# Properties

 Label 512.2.b.a Level $512$ Weight $2$ Character orbit 512.b Analytic conductor $4.088$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$512 = 2^{9}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 512.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.08834058349$$ 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: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} -2 \beta q^{5} -4 q^{7} + q^{9} +O(q^{10})$$ $$q + \beta q^{3} -2 \beta q^{5} -4 q^{7} + q^{9} -\beta q^{11} -2 \beta q^{13} + 4 q^{15} -4 q^{17} -5 \beta q^{19} -4 \beta q^{21} -4 q^{23} -3 q^{25} + 4 \beta q^{27} -6 \beta q^{29} + 8 q^{31} + 2 q^{33} + 8 \beta q^{35} + 2 \beta q^{37} + 4 q^{39} -2 q^{41} -3 \beta q^{43} -2 \beta q^{45} + 9 q^{49} -4 \beta q^{51} + 2 \beta q^{53} -4 q^{55} + 10 q^{57} -3 \beta q^{59} + 6 \beta q^{61} -4 q^{63} -8 q^{65} + 3 \beta q^{67} -4 \beta q^{69} + 4 q^{71} + 4 q^{73} -3 \beta q^{75} + 4 \beta q^{77} -8 q^{79} -5 q^{81} -7 \beta q^{83} + 8 \beta q^{85} + 12 q^{87} -12 q^{89} + 8 \beta q^{91} + 8 \beta q^{93} -20 q^{95} -4 q^{97} -\beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{7} + 2 q^{9} + O(q^{10})$$ $$2 q - 8 q^{7} + 2 q^{9} + 8 q^{15} - 8 q^{17} - 8 q^{23} - 6 q^{25} + 16 q^{31} + 4 q^{33} + 8 q^{39} - 4 q^{41} + 18 q^{49} - 8 q^{55} + 20 q^{57} - 8 q^{63} - 16 q^{65} + 8 q^{71} + 8 q^{73} - 16 q^{79} - 10 q^{81} + 24 q^{87} - 24 q^{89} - 40 q^{95} - 8 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/512\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$511$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 − 1.41421i 1.41421i
0 1.41421i 0 2.82843i 0 −4.00000 0 1.00000 0
257.2 0 1.41421i 0 2.82843i 0 −4.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 512.2.b.a 2
3.b odd 2 1 4608.2.d.a 2
4.b odd 2 1 512.2.b.b 2
8.b even 2 1 inner 512.2.b.a 2
8.d odd 2 1 512.2.b.b 2
12.b even 2 1 4608.2.d.b 2
16.e even 4 2 512.2.a.d yes 2
16.f odd 4 2 512.2.a.c 2
24.f even 2 1 4608.2.d.b 2
24.h odd 2 1 4608.2.d.a 2
32.g even 8 2 1024.2.e.c 2
32.g even 8 2 1024.2.e.d 2
32.h odd 8 2 1024.2.e.b 2
32.h odd 8 2 1024.2.e.e 2
48.i odd 4 2 4608.2.a.m 2
48.k even 4 2 4608.2.a.f 2

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

## Hecke kernels

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

 $$T_{3}^{2} + 2$$ $$T_{5}^{2} + 8$$ $$T_{7} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$2 + T^{2}$$
$5$ $$8 + T^{2}$$
$7$ $$( 4 + T )^{2}$$
$11$ $$2 + T^{2}$$
$13$ $$8 + T^{2}$$
$17$ $$( 4 + T )^{2}$$
$19$ $$50 + T^{2}$$
$23$ $$( 4 + T )^{2}$$
$29$ $$72 + T^{2}$$
$31$ $$( -8 + T )^{2}$$
$37$ $$8 + T^{2}$$
$41$ $$( 2 + T )^{2}$$
$43$ $$18 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$8 + T^{2}$$
$59$ $$18 + T^{2}$$
$61$ $$72 + T^{2}$$
$67$ $$18 + T^{2}$$
$71$ $$( -4 + T )^{2}$$
$73$ $$( -4 + T )^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$98 + T^{2}$$
$89$ $$( 12 + T )^{2}$$
$97$ $$( 4 + T )^{2}$$