# Properties

 Label 512.2.a.b Level $512$ Weight $2$ Character orbit 512.a Self dual yes Analytic conductor $4.088$ Analytic rank $1$ Dimension $2$ CM no 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: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} -2 q^{5} -2 \beta q^{7} - q^{9} +O(q^{10})$$ $$q + \beta q^{3} -2 q^{5} -2 \beta q^{7} - q^{9} -3 \beta q^{11} -6 q^{13} -2 \beta q^{15} + 3 \beta q^{19} -4 q^{21} + 6 \beta q^{23} - q^{25} -4 \beta q^{27} -2 q^{29} + 4 \beta q^{31} -6 q^{33} + 4 \beta q^{35} -6 q^{37} -6 \beta q^{39} + 6 q^{41} + 3 \beta q^{43} + 2 q^{45} + q^{49} + 2 q^{53} + 6 \beta q^{55} + 6 q^{57} -\beta q^{59} -6 q^{61} + 2 \beta q^{63} + 12 q^{65} -9 \beta q^{67} + 12 q^{69} -6 \beta q^{71} -12 q^{73} -\beta q^{75} + 12 q^{77} -4 \beta q^{79} -5 q^{81} -3 \beta q^{83} -2 \beta q^{87} -12 q^{89} + 12 \beta q^{91} + 8 q^{93} -6 \beta q^{95} -8 q^{97} + 3 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{5} - 2q^{9} + O(q^{10})$$ $$2q - 4q^{5} - 2q^{9} - 12q^{13} - 8q^{21} - 2q^{25} - 4q^{29} - 12q^{33} - 12q^{37} + 12q^{41} + 4q^{45} + 2q^{49} + 4q^{53} + 12q^{57} - 12q^{61} + 24q^{65} + 24q^{69} - 24q^{73} + 24q^{77} - 10q^{81} - 24q^{89} + 16q^{93} - 16q^{97} + O(q^{100})$$

## 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 −2.00000 0 2.82843 0 −1.00000 0
1.2 0 1.41421 0 −2.00000 0 −2.82843 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
$$2$$ $$1$$

## 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 512.2.a.b 2
3.b odd 2 1 4608.2.a.p 2
4.b odd 2 1 inner 512.2.a.b 2
8.b even 2 1 512.2.a.e yes 2
8.d odd 2 1 512.2.a.e yes 2
12.b even 2 1 4608.2.a.p 2
16.e even 4 2 512.2.b.e 4
16.f odd 4 2 512.2.b.e 4
24.f even 2 1 4608.2.a.c 2
24.h odd 2 1 4608.2.a.c 2
32.g even 8 2 1024.2.e.h 4
32.g even 8 2 1024.2.e.n 4
32.h odd 8 2 1024.2.e.h 4
32.h odd 8 2 1024.2.e.n 4
48.i odd 4 2 4608.2.d.j 4
48.k even 4 2 4608.2.d.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.a.b 2 1.a even 1 1 trivial
512.2.a.b 2 4.b odd 2 1 inner
512.2.a.e yes 2 8.b even 2 1
512.2.a.e yes 2 8.d odd 2 1
512.2.b.e 4 16.e even 4 2
512.2.b.e 4 16.f odd 4 2
1024.2.e.h 4 32.g even 8 2
1024.2.e.h 4 32.h odd 8 2
1024.2.e.n 4 32.g even 8 2
1024.2.e.n 4 32.h odd 8 2
4608.2.a.c 2 24.f even 2 1
4608.2.a.c 2 24.h odd 2 1
4608.2.a.p 2 3.b odd 2 1
4608.2.a.p 2 12.b even 2 1
4608.2.d.j 4 48.i odd 4 2
4608.2.d.j 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(512))$$:

 $$T_{3}^{2} - 2$$ $$T_{5} + 2$$ $$T_{7}^{2} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-2 + T^{2}$$
$5$ $$( 2 + T )^{2}$$
$7$ $$-8 + T^{2}$$
$11$ $$-18 + T^{2}$$
$13$ $$( 6 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$-18 + T^{2}$$
$23$ $$-72 + T^{2}$$
$29$ $$( 2 + T )^{2}$$
$31$ $$-32 + T^{2}$$
$37$ $$( 6 + T )^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$-18 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$( -2 + T )^{2}$$
$59$ $$-2 + T^{2}$$
$61$ $$( 6 + T )^{2}$$
$67$ $$-162 + T^{2}$$
$71$ $$-72 + T^{2}$$
$73$ $$( 12 + T )^{2}$$
$79$ $$-32 + T^{2}$$
$83$ $$-18 + T^{2}$$
$89$ $$( 12 + T )^{2}$$
$97$ $$( 8 + T )^{2}$$