# Properties

 Label 6400.2.a.bu Level $6400$ Weight $2$ Character orbit 6400.a Self dual yes Analytic conductor $51.104$ Analytic rank $1$ Dimension $2$ CM no 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: $$1$$ 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 640) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} -\beta q^{7} + 3 q^{9} +O(q^{10})$$ $$q + \beta q^{3} -\beta q^{7} + 3 q^{9} + 2 \beta q^{11} -4 q^{17} -2 \beta q^{19} -6 q^{21} -\beta q^{23} -8 q^{29} -4 \beta q^{31} + 12 q^{33} + 4 q^{37} -8 q^{41} + 3 \beta q^{43} -5 \beta q^{47} - q^{49} -4 \beta q^{51} + 8 q^{53} -12 q^{57} + 2 \beta q^{59} -6 q^{61} -3 \beta q^{63} + \beta q^{67} -6 q^{69} + 4 \beta q^{71} -4 q^{73} -12 q^{77} -4 \beta q^{79} -9 q^{81} -\beta q^{83} -8 \beta q^{87} + 2 q^{89} -24 q^{93} + 4 q^{97} + 6 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{9} + O(q^{10})$$ $$2q + 6q^{9} - 8q^{17} - 12q^{21} - 16q^{29} + 24q^{33} + 8q^{37} - 16q^{41} - 2q^{49} + 16q^{53} - 24q^{57} - 12q^{61} - 12q^{69} - 8q^{73} - 24q^{77} - 18q^{81} + 4q^{89} - 48q^{93} + 8q^{97} + O(q^{100})$$

## 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
 −2.44949 2.44949
0 −2.44949 0 0 0 2.44949 0 3.00000 0
1.2 0 2.44949 0 0 0 −2.44949 0 3.00000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6400.2.a.bu 2
4.b odd 2 1 inner 6400.2.a.bu 2
5.b even 2 1 6400.2.a.bw 2
5.c odd 4 2 1280.2.c.e 4
8.b even 2 1 6400.2.a.bv 2
8.d odd 2 1 6400.2.a.bv 2
16.e even 4 2 3200.2.d.k 4
16.f odd 4 2 3200.2.d.k 4
20.d odd 2 1 6400.2.a.bw 2
20.e even 4 2 1280.2.c.e 4
40.e odd 2 1 6400.2.a.bx 2
40.f even 2 1 6400.2.a.bx 2
40.i odd 4 2 1280.2.c.m 4
40.k even 4 2 1280.2.c.m 4
80.i odd 4 2 640.2.f.c 4
80.j even 4 2 640.2.f.g yes 4
80.k odd 4 2 3200.2.d.l 4
80.q even 4 2 3200.2.d.l 4
80.s even 4 2 640.2.f.c 4
80.t odd 4 2 640.2.f.g yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.f.c 4 80.i odd 4 2
640.2.f.c 4 80.s even 4 2
640.2.f.g yes 4 80.j even 4 2
640.2.f.g yes 4 80.t odd 4 2
1280.2.c.e 4 5.c odd 4 2
1280.2.c.e 4 20.e even 4 2
1280.2.c.m 4 40.i odd 4 2
1280.2.c.m 4 40.k even 4 2
3200.2.d.k 4 16.e even 4 2
3200.2.d.k 4 16.f odd 4 2
3200.2.d.l 4 80.k odd 4 2
3200.2.d.l 4 80.q even 4 2
6400.2.a.bu 2 1.a even 1 1 trivial
6400.2.a.bu 2 4.b odd 2 1 inner
6400.2.a.bv 2 8.b even 2 1
6400.2.a.bv 2 8.d odd 2 1
6400.2.a.bw 2 5.b even 2 1
6400.2.a.bw 2 20.d odd 2 1
6400.2.a.bx 2 40.e odd 2 1
6400.2.a.bx 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6400))$$:

 $$T_{3}^{2} - 6$$ $$T_{7}^{2} - 6$$ $$T_{11}^{2} - 24$$ $$T_{13}$$ $$T_{17} + 4$$ $$T_{29} + 8$$ $$T_{31}^{2} - 96$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-6 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$-6 + T^{2}$$
$11$ $$-24 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$( 4 + T )^{2}$$
$19$ $$-24 + T^{2}$$
$23$ $$-6 + T^{2}$$
$29$ $$( 8 + T )^{2}$$
$31$ $$-96 + T^{2}$$
$37$ $$( -4 + T )^{2}$$
$41$ $$( 8 + T )^{2}$$
$43$ $$-54 + T^{2}$$
$47$ $$-150 + T^{2}$$
$53$ $$( -8 + T )^{2}$$
$59$ $$-24 + T^{2}$$
$61$ $$( 6 + T )^{2}$$
$67$ $$-6 + T^{2}$$
$71$ $$-96 + T^{2}$$
$73$ $$( 4 + T )^{2}$$
$79$ $$-96 + T^{2}$$
$83$ $$-6 + T^{2}$$
$89$ $$( -2 + T )^{2}$$
$97$ $$( -4 + T )^{2}$$