# Properties

 Label 1024.2.e.b Level $1024$ Weight $2$ Character orbit 1024.e Analytic conductor $8.177$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 512) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + i ) q^{3} + ( -2 - 2 i ) q^{5} -4 i q^{7} + i q^{9} +O(q^{10})$$ $$q + ( -1 + i ) q^{3} + ( -2 - 2 i ) q^{5} -4 i q^{7} + i q^{9} + ( -1 - i ) q^{11} + ( 2 - 2 i ) q^{13} + 4 q^{15} + 4 q^{17} + ( -5 + 5 i ) q^{19} + ( 4 + 4 i ) q^{21} + 4 i q^{23} + 3 i q^{25} + ( -4 - 4 i ) q^{27} + ( -6 + 6 i ) q^{29} -8 q^{31} + 2 q^{33} + ( -8 + 8 i ) q^{35} + ( 2 + 2 i ) q^{37} + 4 i q^{39} -2 i q^{41} + ( -3 - 3 i ) q^{43} + ( 2 - 2 i ) q^{45} -9 q^{49} + ( -4 + 4 i ) q^{51} + ( -2 - 2 i ) q^{53} + 4 i q^{55} -10 i q^{57} + ( 3 + 3 i ) q^{59} + ( 6 - 6 i ) q^{61} + 4 q^{63} -8 q^{65} + ( -3 + 3 i ) q^{67} + ( -4 - 4 i ) q^{69} + 4 i q^{71} + 4 i q^{73} + ( -3 - 3 i ) q^{75} + ( -4 + 4 i ) q^{77} -8 q^{79} + 5 q^{81} + ( -7 + 7 i ) q^{83} + ( -8 - 8 i ) q^{85} -12 i q^{87} + 12 i q^{89} + ( -8 - 8 i ) q^{91} + ( 8 - 8 i ) q^{93} + 20 q^{95} -4 q^{97} + ( 1 - i ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 4 q^{5} + O(q^{10})$$ $$2 q - 2 q^{3} - 4 q^{5} - 2 q^{11} + 4 q^{13} + 8 q^{15} + 8 q^{17} - 10 q^{19} + 8 q^{21} - 8 q^{27} - 12 q^{29} - 16 q^{31} + 4 q^{33} - 16 q^{35} + 4 q^{37} - 6 q^{43} + 4 q^{45} - 18 q^{49} - 8 q^{51} - 4 q^{53} + 6 q^{59} + 12 q^{61} + 8 q^{63} - 16 q^{65} - 6 q^{67} - 8 q^{69} - 6 q^{75} - 8 q^{77} - 16 q^{79} + 10 q^{81} - 14 q^{83} - 16 q^{85} - 16 q^{91} + 16 q^{93} + 40 q^{95} - 8 q^{97} + 2 q^{99} + O(q^{100})$$

## 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 −1.00000 1.00000i 0 −2.00000 + 2.00000i 0 4.00000i 0 1.00000i 0
769.1 0 −1.00000 + 1.00000i 0 −2.00000 2.00000i 0 4.00000i 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
16.e even 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.b 2
4.b odd 2 1 1024.2.e.d 2
8.b even 2 1 1024.2.e.e 2
8.d odd 2 1 1024.2.e.c 2
16.e even 4 1 inner 1024.2.e.b 2
16.e even 4 1 1024.2.e.e 2
16.f odd 4 1 1024.2.e.c 2
16.f odd 4 1 1024.2.e.d 2
32.g even 8 2 512.2.a.c 2
32.g even 8 2 512.2.b.b 2
32.h odd 8 2 512.2.a.d yes 2
32.h odd 8 2 512.2.b.a 2
96.o even 8 2 4608.2.a.m 2
96.o even 8 2 4608.2.d.a 2
96.p odd 8 2 4608.2.a.f 2
96.p odd 8 2 4608.2.d.b 2

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$2 + 2 T + T^{2}$$
$5$ $$8 + 4 T + T^{2}$$
$7$ $$16 + T^{2}$$
$11$ $$2 + 2 T + T^{2}$$
$13$ $$8 - 4 T + T^{2}$$
$17$ $$( -4 + T )^{2}$$
$19$ $$50 + 10 T + T^{2}$$
$23$ $$16 + T^{2}$$
$29$ $$72 + 12 T + T^{2}$$
$31$ $$( 8 + T )^{2}$$
$37$ $$8 - 4 T + T^{2}$$
$41$ $$4 + T^{2}$$
$43$ $$18 + 6 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$8 + 4 T + T^{2}$$
$59$ $$18 - 6 T + T^{2}$$
$61$ $$72 - 12 T + T^{2}$$
$67$ $$18 + 6 T + T^{2}$$
$71$ $$16 + T^{2}$$
$73$ $$16 + T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$98 + 14 T + T^{2}$$
$89$ $$144 + T^{2}$$
$97$ $$( 4 + T )^{2}$$