# Properties

 Label 512.2.a.a Level $512$ Weight $2$ Character orbit 512.a Self dual yes Analytic conductor $4.088$ Analytic rank $1$ Dimension $2$ CM discriminant -8 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$512 = 2^{9}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 512.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.08834058349$$ 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: yes Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -2 + \beta ) q^{3} + ( 3 - 4 \beta ) q^{9} +O(q^{10})$$ $$q + ( -2 + \beta ) q^{3} + ( 3 - 4 \beta ) q^{9} + ( -2 - 3 \beta ) q^{11} + 4 \beta q^{17} + ( -6 - \beta ) q^{19} -5 q^{25} + ( -8 + 8 \beta ) q^{27} + ( -2 + 4 \beta ) q^{33} -6 q^{41} + ( -6 - 5 \beta ) q^{43} -7 q^{49} + ( 8 - 8 \beta ) q^{51} + ( 10 - 4 \beta ) q^{57} + ( 10 + 3 \beta ) q^{59} + ( -6 + 7 \beta ) q^{67} + 12 \beta q^{73} + ( 10 - 5 \beta ) q^{75} + ( 23 - 12 \beta ) q^{81} + ( -2 + 9 \beta ) q^{83} -4 \beta q^{89} -12 \beta q^{97} + ( 18 - \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{3} + 6q^{9} + O(q^{10})$$ $$2q - 4q^{3} + 6q^{9} - 4q^{11} - 12q^{19} - 10q^{25} - 16q^{27} - 4q^{33} - 12q^{41} - 12q^{43} - 14q^{49} + 16q^{51} + 20q^{57} + 20q^{59} - 12q^{67} + 20q^{75} + 46q^{81} - 4q^{83} + 36q^{99} + O(q^{100})$$

## 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
 −1.41421 1.41421
0 −3.41421 0 0 0 0 0 8.65685 0
1.2 0 −0.585786 0 0 0 0 0 −2.65685 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
$$2$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 512.2.a.a 2
3.b odd 2 1 4608.2.a.k 2
4.b odd 2 1 512.2.a.f yes 2
8.b even 2 1 512.2.a.f yes 2
8.d odd 2 1 CM 512.2.a.a 2
12.b even 2 1 4608.2.a.i 2
16.e even 4 2 512.2.b.c 4
16.f odd 4 2 512.2.b.c 4
24.f even 2 1 4608.2.a.k 2
24.h odd 2 1 4608.2.a.i 2
32.g even 8 2 1024.2.e.g 4
32.g even 8 2 1024.2.e.o 4
32.h odd 8 2 1024.2.e.g 4
32.h odd 8 2 1024.2.e.o 4
48.i odd 4 2 4608.2.d.k 4
48.k even 4 2 4608.2.d.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.a.a 2 1.a even 1 1 trivial
512.2.a.a 2 8.d odd 2 1 CM
512.2.a.f yes 2 4.b odd 2 1
512.2.a.f yes 2 8.b even 2 1
512.2.b.c 4 16.e even 4 2
512.2.b.c 4 16.f odd 4 2
1024.2.e.g 4 32.g even 8 2
1024.2.e.g 4 32.h odd 8 2
1024.2.e.o 4 32.g even 8 2
1024.2.e.o 4 32.h odd 8 2
4608.2.a.i 2 12.b even 2 1
4608.2.a.i 2 24.h odd 2 1
4608.2.a.k 2 3.b odd 2 1
4608.2.a.k 2 24.f even 2 1
4608.2.d.k 4 48.i odd 4 2
4608.2.d.k 4 48.k even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(512))$$:

 $$T_{3}^{2} + 4 T_{3} + 2$$ $$T_{5}$$ $$T_{7}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$2 + 4 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$-14 + 4 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$-32 + T^{2}$$
$19$ $$34 + 12 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$-14 + 12 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$82 - 20 T + T^{2}$$
$61$ $$T^{2}$$
$67$ $$-62 + 12 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$-288 + T^{2}$$
$79$ $$T^{2}$$
$83$ $$-158 + 4 T + T^{2}$$
$89$ $$-32 + T^{2}$$
$97$ $$-288 + T^{2}$$