# Properties

 Label 256.2.a.d Level $256$ Weight $2$ Character orbit 256.a Self dual yes Analytic conductor $2.044$ Analytic rank $0$ Dimension $1$ CM discriminant -8 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 256.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.04417029174$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 64) Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

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

## 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
 0
0 2.00000 0 0 0 0 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
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 256.2.a.d 1
3.b odd 2 1 2304.2.a.h 1
4.b odd 2 1 256.2.a.a 1
5.b even 2 1 6400.2.a.a 1
8.b even 2 1 256.2.a.a 1
8.d odd 2 1 CM 256.2.a.d 1
12.b even 2 1 2304.2.a.i 1
16.e even 4 2 64.2.b.a 2
16.f odd 4 2 64.2.b.a 2
20.d odd 2 1 6400.2.a.x 1
24.f even 2 1 2304.2.a.h 1
24.h odd 2 1 2304.2.a.i 1
32.g even 8 4 1024.2.e.l 4
32.h odd 8 4 1024.2.e.l 4
40.e odd 2 1 6400.2.a.a 1
40.f even 2 1 6400.2.a.x 1
48.i odd 4 2 576.2.d.a 2
48.k even 4 2 576.2.d.a 2
80.i odd 4 2 1600.2.f.a 2
80.j even 4 2 1600.2.f.a 2
80.k odd 4 2 1600.2.d.a 2
80.q even 4 2 1600.2.d.a 2
80.s even 4 2 1600.2.f.b 2
80.t odd 4 2 1600.2.f.b 2
112.j even 4 2 3136.2.b.b 2
112.l odd 4 2 3136.2.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.2.b.a 2 16.e even 4 2
64.2.b.a 2 16.f odd 4 2
256.2.a.a 1 4.b odd 2 1
256.2.a.a 1 8.b even 2 1
256.2.a.d 1 1.a even 1 1 trivial
256.2.a.d 1 8.d odd 2 1 CM
576.2.d.a 2 48.i odd 4 2
576.2.d.a 2 48.k even 4 2
1024.2.e.l 4 32.g even 8 4
1024.2.e.l 4 32.h odd 8 4
1600.2.d.a 2 80.k odd 4 2
1600.2.d.a 2 80.q even 4 2
1600.2.f.a 2 80.i odd 4 2
1600.2.f.a 2 80.j even 4 2
1600.2.f.b 2 80.s even 4 2
1600.2.f.b 2 80.t odd 4 2
2304.2.a.h 1 3.b odd 2 1
2304.2.a.h 1 24.f even 2 1
2304.2.a.i 1 12.b even 2 1
2304.2.a.i 1 24.h odd 2 1
3136.2.b.b 2 112.j even 4 2
3136.2.b.b 2 112.l odd 4 2
6400.2.a.a 1 5.b even 2 1
6400.2.a.a 1 40.e odd 2 1
6400.2.a.x 1 20.d odd 2 1
6400.2.a.x 1 40.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(256))$$:

 $$T_{3} - 2$$ $$T_{5}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-2 + T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$-6 + T$$
$13$ $$T$$
$17$ $$6 + T$$
$19$ $$-2 + T$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$-6 + T$$
$43$ $$10 + T$$
$47$ $$T$$
$53$ $$T$$
$59$ $$-6 + T$$
$61$ $$T$$
$67$ $$14 + T$$
$71$ $$T$$
$73$ $$2 + T$$
$79$ $$T$$
$83$ $$-18 + T$$
$89$ $$18 + T$$
$97$ $$-10 + T$$