# Properties

 Label 1024.2.b.g Level $1024$ Weight $2$ Character orbit 1024.b Analytic conductor $8.177$ Analytic rank $0$ Dimension $8$ CM no 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: $$8$$ Coefficient field: $$\Q(\zeta_{16})$$ Defining polynomial: $$x^{8} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{11}$$ Twist minimal: no (minimal twist has level 512) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{16}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{16} - \zeta_{16}^{3} - \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{3} + ( \zeta_{16}^{2} + 2 \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{5} + ( -2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} ) q^{7} + ( -1 - 2 \zeta_{16}^{2} + 2 \zeta_{16}^{6} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{16} - \zeta_{16}^{3} - \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{3} + ( \zeta_{16}^{2} + 2 \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{5} + ( -2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} ) q^{7} + ( -1 - 2 \zeta_{16}^{2} + 2 \zeta_{16}^{6} ) q^{9} + ( \zeta_{16} - 3 \zeta_{16}^{3} - 3 \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{11} + ( \zeta_{16}^{2} - 2 \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{13} + ( 4 \zeta_{16} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} - 4 \zeta_{16}^{7} ) q^{15} + ( -2 \zeta_{16}^{2} + 2 \zeta_{16}^{6} ) q^{17} + ( 3 \zeta_{16} - \zeta_{16}^{3} - \zeta_{16}^{5} + 3 \zeta_{16}^{7} ) q^{19} + 4 \zeta_{16}^{4} q^{21} + ( -4 \zeta_{16} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} + 4 \zeta_{16}^{7} ) q^{23} + ( -1 - 4 \zeta_{16}^{2} + 4 \zeta_{16}^{6} ) q^{25} + ( -2 \zeta_{16} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{27} + ( \zeta_{16}^{2} - 6 \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{29} + ( -4 \zeta_{16} + 4 \zeta_{16}^{7} ) q^{31} + ( -4 - 6 \zeta_{16}^{2} + 6 \zeta_{16}^{6} ) q^{33} + ( -2 \zeta_{16} - 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} - 2 \zeta_{16}^{7} ) q^{35} + ( -5 \zeta_{16}^{2} + 2 \zeta_{16}^{4} - 5 \zeta_{16}^{6} ) q^{37} + ( -2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} ) q^{39} + 4 q^{41} + ( -\zeta_{16} - \zeta_{16}^{3} - \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{43} + ( -5 \zeta_{16}^{2} - 6 \zeta_{16}^{4} - 5 \zeta_{16}^{6} ) q^{45} + ( -4 \zeta_{16} + 4 \zeta_{16}^{7} ) q^{47} + ( 1 - 4 \zeta_{16}^{2} + 4 \zeta_{16}^{6} ) q^{49} + ( 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} ) q^{51} + ( \zeta_{16}^{2} + 6 \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{53} + ( 8 \zeta_{16} + 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} - 8 \zeta_{16}^{7} ) q^{55} + ( 4 - 2 \zeta_{16}^{2} + 2 \zeta_{16}^{6} ) q^{57} + ( \zeta_{16} + \zeta_{16}^{3} + \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{59} + ( -5 \zeta_{16}^{2} - 6 \zeta_{16}^{4} - 5 \zeta_{16}^{6} ) q^{61} + ( 4 \zeta_{16} - 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} - 4 \zeta_{16}^{7} ) q^{63} + 2 q^{65} + ( \zeta_{16} + 5 \zeta_{16}^{3} + 5 \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{67} + ( 8 \zeta_{16}^{2} + 4 \zeta_{16}^{4} + 8 \zeta_{16}^{6} ) q^{69} + ( 4 \zeta_{16} + 6 \zeta_{16}^{3} - 6 \zeta_{16}^{5} - 4 \zeta_{16}^{7} ) q^{71} + ( -2 - 6 \zeta_{16}^{2} + 6 \zeta_{16}^{6} ) q^{73} + ( \zeta_{16} + 9 \zeta_{16}^{3} + 9 \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{75} + ( 8 \zeta_{16}^{2} - 4 \zeta_{16}^{4} + 8 \zeta_{16}^{6} ) q^{77} + ( -8 \zeta_{16}^{3} + 8 \zeta_{16}^{5} ) q^{79} + ( -3 - 2 \zeta_{16}^{2} + 2 \zeta_{16}^{6} ) q^{81} + ( -7 \zeta_{16} + \zeta_{16}^{3} + \zeta_{16}^{5} - 7 \zeta_{16}^{7} ) q^{83} + ( -4 \zeta_{16}^{2} - 4 \zeta_{16}^{4} - 4 \zeta_{16}^{6} ) q^{85} + ( -4 \zeta_{16} - 6 \zeta_{16}^{3} + 6 \zeta_{16}^{5} + 4 \zeta_{16}^{7} ) q^{87} + ( -2 + 2 \zeta_{16}^{2} - 2 \zeta_{16}^{6} ) q^{89} + ( 6 \zeta_{16} - 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} + 6 \zeta_{16}^{7} ) q^{91} + ( 8 \zeta_{16}^{2} + 8 \zeta_{16}^{4} + 8 \zeta_{16}^{6} ) q^{93} + ( -2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} ) q^{95} + ( 8 + 2 \zeta_{16}^{2} - 2 \zeta_{16}^{6} ) q^{97} + ( 7 \zeta_{16} + 7 \zeta_{16}^{3} + 7 \zeta_{16}^{5} + 7 \zeta_{16}^{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{9} + O(q^{10})$$ $$8q - 8q^{9} - 8q^{25} - 32q^{33} + 32q^{41} + 8q^{49} + 32q^{57} + 16q^{65} - 16q^{73} - 24q^{81} - 16q^{89} + 64q^{97} + O(q^{100})$$

