# Properties

 Label 1024.2.b.c Level $1024$ Weight $2$ Character orbit 1024.b Analytic conductor $8.177$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$8.17668116698$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 512) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta q^{5} + 3 q^{9} +O(q^{10})$$ $$q -\beta q^{5} + 3 q^{9} -5 \beta q^{13} -8 q^{17} + 3 q^{25} + 3 \beta q^{29} -7 \beta q^{37} + 8 q^{41} -3 \beta q^{45} -7 q^{49} -9 \beta q^{53} -11 \beta q^{61} -10 q^{65} -6 q^{73} + 9 q^{81} + 8 \beta q^{85} + 10 q^{89} -8 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{9} + O(q^{10})$$ $$2 q + 6 q^{9} - 16 q^{17} + 6 q^{25} + 16 q^{41} - 14 q^{49} - 20 q^{65} - 12 q^{73} + 18 q^{81} + 20 q^{89} - 16 q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
513.1
 1.41421i − 1.41421i
0 0 0 1.41421i 0 0 0 3.00000 0
513.2 0 0 0 1.41421i 0 0 0 3.00000 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 inner
8.d odd 2 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1024, [\chi])$$:

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

## Hecke characteristic polynomials

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