# Properties

 Label 6400.2.a.cf Level $6400$ Weight $2$ Character orbit 6400.a Self dual yes Analytic conductor $51.104$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1600) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} -\beta q^{7} -2 q^{9} +O(q^{10})$$ $$q + q^{3} -\beta q^{7} -2 q^{9} -3 q^{11} + \beta q^{13} -3 q^{17} + q^{19} -\beta q^{21} -5 q^{27} -3 \beta q^{29} + 2 \beta q^{31} -3 q^{33} -3 \beta q^{37} + \beta q^{39} + 9 q^{41} + 4 q^{43} + 3 \beta q^{47} + 5 q^{49} -3 q^{51} + q^{57} + 12 q^{59} + \beta q^{61} + 2 \beta q^{63} + 11 q^{67} -3 \beta q^{71} + 7 q^{73} + 3 \beta q^{77} -3 \beta q^{79} + q^{81} + 15 q^{83} -3 \beta q^{87} + 3 q^{89} -12 q^{91} + 2 \beta q^{93} -14 q^{97} + 6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - 4q^{9} + O(q^{10})$$ $$2q + 2q^{3} - 4q^{9} - 6q^{11} - 6q^{17} + 2q^{19} - 10q^{27} - 6q^{33} + 18q^{41} + 8q^{43} + 10q^{49} - 6q^{51} + 2q^{57} + 24q^{59} + 22q^{67} + 14q^{73} + 2q^{81} + 30q^{83} + 6q^{89} - 24q^{91} - 28q^{97} + 12q^{99} + O(q^{100})$$

## 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
 1.73205 −1.73205
0 1.00000 0 0 0 −3.46410 0 −2.00000 0
1.2 0 1.00000 0 0 0 3.46410 0 −2.00000 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
8.d 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.cf 2
4.b odd 2 1 6400.2.a.bb 2
5.b even 2 1 6400.2.a.ba 2
8.b even 2 1 6400.2.a.bb 2
8.d odd 2 1 inner 6400.2.a.cf 2
16.e even 4 2 1600.2.d.e 4
16.f odd 4 2 1600.2.d.e 4
20.d odd 2 1 6400.2.a.cg 2
40.e odd 2 1 6400.2.a.ba 2
40.f even 2 1 6400.2.a.cg 2
80.i odd 4 2 1600.2.f.c 4
80.j even 4 2 1600.2.f.c 4
80.k odd 4 2 1600.2.d.f yes 4
80.q even 4 2 1600.2.d.f yes 4
80.s even 4 2 1600.2.f.g 4
80.t odd 4 2 1600.2.f.g 4

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

 $$T_{3} - 1$$ $$T_{7}^{2} - 12$$ $$T_{11} + 3$$ $$T_{13}^{2} - 12$$ $$T_{17} + 3$$ $$T_{29}^{2} - 108$$ $$T_{31}^{2} - 48$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$-12 + T^{2}$$
$11$ $$( 3 + T )^{2}$$
$13$ $$-12 + T^{2}$$
$17$ $$( 3 + T )^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$-108 + T^{2}$$
$31$ $$-48 + T^{2}$$
$37$ $$-108 + T^{2}$$
$41$ $$( -9 + T )^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$-108 + T^{2}$$
$53$ $$T^{2}$$
$59$ $$( -12 + T )^{2}$$
$61$ $$-12 + T^{2}$$
$67$ $$( -11 + T )^{2}$$
$71$ $$-108 + T^{2}$$
$73$ $$( -7 + T )^{2}$$
$79$ $$-108 + T^{2}$$
$83$ $$( -15 + T )^{2}$$
$89$ $$( -3 + T )^{2}$$
$97$ $$( 14 + T )^{2}$$