# Properties

 Label 1024.2.b.e Level $1024$ Weight $2$ Character orbit 1024.b Analytic conductor $8.177$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 16) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + \beta q^{5} + 2 q^{7} + q^{9} +O(q^{10})$$ $$q + \beta q^{3} + \beta q^{5} + 2 q^{7} + q^{9} + \beta q^{11} -\beta q^{13} -2 q^{15} + 2 q^{17} + 3 \beta q^{19} + 2 \beta q^{21} + 6 q^{23} + 3 q^{25} + 4 \beta q^{27} -3 \beta q^{29} -8 q^{31} -2 q^{33} + 2 \beta q^{35} -3 \beta q^{37} + 2 q^{39} + 5 \beta q^{43} + \beta q^{45} -8 q^{47} -3 q^{49} + 2 \beta q^{51} -5 \beta q^{53} -2 q^{55} -6 q^{57} + 3 \beta q^{59} + 9 \beta q^{61} + 2 q^{63} + 2 q^{65} + 5 \beta q^{67} + 6 \beta q^{69} + 10 q^{71} -4 q^{73} + 3 \beta q^{75} + 2 \beta q^{77} -5 q^{81} -\beta q^{83} + 2 \beta q^{85} + 6 q^{87} -4 q^{89} -2 \beta q^{91} -8 \beta q^{93} -6 q^{95} -2 q^{97} + \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{7} + 2q^{9} + O(q^{10})$$ $$2q + 4q^{7} + 2q^{9} - 4q^{15} + 4q^{17} + 12q^{23} + 6q^{25} - 16q^{31} - 4q^{33} + 4q^{39} - 16q^{47} - 6q^{49} - 4q^{55} - 12q^{57} + 4q^{63} + 4q^{65} + 20q^{71} - 8q^{73} - 10q^{81} + 12q^{87} - 8q^{89} - 12q^{95} - 4q^{97} + O(q^{100})$$

## Character values

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

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

## 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 1.41421i 0 1.41421i 0 2.00000 0 1.00000 0
513.2 0 1.41421i 0 1.41421i 0 2.00000 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
8.b even 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.e 2
4.b odd 2 1 1024.2.b.b 2
8.b even 2 1 inner 1024.2.b.e 2
8.d odd 2 1 1024.2.b.b 2
16.e even 4 2 1024.2.a.b 2
16.f odd 4 2 1024.2.a.e 2
32.g even 8 2 16.2.e.a 2
32.g even 8 2 128.2.e.b 2
32.h odd 8 2 64.2.e.a 2
32.h odd 8 2 128.2.e.a 2
48.i odd 4 2 9216.2.a.d 2
48.k even 4 2 9216.2.a.s 2
96.o even 8 2 576.2.k.a 2
96.o even 8 2 1152.2.k.a 2
96.p odd 8 2 144.2.k.a 2
96.p odd 8 2 1152.2.k.b 2
160.u even 8 2 1600.2.q.a 2
160.v odd 8 2 400.2.q.a 2
160.y odd 8 2 1600.2.l.a 2
160.z even 8 2 400.2.l.c 2
160.ba even 8 2 1600.2.q.b 2
160.bb odd 8 2 400.2.q.b 2
224.v odd 8 2 784.2.m.b 2
224.bc odd 24 4 784.2.x.c 4
224.bd even 24 4 784.2.x.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.2.e.a 2 32.g even 8 2
64.2.e.a 2 32.h odd 8 2
128.2.e.a 2 32.h odd 8 2
128.2.e.b 2 32.g even 8 2
144.2.k.a 2 96.p odd 8 2
400.2.l.c 2 160.z even 8 2
400.2.q.a 2 160.v odd 8 2
400.2.q.b 2 160.bb odd 8 2
576.2.k.a 2 96.o even 8 2
784.2.m.b 2 224.v odd 8 2
784.2.x.c 4 224.bc odd 24 4
784.2.x.f 4 224.bd even 24 4
1024.2.a.b 2 16.e even 4 2
1024.2.a.e 2 16.f odd 4 2
1024.2.b.b 2 4.b odd 2 1
1024.2.b.b 2 8.d odd 2 1
1024.2.b.e 2 1.a even 1 1 trivial
1024.2.b.e 2 8.b even 2 1 inner
1152.2.k.a 2 96.o even 8 2
1152.2.k.b 2 96.p odd 8 2
1600.2.l.a 2 160.y odd 8 2
1600.2.q.a 2 160.u even 8 2
1600.2.q.b 2 160.ba even 8 2
9216.2.a.d 2 48.i odd 4 2
9216.2.a.s 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 $$S_{2}^{\mathrm{new}}(1024, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$2 + T^{2}$$
$5$ $$2 + T^{2}$$
$7$ $$( -2 + T )^{2}$$
$11$ $$2 + T^{2}$$
$13$ $$2 + T^{2}$$
$17$ $$( -2 + T )^{2}$$
$19$ $$18 + T^{2}$$
$23$ $$( -6 + T )^{2}$$
$29$ $$18 + T^{2}$$
$31$ $$( 8 + T )^{2}$$
$37$ $$18 + T^{2}$$
$41$ $$T^{2}$$
$43$ $$50 + T^{2}$$
$47$ $$( 8 + T )^{2}$$
$53$ $$50 + T^{2}$$
$59$ $$18 + T^{2}$$
$61$ $$162 + T^{2}$$
$67$ $$50 + T^{2}$$
$71$ $$( -10 + T )^{2}$$
$73$ $$( 4 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$2 + T^{2}$$
$89$ $$( 4 + T )^{2}$$
$97$ $$( 2 + T )^{2}$$