# Properties

 Label 512.2.a.g Level $512$ Weight $2$ Character orbit 512.a Self dual yes Analytic conductor $4.088$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{24})^+$$ Defining polynomial: $$x^{4} - 4 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} -\beta_{2} q^{5} -\beta_{3} q^{7} + 3 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} -\beta_{2} q^{5} -\beta_{3} q^{7} + 3 q^{9} + \beta_{1} q^{11} + \beta_{2} q^{13} + 3 \beta_{3} q^{15} + 4 q^{17} -\beta_{1} q^{19} + 2 \beta_{2} q^{21} -\beta_{3} q^{23} + 7 q^{25} -\beta_{2} q^{29} -2 \beta_{3} q^{31} + 6 q^{33} -4 \beta_{1} q^{35} + \beta_{2} q^{37} -3 \beta_{3} q^{39} -2 q^{41} -5 \beta_{1} q^{43} -3 \beta_{2} q^{45} + 4 \beta_{3} q^{47} + q^{49} + 4 \beta_{1} q^{51} -3 \beta_{2} q^{53} + 3 \beta_{3} q^{55} -6 q^{57} -\beta_{1} q^{59} + \beta_{2} q^{61} -3 \beta_{3} q^{63} -12 q^{65} + 3 \beta_{1} q^{67} + 2 \beta_{2} q^{69} + \beta_{3} q^{71} + 8 q^{73} + 7 \beta_{1} q^{75} + 2 \beta_{2} q^{77} -2 \beta_{3} q^{79} -9 q^{81} -3 \beta_{1} q^{83} -4 \beta_{2} q^{85} + 3 \beta_{3} q^{87} -8 q^{89} + 4 \beta_{1} q^{91} + 4 \beta_{2} q^{93} -3 \beta_{3} q^{95} + 12 q^{97} + 3 \beta_{1} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{9} + O(q^{10})$$ $$4q + 12q^{9} + 16q^{17} + 28q^{25} + 24q^{33} - 8q^{41} + 4q^{49} - 24q^{57} - 48q^{65} + 32q^{73} - 36q^{81} - 32q^{89} + 48q^{97} + O(q^{100})$$

## 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
 1.93185 0.517638 −1.93185 −0.517638
0 −2.44949 0 −3.46410 0 −2.82843 0 3.00000 0
1.2 0 −2.44949 0 3.46410 0 2.82843 0 3.00000 0
1.3 0 2.44949 0 −3.46410 0 2.82843 0 3.00000 0
1.4 0 2.44949 0 3.46410 0 −2.82843 0 3.00000 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
8.b even 2 1 inner
8.d 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.g 4
3.b odd 2 1 4608.2.a.w 4
4.b odd 2 1 inner 512.2.a.g 4
8.b even 2 1 inner 512.2.a.g 4
8.d odd 2 1 inner 512.2.a.g 4
12.b even 2 1 4608.2.a.w 4
16.e even 4 2 512.2.b.d 4
16.f odd 4 2 512.2.b.d 4
24.f even 2 1 4608.2.a.w 4
24.h odd 2 1 4608.2.a.w 4
32.g even 8 4 1024.2.e.p 8
32.h odd 8 4 1024.2.e.p 8
48.i odd 4 2 4608.2.d.d 4
48.k even 4 2 4608.2.d.d 4

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

 $$T_{3}^{2} - 6$$ $$T_{5}^{2} - 12$$ $$T_{7}^{2} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -6 + T^{2} )^{2}$$
$5$ $$( -12 + T^{2} )^{2}$$
$7$ $$( -8 + T^{2} )^{2}$$
$11$ $$( -6 + T^{2} )^{2}$$
$13$ $$( -12 + T^{2} )^{2}$$
$17$ $$( -4 + T )^{4}$$
$19$ $$( -6 + T^{2} )^{2}$$
$23$ $$( -8 + T^{2} )^{2}$$
$29$ $$( -12 + T^{2} )^{2}$$
$31$ $$( -32 + T^{2} )^{2}$$
$37$ $$( -12 + T^{2} )^{2}$$
$41$ $$( 2 + T )^{4}$$
$43$ $$( -150 + T^{2} )^{2}$$
$47$ $$( -128 + T^{2} )^{2}$$
$53$ $$( -108 + T^{2} )^{2}$$
$59$ $$( -6 + T^{2} )^{2}$$
$61$ $$( -12 + T^{2} )^{2}$$
$67$ $$( -54 + T^{2} )^{2}$$
$71$ $$( -8 + T^{2} )^{2}$$
$73$ $$( -8 + T )^{4}$$
$79$ $$( -32 + T^{2} )^{2}$$
$83$ $$( -54 + T^{2} )^{2}$$
$89$ $$( 8 + T )^{4}$$
$97$ $$( -12 + T )^{4}$$