## 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
 −0.923880 + 0.382683i 0.923880 + 0.382683i 0.382683 + 0.923880i −0.382683 + 0.923880i −0.382683 − 0.923880i 0.382683 − 0.923880i 0.923880 − 0.382683i −0.923880 − 0.382683i
0 2.61313i 0 3.41421i 0 1.53073 0 −3.82843 0
513.2 0 2.61313i 0 3.41421i 0 −1.53073 0 −3.82843 0
513.3 0 1.08239i 0 0.585786i 0 3.69552 0 1.82843 0
513.4 0 1.08239i 0 0.585786i 0 −3.69552 0 1.82843 0
513.5 0 1.08239i 0 0.585786i 0 −3.69552 0 1.82843 0
513.6 0 1.08239i 0 0.585786i 0 3.69552 0 1.82843 0
513.7 0 2.61313i 0 3.41421i 0 −1.53073 0 −3.82843 0
513.8 0 2.61313i 0 3.41421i 0 1.53073 0 −3.82843 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 513.8 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 inner
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.g 8
4.b odd 2 1 inner 1024.2.b.g 8
8.b even 2 1 inner 1024.2.b.g 8
8.d odd 2 1 inner 1024.2.b.g 8
16.e even 4 1 1024.2.a.h 4
16.e even 4 1 1024.2.a.i 4
16.f odd 4 1 1024.2.a.h 4
16.f odd 4 1 1024.2.a.i 4
32.g even 8 2 512.2.e.i 8
32.g even 8 2 512.2.e.j yes 8
32.h odd 8 2 512.2.e.i 8
32.h odd 8 2 512.2.e.j yes 8
48.i odd 4 1 9216.2.a.w 4
48.i odd 4 1 9216.2.a.bp 4
48.k even 4 1 9216.2.a.w 4
48.k even 4 1 9216.2.a.bp 4
96.o even 8 2 4608.2.k.bd 8
96.o even 8 2 4608.2.k.bi 8
96.p odd 8 2 4608.2.k.bd 8
96.p odd 8 2 4608.2.k.bi 8

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

## Hecke kernels

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

 $$T_{3}^{4} + 8 T_{3}^{2} + 8$$ $$T_{5}^{4} + 12 T_{5}^{2} + 4$$ $$T_{7}^{4} - 16 T_{7}^{2} + 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 8 + 8 T^{2} + T^{4} )^{2}$$
$5$ $$( 4 + 12 T^{2} + T^{4} )^{2}$$
$7$ $$( 32 - 16 T^{2} + T^{4} )^{2}$$
$11$ $$( 392 + 40 T^{2} + T^{4} )^{2}$$
$13$ $$( 4 + 12 T^{2} + T^{4} )^{2}$$
$17$ $$( -8 + T^{2} )^{4}$$
$19$ $$( 8 + 40 T^{2} + T^{4} )^{2}$$
$23$ $$( 1568 - 80 T^{2} + T^{4} )^{2}$$
$29$ $$( 1156 + 76 T^{2} + T^{4} )^{2}$$
$31$ $$( 512 - 64 T^{2} + T^{4} )^{2}$$
$37$ $$( 2116 + 108 T^{2} + T^{4} )^{2}$$
$41$ $$( -4 + T )^{8}$$
$43$ $$( 8 + 8 T^{2} + T^{4} )^{2}$$
$47$ $$( 512 - 64 T^{2} + T^{4} )^{2}$$
$53$ $$( 1156 + 76 T^{2} + T^{4} )^{2}$$
$59$ $$( 8 + 8 T^{2} + T^{4} )^{2}$$
$61$ $$( 196 + 172 T^{2} + T^{4} )^{2}$$
$67$ $$( 392 + 104 T^{2} + T^{4} )^{2}$$
$71$ $$( 9248 - 208 T^{2} + T^{4} )^{2}$$
$73$ $$( -68 + 4 T + T^{2} )^{4}$$
$79$ $$( 8192 - 256 T^{2} + T^{4} )^{2}$$
$83$ $$( 2312 + 200 T^{2} + T^{4} )^{2}$$
$89$ $$( -4 + 4 T + T^{2} )^{4}$$
$97$ $$( 56 - 16 T + T^{2} )^{4}$$