# Properties

 Label 1024.1.f.b Level $1024$ Weight $1$ Character orbit 1024.f Analytic conductor $0.511$ Analytic rank $0$ Dimension $2$ Projective image $D_{2}$ CM/RM discs -4, -8, 8 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1024 = 2^{10}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1024.f (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.511042572936$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ 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 $OD_{16}$ Artin field Galois closure of 8.4.2147483648.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -i q^{9} +O(q^{10})$$ $$q -i q^{9} + 2 q^{17} -i q^{25} + 2 i q^{41} - q^{49} -2 i q^{73} - q^{81} + 2 i q^{89} -2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 4q^{17} - 2q^{49} - 2q^{81} - 4q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$1023$$ $$\chi(n)$$ $$i$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
255.1
 1.00000i − 1.00000i
0 0 0 0 0 0 0 1.00000i 0
767.1 0 0 0 0 0 0 0 1.00000i 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 $$\Q(\sqrt{-1})$$
8.b even 2 1 RM by $$\Q(\sqrt{2})$$
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
16.e even 4 2 inner
16.f odd 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1024.1.f.b 2
4.b odd 2 1 CM 1024.1.f.b 2
8.b even 2 1 RM 1024.1.f.b 2
8.d odd 2 1 CM 1024.1.f.b 2
16.e even 4 2 inner 1024.1.f.b 2
16.f odd 4 2 inner 1024.1.f.b 2
32.g even 8 2 128.1.d.a 1
32.g even 8 2 256.1.c.a 1
32.h odd 8 2 128.1.d.a 1
32.h odd 8 2 256.1.c.a 1
96.o even 8 2 1152.1.b.a 1
96.o even 8 2 2304.1.g.b 1
96.p odd 8 2 1152.1.b.a 1
96.p odd 8 2 2304.1.g.b 1
160.u even 8 2 3200.1.e.a 2
160.v odd 8 2 3200.1.e.a 2
160.y odd 8 2 3200.1.g.a 1
160.z even 8 2 3200.1.g.a 1
160.ba even 8 2 3200.1.e.a 2
160.bb odd 8 2 3200.1.e.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.1.d.a 1 32.g even 8 2
128.1.d.a 1 32.h odd 8 2
256.1.c.a 1 32.g even 8 2
256.1.c.a 1 32.h odd 8 2
1024.1.f.b 2 1.a even 1 1 trivial
1024.1.f.b 2 4.b odd 2 1 CM
1024.1.f.b 2 8.b even 2 1 RM
1024.1.f.b 2 8.d odd 2 1 CM
1024.1.f.b 2 16.e even 4 2 inner
1024.1.f.b 2 16.f odd 4 2 inner
1152.1.b.a 1 96.o even 8 2
1152.1.b.a 1 96.p odd 8 2
2304.1.g.b 1 96.o even 8 2
2304.1.g.b 1 96.p odd 8 2
3200.1.e.a 2 160.u even 8 2
3200.1.e.a 2 160.v odd 8 2
3200.1.e.a 2 160.ba even 8 2
3200.1.e.a 2 160.bb odd 8 2
3200.1.g.a 1 160.y odd 8 2
3200.1.g.a 1 160.z even 8 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{1}^{\mathrm{new}}(1024, [\chi])$$.

## Hecke characteristic polynomials

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