# Properties

 Label 256.1.c.a Level $256$ Weight $1$ Character orbit 256.c Self dual yes Analytic conductor $0.128$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -4, -8, 8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 256.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.127760643234$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 128) Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\zeta_{8})$$ Artin image $D_4$ Artin field Galois closure of 4.2.2048.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{9} + O(q^{10})$$ $$q + q^{9} - 2q^{17} - q^{25} - 2q^{41} + q^{49} + 2q^{73} + q^{81} + 2q^{89} - 2q^{97} + O(q^{100})$$

## Expression as an eta quotient

$$f(z) = \dfrac{\eta(16z)^{4}}{\eta(8z)\eta(32z)}=q\prod_{n=1}^\infty(1 - q^{8n})^{-1}(1 - q^{16n})^{4}(1 - q^{32n})^{-1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$255$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
255.1
 0
0 0 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

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
8.b even 2 1 RM by $$\Q(\sqrt{2})$$
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 256.1.c.a 1
3.b odd 2 1 2304.1.g.b 1
4.b odd 2 1 CM 256.1.c.a 1
8.b even 2 1 RM 256.1.c.a 1
8.d odd 2 1 CM 256.1.c.a 1
12.b even 2 1 2304.1.g.b 1
16.e even 4 2 128.1.d.a 1
16.f odd 4 2 128.1.d.a 1
24.f even 2 1 2304.1.g.b 1
24.h odd 2 1 2304.1.g.b 1
32.g even 8 4 1024.1.f.b 2
32.h odd 8 4 1024.1.f.b 2
48.i odd 4 2 1152.1.b.a 1
48.k even 4 2 1152.1.b.a 1
80.i odd 4 2 3200.1.e.a 2
80.j even 4 2 3200.1.e.a 2
80.k odd 4 2 3200.1.g.a 1
80.q even 4 2 3200.1.g.a 1
80.s even 4 2 3200.1.e.a 2
80.t odd 4 2 3200.1.e.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.1.d.a 1 16.e even 4 2
128.1.d.a 1 16.f odd 4 2
256.1.c.a 1 1.a even 1 1 trivial
256.1.c.a 1 4.b odd 2 1 CM
256.1.c.a 1 8.b even 2 1 RM
256.1.c.a 1 8.d odd 2 1 CM
1024.1.f.b 2 32.g even 8 4
1024.1.f.b 2 32.h odd 8 4
1152.1.b.a 1 48.i odd 4 2
1152.1.b.a 1 48.k even 4 2
2304.1.g.b 1 3.b odd 2 1
2304.1.g.b 1 12.b even 2 1
2304.1.g.b 1 24.f even 2 1
2304.1.g.b 1 24.h odd 2 1
3200.1.e.a 2 80.i odd 4 2
3200.1.e.a 2 80.j even 4 2
3200.1.e.a 2 80.s even 4 2
3200.1.e.a 2 80.t odd 4 2
3200.1.g.a 1 80.k odd 4 2
3200.1.g.a 1 80.q even 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(256, [\chi])$$.

## Hecke characteristic polynomials

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