# Properties

 Label 6400.2.a.bm 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: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + \beta q^{7} - q^{9} +O(q^{10})$$ $$q + \beta q^{3} + \beta q^{7} - q^{9} + 2 \beta q^{11} -2 q^{13} -2 q^{17} -4 \beta q^{19} + 2 q^{21} + \beta q^{23} -4 \beta q^{27} + 8 q^{29} -6 \beta q^{31} + 4 q^{33} -10 q^{37} -2 \beta q^{39} -5 \beta q^{43} -7 \beta q^{47} -5 q^{49} -2 \beta q^{51} -2 q^{53} -8 q^{57} -4 \beta q^{59} + 10 q^{61} -\beta q^{63} + \beta q^{67} + 2 q^{69} + 10 \beta q^{71} + 6 q^{73} + 4 q^{77} + 8 \beta q^{79} -5 q^{81} -9 \beta q^{83} + 8 \beta q^{87} -10 q^{89} -2 \beta q^{91} -12 q^{93} -14 q^{97} -2 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{9} - 4 q^{13} - 4 q^{17} + 4 q^{21} + 16 q^{29} + 8 q^{33} - 20 q^{37} - 10 q^{49} - 4 q^{53} - 16 q^{57} + 20 q^{61} + 4 q^{69} + 12 q^{73} + 8 q^{77} - 10 q^{81} - 20 q^{89} - 24 q^{93} - 28 q^{97} + O(q^{100})$$

## 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
 −1.41421 1.41421
0 −1.41421 0 0 0 −1.41421 0 −1.00000 0
1.2 0 1.41421 0 0 0 1.41421 0 −1.00000 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.bm 2
4.b odd 2 1 inner 6400.2.a.bm 2
5.b even 2 1 1280.2.a.k 2
8.b even 2 1 6400.2.a.bo 2
8.d odd 2 1 6400.2.a.bo 2
16.e even 4 2 3200.2.d.t 4
16.f odd 4 2 3200.2.d.t 4
20.d odd 2 1 1280.2.a.k 2
40.e odd 2 1 1280.2.a.e 2
40.f even 2 1 1280.2.a.e 2
80.i odd 4 2 3200.2.f.k 4
80.j even 4 2 3200.2.f.h 4
80.k odd 4 2 640.2.d.c 4
80.q even 4 2 640.2.d.c 4
80.s even 4 2 3200.2.f.k 4
80.t odd 4 2 3200.2.f.h 4
240.t even 4 2 5760.2.k.q 4
240.bm odd 4 2 5760.2.k.q 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.d.c 4 80.k odd 4 2
640.2.d.c 4 80.q even 4 2
1280.2.a.e 2 40.e odd 2 1
1280.2.a.e 2 40.f even 2 1
1280.2.a.k 2 5.b even 2 1
1280.2.a.k 2 20.d odd 2 1
3200.2.d.t 4 16.e even 4 2
3200.2.d.t 4 16.f odd 4 2
3200.2.f.h 4 80.j even 4 2
3200.2.f.h 4 80.t odd 4 2
3200.2.f.k 4 80.i odd 4 2
3200.2.f.k 4 80.s even 4 2
5760.2.k.q 4 240.t even 4 2
5760.2.k.q 4 240.bm odd 4 2
6400.2.a.bm 2 1.a even 1 1 trivial
6400.2.a.bm 2 4.b odd 2 1 inner
6400.2.a.bo 2 8.b even 2 1
6400.2.a.bo 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6400))$$:

 $$T_{3}^{2} - 2$$ $$T_{7}^{2} - 2$$ $$T_{11}^{2} - 8$$ $$T_{13} + 2$$ $$T_{17} + 2$$ $$T_{29} - 8$$ $$T_{31}^{2} - 72$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-2 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$-2 + T^{2}$$
$11$ $$-8 + T^{2}$$
$13$ $$( 2 + T )^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$-32 + T^{2}$$
$23$ $$-2 + T^{2}$$
$29$ $$( -8 + T )^{2}$$
$31$ $$-72 + T^{2}$$
$37$ $$( 10 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$-50 + T^{2}$$
$47$ $$-98 + T^{2}$$
$53$ $$( 2 + T )^{2}$$
$59$ $$-32 + T^{2}$$
$61$ $$( -10 + T )^{2}$$
$67$ $$-2 + T^{2}$$
$71$ $$-200 + T^{2}$$
$73$ $$( -6 + T )^{2}$$
$79$ $$-128 + T^{2}$$
$83$ $$-162 + T^{2}$$
$89$ $$( 10 + T )^{2}$$
$97$ $$( 14 + T )^{2}$$