# Properties

 Label 6400.2.a.by 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 $4$

# 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: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 128) Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + 5 q^{9} +O(q^{10})$$ $$q + \beta q^{3} + 5 q^{9} + \beta q^{11} -6 q^{17} + 3 \beta q^{19} + 2 \beta q^{27} + 8 q^{33} + 6 q^{41} + 3 \beta q^{43} -7 q^{49} -6 \beta q^{51} + 24 q^{57} + 5 \beta q^{59} -3 \beta q^{67} -2 q^{73} + q^{81} + \beta q^{83} + 18 q^{89} + 10 q^{97} + 5 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 10q^{9} + O(q^{10})$$ $$2q + 10q^{9} - 12q^{17} + 16q^{33} + 12q^{41} - 14q^{49} + 48q^{57} - 4q^{73} + 2q^{81} + 36q^{89} + 20q^{97} + O(q^{100})$$

## 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 −2.82843 0 0 0 0 0 5.00000 0
1.2 0 2.82843 0 0 0 0 0 5.00000 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 CM by $$\Q(\sqrt{-2})$$
4.b odd 2 1 inner
8.b even 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.by 2
4.b odd 2 1 inner 6400.2.a.by 2
5.b even 2 1 256.2.a.e 2
8.b even 2 1 inner 6400.2.a.by 2
8.d odd 2 1 CM 6400.2.a.by 2
15.d odd 2 1 2304.2.a.t 2
16.e even 4 2 3200.2.d.c 2
16.f odd 4 2 3200.2.d.c 2
20.d odd 2 1 256.2.a.e 2
40.e odd 2 1 256.2.a.e 2
40.f even 2 1 256.2.a.e 2
60.h even 2 1 2304.2.a.t 2
80.i odd 4 2 3200.2.f.o 4
80.j even 4 2 3200.2.f.o 4
80.k odd 4 2 128.2.b.a 2
80.q even 4 2 128.2.b.a 2
80.s even 4 2 3200.2.f.o 4
80.t odd 4 2 3200.2.f.o 4
120.i odd 2 1 2304.2.a.t 2
120.m even 2 1 2304.2.a.t 2
160.y odd 8 2 1024.2.e.a 2
160.y odd 8 2 1024.2.e.f 2
160.z even 8 2 1024.2.e.a 2
160.z even 8 2 1024.2.e.f 2
240.t even 4 2 1152.2.d.c 2
240.bm odd 4 2 1152.2.d.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.b.a 2 80.k odd 4 2
128.2.b.a 2 80.q even 4 2
256.2.a.e 2 5.b even 2 1
256.2.a.e 2 20.d odd 2 1
256.2.a.e 2 40.e odd 2 1
256.2.a.e 2 40.f even 2 1
1024.2.e.a 2 160.y odd 8 2
1024.2.e.a 2 160.z even 8 2
1024.2.e.f 2 160.y odd 8 2
1024.2.e.f 2 160.z even 8 2
1152.2.d.c 2 240.t even 4 2
1152.2.d.c 2 240.bm odd 4 2
2304.2.a.t 2 15.d odd 2 1
2304.2.a.t 2 60.h even 2 1
2304.2.a.t 2 120.i odd 2 1
2304.2.a.t 2 120.m even 2 1
3200.2.d.c 2 16.e even 4 2
3200.2.d.c 2 16.f odd 4 2
3200.2.f.o 4 80.i odd 4 2
3200.2.f.o 4 80.j even 4 2
3200.2.f.o 4 80.s even 4 2
3200.2.f.o 4 80.t odd 4 2
6400.2.a.by 2 1.a even 1 1 trivial
6400.2.a.by 2 4.b odd 2 1 inner
6400.2.a.by 2 8.b even 2 1 inner
6400.2.a.by 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6400))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-8 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$-8 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$( 6 + T )^{2}$$
$19$ $$-72 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$-72 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$-200 + T^{2}$$
$61$ $$T^{2}$$
$67$ $$-72 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 2 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$-8 + T^{2}$$
$89$ $$( -18 + T )^{2}$$
$97$ $$( -10 + T )^{2}$$