# Properties

 Label 512.2.e.g Level $512$ Weight $2$ Character orbit 512.e Analytic conductor $4.088$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$512 = 2^{9}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 512.e (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.08834058349$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( 2 - 2 i ) q^{3} + ( -1 - i ) q^{5} -4 i q^{7} -5 i q^{9} +O(q^{10})$$ $$q + ( 2 - 2 i ) q^{3} + ( -1 - i ) q^{5} -4 i q^{7} -5 i q^{9} + ( 2 + 2 i ) q^{11} + ( -3 + 3 i ) q^{13} -4 q^{15} + ( -2 + 2 i ) q^{19} + ( -8 - 8 i ) q^{21} + 4 i q^{23} -3 i q^{25} + ( -4 - 4 i ) q^{27} + ( 3 - 3 i ) q^{29} + 8 q^{31} + 8 q^{33} + ( -4 + 4 i ) q^{35} + ( 1 + i ) q^{37} + 12 i q^{39} -8 i q^{41} + ( -2 - 2 i ) q^{43} + ( -5 + 5 i ) q^{45} + 8 q^{47} -9 q^{49} + ( 1 + i ) q^{53} -4 i q^{55} + 8 i q^{57} + ( 6 + 6 i ) q^{59} + ( -3 + 3 i ) q^{61} -20 q^{63} + 6 q^{65} + ( -2 + 2 i ) q^{67} + ( 8 + 8 i ) q^{69} -12 i q^{71} + 2 i q^{73} + ( -6 - 6 i ) q^{75} + ( 8 - 8 i ) q^{77} - q^{81} + ( 10 - 10 i ) q^{83} -12 i q^{87} + 14 i q^{89} + ( 12 + 12 i ) q^{91} + ( 16 - 16 i ) q^{93} + 4 q^{95} -16 q^{97} + ( 10 - 10 i ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{3} - 2q^{5} + O(q^{10})$$ $$2q + 4q^{3} - 2q^{5} + 4q^{11} - 6q^{13} - 8q^{15} - 4q^{19} - 16q^{21} - 8q^{27} + 6q^{29} + 16q^{31} + 16q^{33} - 8q^{35} + 2q^{37} - 4q^{43} - 10q^{45} + 16q^{47} - 18q^{49} + 2q^{53} + 12q^{59} - 6q^{61} - 40q^{63} + 12q^{65} - 4q^{67} + 16q^{69} - 12q^{75} + 16q^{77} - 2q^{81} + 20q^{83} + 24q^{91} + 32q^{93} + 8q^{95} - 32q^{97} + 20q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/512\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$511$$ $$\chi(n)$$ $$i$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
129.1
 − 1.00000i 1.00000i
0 2.00000 + 2.00000i 0 −1.00000 + 1.00000i 0 4.00000i 0 5.00000i 0
385.1 0 2.00000 2.00000i 0 −1.00000 1.00000i 0 4.00000i 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
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 512.2.e.g yes 2
3.b odd 2 1 4608.2.k.o 2
4.b odd 2 1 512.2.e.a 2
8.b even 2 1 512.2.e.b yes 2
8.d odd 2 1 512.2.e.h yes 2
12.b even 2 1 4608.2.k.s 2
16.e even 4 1 512.2.e.b yes 2
16.e even 4 1 inner 512.2.e.g yes 2
16.f odd 4 1 512.2.e.a 2
16.f odd 4 1 512.2.e.h yes 2
24.f even 2 1 4608.2.k.f 2
24.h odd 2 1 4608.2.k.j 2
32.g even 8 2 1024.2.a.a 2
32.g even 8 2 1024.2.b.f 2
32.h odd 8 2 1024.2.a.f 2
32.h odd 8 2 1024.2.b.a 2
48.i odd 4 1 4608.2.k.j 2
48.i odd 4 1 4608.2.k.o 2
48.k even 4 1 4608.2.k.f 2
48.k even 4 1 4608.2.k.s 2
96.o even 8 2 9216.2.a.u 2
96.p odd 8 2 9216.2.a.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.e.a 2 4.b odd 2 1
512.2.e.a 2 16.f odd 4 1
512.2.e.b yes 2 8.b even 2 1
512.2.e.b yes 2 16.e even 4 1
512.2.e.g yes 2 1.a even 1 1 trivial
512.2.e.g yes 2 16.e even 4 1 inner
512.2.e.h yes 2 8.d odd 2 1
512.2.e.h yes 2 16.f odd 4 1
1024.2.a.a 2 32.g even 8 2
1024.2.a.f 2 32.h odd 8 2
1024.2.b.a 2 32.h odd 8 2
1024.2.b.f 2 32.g even 8 2
4608.2.k.f 2 24.f even 2 1
4608.2.k.f 2 48.k even 4 1
4608.2.k.j 2 24.h odd 2 1
4608.2.k.j 2 48.i odd 4 1
4608.2.k.o 2 3.b odd 2 1
4608.2.k.o 2 48.i odd 4 1
4608.2.k.s 2 12.b even 2 1
4608.2.k.s 2 48.k even 4 1
9216.2.a.b 2 96.p odd 8 2
9216.2.a.u 2 96.o even 8 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(512, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$8 - 4 T + T^{2}$$
$5$ $$2 + 2 T + T^{2}$$
$7$ $$16 + T^{2}$$
$11$ $$8 - 4 T + T^{2}$$
$13$ $$18 + 6 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$8 + 4 T + T^{2}$$
$23$ $$16 + T^{2}$$
$29$ $$18 - 6 T + T^{2}$$
$31$ $$( -8 + T )^{2}$$
$37$ $$2 - 2 T + T^{2}$$
$41$ $$64 + T^{2}$$
$43$ $$8 + 4 T + T^{2}$$
$47$ $$( -8 + T )^{2}$$
$53$ $$2 - 2 T + T^{2}$$
$59$ $$72 - 12 T + T^{2}$$
$61$ $$18 + 6 T + T^{2}$$
$67$ $$8 + 4 T + T^{2}$$
$71$ $$144 + T^{2}$$
$73$ $$4 + T^{2}$$
$79$ $$T^{2}$$
$83$ $$200 - 20 T + T^{2}$$
$89$ $$196 + T^{2}$$
$97$ $$( 16 + T )^{2}$$