# Properties

 Label 1024.2.e.f Level 1024 Weight 2 Character orbit 1024.e Analytic conductor 8.177 Analytic rank 0 Dimension 2 CM discriminant -8 Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$8.17668116698$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ 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) 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.

 $$f(q)$$ $$=$$ $$q + ( 2 - 2 i ) q^{3} -5 i q^{9} +O(q^{10})$$ $$q + ( 2 - 2 i ) q^{3} -5 i q^{9} + ( -2 - 2 i ) q^{11} -6 q^{17} + ( 6 - 6 i ) q^{19} -5 i q^{25} + ( -4 - 4 i ) q^{27} -8 q^{33} -6 i q^{41} + ( 6 + 6 i ) q^{43} + 7 q^{49} + ( -12 + 12 i ) q^{51} -24 i q^{57} + ( 10 + 10 i ) q^{59} + ( -6 + 6 i ) q^{67} -2 i q^{73} + ( -10 - 10 i ) q^{75} - q^{81} + ( -2 + 2 i ) q^{83} + 18 i q^{89} -10 q^{97} + ( -10 + 10 i ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{3} + O(q^{10})$$ $$2q + 4q^{3} - 4q^{11} - 12q^{17} + 12q^{19} - 8q^{27} - 16q^{33} + 12q^{43} + 14q^{49} - 24q^{51} + 20q^{59} - 12q^{67} - 20q^{75} - 2q^{81} - 4q^{83} - 20q^{97} - 20q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 − 1.00000i 1.00000i
0 2.00000 + 2.00000i 0 0 0 0 0 5.00000i 0
769.1 0 2.00000 2.00000i 0 0 0 0 0 5.00000i 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
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 1024.2.e.f 2
4.b odd 2 1 1024.2.e.a 2
8.b even 2 1 1024.2.e.a 2
8.d odd 2 1 CM 1024.2.e.f 2
16.e even 4 1 1024.2.e.a 2
16.e even 4 1 inner 1024.2.e.f 2
16.f odd 4 1 1024.2.e.a 2
16.f odd 4 1 inner 1024.2.e.f 2
32.g even 8 2 128.2.b.a 2
32.g even 8 2 256.2.a.e 2
32.h odd 8 2 128.2.b.a 2
32.h odd 8 2 256.2.a.e 2
96.o even 8 2 1152.2.d.c 2
96.o even 8 2 2304.2.a.t 2
96.p odd 8 2 1152.2.d.c 2
96.p odd 8 2 2304.2.a.t 2
160.u even 8 2 3200.2.f.o 4
160.v odd 8 2 3200.2.f.o 4
160.y odd 8 2 3200.2.d.c 2
160.y odd 8 2 6400.2.a.by 2
160.z even 8 2 3200.2.d.c 2
160.z even 8 2 6400.2.a.by 2
160.ba even 8 2 3200.2.f.o 4
160.bb odd 8 2 3200.2.f.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.b.a 2 32.g even 8 2
128.2.b.a 2 32.h odd 8 2
256.2.a.e 2 32.g even 8 2
256.2.a.e 2 32.h odd 8 2
1024.2.e.a 2 4.b odd 2 1
1024.2.e.a 2 8.b even 2 1
1024.2.e.a 2 16.e even 4 1
1024.2.e.a 2 16.f odd 4 1
1024.2.e.f 2 1.a even 1 1 trivial
1024.2.e.f 2 8.d odd 2 1 CM
1024.2.e.f 2 16.e even 4 1 inner
1024.2.e.f 2 16.f odd 4 1 inner
1152.2.d.c 2 96.o even 8 2
1152.2.d.c 2 96.p odd 8 2
2304.2.a.t 2 96.o even 8 2
2304.2.a.t 2 96.p odd 8 2
3200.2.d.c 2 160.y odd 8 2
3200.2.d.c 2 160.z even 8 2
3200.2.f.o 4 160.u even 8 2
3200.2.f.o 4 160.v odd 8 2
3200.2.f.o 4 160.ba even 8 2
3200.2.f.o 4 160.bb odd 8 2
6400.2.a.by 2 160.y odd 8 2
6400.2.a.by 2 160.z even 8 2

## Hecke kernels

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

 $$T_{3}^{2} - 4 T_{3} + 8$$ $$T_{5}$$ $$T_{47}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 4 T + 8 T^{2} - 12 T^{3} + 9 T^{4}$$
$5$ $$1 + 25 T^{4}$$
$7$ $$( 1 - 7 T^{2} )^{2}$$
$11$ $$1 + 4 T + 8 T^{2} + 44 T^{3} + 121 T^{4}$$
$13$ $$1 + 169 T^{4}$$
$17$ $$( 1 + 6 T + 17 T^{2} )^{2}$$
$19$ $$1 - 12 T + 72 T^{2} - 228 T^{3} + 361 T^{4}$$
$23$ $$( 1 - 23 T^{2} )^{2}$$
$29$ $$1 + 841 T^{4}$$
$31$ $$( 1 + 31 T^{2} )^{2}$$
$37$ $$1 + 1369 T^{4}$$
$41$ $$1 - 46 T^{2} + 1681 T^{4}$$
$43$ $$1 - 12 T + 72 T^{2} - 516 T^{3} + 1849 T^{4}$$
$47$ $$( 1 + 47 T^{2} )^{2}$$
$53$ $$1 + 2809 T^{4}$$
$59$ $$1 - 20 T + 200 T^{2} - 1180 T^{3} + 3481 T^{4}$$
$61$ $$1 + 3721 T^{4}$$
$67$ $$1 + 12 T + 72 T^{2} + 804 T^{3} + 4489 T^{4}$$
$71$ $$( 1 - 71 T^{2} )^{2}$$
$73$ $$1 - 142 T^{2} + 5329 T^{4}$$
$79$ $$( 1 + 79 T^{2} )^{2}$$
$83$ $$1 + 4 T + 8 T^{2} + 332 T^{3} + 6889 T^{4}$$
$89$ $$1 + 146 T^{2} + 7921 T^{4}$$
$97$ $$( 1 + 10 T + 97 T^{2} )^{2}$$