# 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: $$x^{2} - 6$$ 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.

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

## 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
 −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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6400))$$:

 $$T_{3}^{2} - 2 T_{3} - 5$$ $$T_{7}$$ $$T_{11}^{2} - 6 T_{11} + 3$$ $$T_{13}$$ $$T_{17}^{2} - 6 T_{17} - 15$$ $$T_{29}$$ $$T_{31}$$

## Hecke characteristic polynomials

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