# Properties

 Label 512.2.e.d Level $512$ Weight $2$ Character orbit 512.e Analytic conductor $4.088$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$512 = 2^{9}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 512.e (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.08834058349$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - i ) q^{5} + 3 i q^{9} +O(q^{10})$$ $$q + ( -1 - i ) q^{5} + 3 i q^{9} + ( 5 - 5 i ) q^{13} + 8 q^{17} -3 i q^{25} + ( 3 - 3 i ) q^{29} + ( -7 - 7 i ) q^{37} + 8 i q^{41} + ( 3 - 3 i ) q^{45} + 7 q^{49} + ( 9 + 9 i ) q^{53} + ( -11 + 11 i ) q^{61} -10 q^{65} -6 i q^{73} -9 q^{81} + ( -8 - 8 i ) q^{85} -10 i q^{89} -8 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} + O(q^{10})$$ $$2 q - 2 q^{5} + 10 q^{13} + 16 q^{17} + 6 q^{29} - 14 q^{37} + 6 q^{45} + 14 q^{49} + 18 q^{53} - 22 q^{61} - 20 q^{65} - 18 q^{81} - 16 q^{85} - 16 q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
129.1
 − 1.00000i 1.00000i
0 0 0 −1.00000 + 1.00000i 0 0 0 3.00000i 0
385.1 0 0 0 −1.00000 1.00000i 0 0 0 3.00000i 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
16.e even 4 1 inner
16.f odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 512.2.e.d 2
3.b odd 2 1 4608.2.k.r 2
4.b odd 2 1 CM 512.2.e.d 2
8.b even 2 1 512.2.e.e yes 2
8.d odd 2 1 512.2.e.e yes 2
12.b even 2 1 4608.2.k.r 2
16.e even 4 1 inner 512.2.e.d 2
16.e even 4 1 512.2.e.e yes 2
16.f odd 4 1 inner 512.2.e.d 2
16.f odd 4 1 512.2.e.e yes 2
24.f even 2 1 4608.2.k.g 2
24.h odd 2 1 4608.2.k.g 2
32.g even 8 2 1024.2.a.c 2
32.g even 8 2 1024.2.b.c 2
32.h odd 8 2 1024.2.a.c 2
32.h odd 8 2 1024.2.b.c 2
48.i odd 4 1 4608.2.k.g 2
48.i odd 4 1 4608.2.k.r 2
48.k even 4 1 4608.2.k.g 2
48.k even 4 1 4608.2.k.r 2
96.o even 8 2 9216.2.a.p 2
96.p odd 8 2 9216.2.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.e.d 2 1.a even 1 1 trivial
512.2.e.d 2 4.b odd 2 1 CM
512.2.e.d 2 16.e even 4 1 inner
512.2.e.d 2 16.f odd 4 1 inner
512.2.e.e yes 2 8.b even 2 1
512.2.e.e yes 2 8.d odd 2 1
512.2.e.e yes 2 16.e even 4 1
512.2.e.e yes 2 16.f odd 4 1
1024.2.a.c 2 32.g even 8 2
1024.2.a.c 2 32.h odd 8 2
1024.2.b.c 2 32.g even 8 2
1024.2.b.c 2 32.h odd 8 2
4608.2.k.g 2 24.f even 2 1
4608.2.k.g 2 24.h odd 2 1
4608.2.k.g 2 48.i odd 4 1
4608.2.k.g 2 48.k even 4 1
4608.2.k.r 2 3.b odd 2 1
4608.2.k.r 2 12.b even 2 1
4608.2.k.r 2 48.i odd 4 1
4608.2.k.r 2 48.k even 4 1
9216.2.a.p 2 96.o even 8 2
9216.2.a.p 2 96.p odd 8 2

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$2 + 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$50 - 10 T + T^{2}$$
$17$ $$( -8 + T )^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$18 - 6 T + T^{2}$$
$31$ $$T^{2}$$
$37$ $$98 + 14 T + T^{2}$$
$41$ $$64 + T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$162 - 18 T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$242 + 22 T + T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$36 + T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$100 + T^{2}$$
$97$ $$( 8 + T )^{2}$$