 17) * q^57 - 6 * q^59 + (3*b - 7) * q^67 + (6*b + 1) * q^73 + (10*b + 19) * q^81 + (-b + 9) * q^83 + (-2*b + 9) * q^89 + 10 * q^97 + (10*b + 24) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 8 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 8 * q^9 $$2 q + 2 q^{3} + 8 q^{9} + 6 q^{11} + 6 q^{17} + 2 q^{19} + 26 q^{27} + 18 q^{33} - 6 q^{41} + 20 q^{43} - 14 q^{49} - 18 q^{51} - 34 q^{57} - 12 q^{59} - 14 q^{67} + 2 q^{73} + 38 q^{81} + 18 q^{83} + 18 q^{89} + 20 q^{97} + 48 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 8 * q^9 + 6 * q^11 + 6 * q^17 + 2 * q^19 + 26 * q^27 + 18 * q^33 - 6 * q^41 + 20 * q^43 - 14 * q^49 - 18 * q^51 - 34 * q^57 - 12 * q^59 - 14 * q^67 + 2 * q^73 + 38 * q^81 + 18 * q^83 + 18 * q^89 + 20 * q^97 + 48 * q^99

## 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
 −2.44949 2.44949
0 −1.44949 0 0 0 0 0 −0.898979 0
1.2 0 3.44949 0 0 0 0 0 8.89898 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 $$\Q(\sqrt{-2})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6400.2.a.ci 2
4.b odd 2 1 6400.2.a.bc 2
5.b even 2 1 6400.2.a.bd 2
8.b even 2 1 6400.2.a.bc 2
8.d odd 2 1 CM 6400.2.a.ci 2
16.e even 4 2 1600.2.d.d yes 4
16.f odd 4 2 1600.2.d.d yes 4
20.d odd 2 1 6400.2.a.ch 2
40.e odd 2 1 6400.2.a.bd 2
40.f even 2 1 6400.2.a.ch 2
80.i odd 4 2 1600.2.f.f 4
80.j even 4 2 1600.2.f.f 4
80.k odd 4 2 1600.2.d.c 4
80.q even 4 2 1600.2.d.c 4
80.s even 4 2 1600.2.f.j 4
80.t odd 4 2 1600.2.f.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1600.2.d.c 4 80.k odd 4 2
1600.2.d.c 4 80.q even 4 2
1600.2.d.d yes 4 16.e even 4 2
1600.2.d.d yes 4 16.f odd 4 2
1600.2.f.f 4 80.i odd 4 2
1600.2.f.f 4 80.j even 4 2
1600.2.f.j 4 80.s even 4 2
1600.2.f.j 4 80.t odd 4 2
6400.2.a.bc 2 4.b odd 2 1
6400.2.a.bc 2 8.b even 2 1
6400.2.a.bd 2 5.b even 2 1
6400.2.a.bd 2 40.e odd 2 1
6400.2.a.ch 2 20.d odd 2 1
6400.2.a.ch 2 40.f even 2 1
6400.2.a.ci 2 1.a even 1 1 trivial
6400.2.a.ci 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6400))$$:

 $$T_{3}^{2} - 2T_{3} - 5$$ T3^2 - 2*T3 - 5 $$T_{7}$$ T7 $$T_{11}^{2} - 6T_{11} + 3$$ T11^2 - 6*T11 + 3 $$T_{13}$$ T13 $$T_{17}^{2} - 6T_{17} - 15$$ T17^2 - 6*T17 - 15 $$T_{29}$$ T29 $$T_{31}$$ T31

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T - 5$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 6T + 3$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 6T - 15$$
$19$ $$T^{2} - 2T - 53$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2} + 6T - 87$$
$43$ $$(T - 10)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$(T + 6)^{2}$$
$61$ $$T^{2}$$
$67$ $$T^{2} + 14T - 5$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 2T - 215$$
$79$ $$T^{2}$$
$83$ $$T^{2} - 18T + 75$$
$89$ $$T^{2} - 18T + 57$$
$97$ $$(T - 10)^{2}$$