# Properties

 Label 6400.2.a.br 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: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 3200) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{3} + 2 q^{9} +O(q^{10})$$ $$q -\beta q^{3} + 2 q^{9} + \beta q^{11} -4 q^{13} + 3 q^{17} + \beta q^{19} -4 \beta q^{23} + \beta q^{27} -4 q^{29} + 4 \beta q^{31} -5 q^{33} -8 q^{37} + 4 \beta q^{39} + 5 q^{41} + 4 \beta q^{43} + 4 \beta q^{47} -7 q^{49} -3 \beta q^{51} -4 q^{53} -5 q^{57} -4 \beta q^{59} -8 q^{61} -3 \beta q^{67} + 20 q^{69} + 4 \beta q^{71} + 9 q^{73} -11 q^{81} -3 \beta q^{83} + 4 \beta q^{87} + 15 q^{89} -20 q^{93} + 2 q^{97} + 2 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{9} + O(q^{10})$$ $$2 q + 4 q^{9} - 8 q^{13} + 6 q^{17} - 8 q^{29} - 10 q^{33} - 16 q^{37} + 10 q^{41} - 14 q^{49} - 8 q^{53} - 10 q^{57} - 16 q^{61} + 40 q^{69} + 18 q^{73} - 22 q^{81} + 30 q^{89} - 40 q^{93} + 4 q^{97} + O(q^{100})$$

## 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
 1.61803 −0.618034
0 −2.23607 0 0 0 0 0 2.00000 0
1.2 0 2.23607 0 0 0 0 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
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.br 2
4.b odd 2 1 inner 6400.2.a.br 2
5.b even 2 1 6400.2.a.bs 2
8.b even 2 1 6400.2.a.bt 2
8.d odd 2 1 6400.2.a.bt 2
16.e even 4 2 3200.2.d.p yes 4
16.f odd 4 2 3200.2.d.p yes 4
20.d odd 2 1 6400.2.a.bs 2
40.e odd 2 1 6400.2.a.bq 2
40.f even 2 1 6400.2.a.bq 2
80.i odd 4 2 3200.2.f.n 4
80.j even 4 2 3200.2.f.m 4
80.k odd 4 2 3200.2.d.o 4
80.q even 4 2 3200.2.d.o 4
80.s even 4 2 3200.2.f.n 4
80.t odd 4 2 3200.2.f.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.2.d.o 4 80.k odd 4 2
3200.2.d.o 4 80.q even 4 2
3200.2.d.p yes 4 16.e even 4 2
3200.2.d.p yes 4 16.f odd 4 2
3200.2.f.m 4 80.j even 4 2
3200.2.f.m 4 80.t odd 4 2
3200.2.f.n 4 80.i odd 4 2
3200.2.f.n 4 80.s even 4 2
6400.2.a.bq 2 40.e odd 2 1
6400.2.a.bq 2 40.f even 2 1
6400.2.a.br 2 1.a even 1 1 trivial
6400.2.a.br 2 4.b odd 2 1 inner
6400.2.a.bs 2 5.b even 2 1
6400.2.a.bs 2 20.d odd 2 1
6400.2.a.bt 2 8.b even 2 1
6400.2.a.bt 2 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6400))$$:

 $$T_{3}^{2} - 5$$ $$T_{7}$$ $$T_{11}^{2} - 5$$ $$T_{13} + 4$$ $$T_{17} - 3$$ $$T_{29} + 4$$ $$T_{31}^{2} - 80$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-5 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$-5 + T^{2}$$
$13$ $$( 4 + T )^{2}$$
$17$ $$( -3 + T )^{2}$$
$19$ $$-5 + T^{2}$$
$23$ $$-80 + T^{2}$$
$29$ $$( 4 + T )^{2}$$
$31$ $$-80 + T^{2}$$
$37$ $$( 8 + T )^{2}$$
$41$ $$( -5 + T )^{2}$$
$43$ $$-80 + T^{2}$$
$47$ $$-80 + T^{2}$$
$53$ $$( 4 + T )^{2}$$
$59$ $$-80 + T^{2}$$
$61$ $$( 8 + T )^{2}$$
$67$ $$-45 + T^{2}$$
$71$ $$-80 + T^{2}$$
$73$ $$( -9 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$-45 + T^{2}$$
$89$ $$( -15 + T )^{2}$$
$97$ $$( -2 + T )^{2}$$