# Properties

 Label 512.2.e.i Level $512$ Weight $2$ Character orbit 512.e Analytic conductor $4.088$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# 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: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{16})$$ Defining polynomial: $$x^{8} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{7}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{3} + ( -1 + \zeta_{16}^{4} - 2 \zeta_{16}^{6} ) q^{5} + ( -2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} ) q^{7} + ( -2 \zeta_{16}^{2} + \zeta_{16}^{4} - 2 \zeta_{16}^{6} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{16} - \zeta_{16}^{3} + \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{3} + ( -1 + \zeta_{16}^{4} - 2 \zeta_{16}^{6} ) q^{5} + ( -2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} ) q^{7} + ( -2 \zeta_{16}^{2} + \zeta_{16}^{4} - 2 \zeta_{16}^{6} ) q^{9} + ( -3 \zeta_{16} + 3 \zeta_{16}^{3} + \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{11} + ( 1 + 2 \zeta_{16}^{2} + \zeta_{16}^{4} ) q^{13} + ( -2 \zeta_{16} + 4 \zeta_{16}^{3} - 4 \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{15} + ( -2 \zeta_{16}^{2} + 2 \zeta_{16}^{6} ) q^{17} + ( -3 \zeta_{16} - 3 \zeta_{16}^{3} - \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{19} + 4 \zeta_{16}^{6} q^{21} + ( -4 \zeta_{16} - 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} - 4 \zeta_{16}^{7} ) q^{23} + ( 4 \zeta_{16}^{2} - \zeta_{16}^{4} + 4 \zeta_{16}^{6} ) q^{25} + ( -2 \zeta_{16} + 2 \zeta_{16}^{3} + 2 \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{27} + ( -1 - 6 \zeta_{16}^{2} - \zeta_{16}^{4} ) q^{29} + ( 4 \zeta_{16}^{3} - 4 \zeta_{16}^{5} ) 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 - 5 \zeta_{16}^{4} - 2 \zeta_{16}^{6} ) q^{37} + ( -2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} ) q^{39} -4 \zeta_{16}^{4} q^{41} + ( -\zeta_{16} + \zeta_{16}^{3} - \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{43} + ( -5 + 6 \zeta_{16}^{2} - 5 \zeta_{16}^{4} ) q^{45} + ( -4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} ) q^{47} + ( -1 - 4 \zeta_{16}^{2} + 4 \zeta_{16}^{6} ) q^{49} + ( 4 \zeta_{16}^{5} - 4 \zeta_{16}^{7} ) q^{51} + ( 1 - \zeta_{16}^{4} + 6 \zeta_{16}^{6} ) q^{53} + ( 8 \zeta_{16} - 2 \zeta_{16}^{3} - 2 \zeta_{16}^{5} + 8 \zeta_{16}^{7} ) q^{55} + ( 2 \zeta_{16}^{2} + 4 \zeta_{16}^{4} + 2 \zeta_{16}^{6} ) q^{57} + ( -\zeta_{16} + \zeta_{16}^{3} - \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{59} + ( 5 - 6 \zeta_{16}^{2} + 5 \zeta_{16}^{4} ) q^{61} + ( -2 \zeta_{16} - 4 \zeta_{16}^{3} + 4 \zeta_{16}^{5} + 2 \zeta_{16}^{7} ) q^{63} + 2 q^{65} + ( \zeta_{16} + \zeta_{16}^{3} - 5 \zeta_{16}^{5} + 5 \zeta_{16}^{7} ) q^{67} + ( -8 + 8 \zeta_{16}^{4} - 4 \zeta_{16}^{6} ) q^{69} + ( -4 \zeta_{16} + 6 \zeta_{16}^{3} + 6 \zeta_{16}^{5} - 4 \zeta_{16}^{7} ) q^{71} + ( -6 \zeta_{16}^{2} + 2 \zeta_{16}^{4} - 6 \zeta_{16}^{6} ) q^{73} + ( 9 \zeta_{16} - 9 \zeta_{16}^{3} + \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{75} + ( 8 + 4 \zeta_{16}^{2} + 8 \zeta_{16}^{4} ) q^{77} + ( 8 \zeta_{16} - 8 \zeta_{16}^{7} ) q^{79} + ( 3 - 2 \zeta_{16}^{2} + 2 \zeta_{16}^{6} ) q^{81} + ( 7 \zeta_{16} + 7 \zeta_{16}^{3} + \zeta_{16}^{5} - \zeta_{16}^{7} ) q^{83} + ( -4 + 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 \zeta_{16}^{2} - 2 \zeta_{16}^{4} - 2 \zeta_{16}^{6} ) q^{89} + ( 2 \zeta_{16} - 2 \zeta_{16}^{3} - 6 \zeta_{16}^{5} - 6 \zeta_{16}^{7} ) q^{91} + ( -8 + 8 \zeta_{16}^{2} - 8 \zeta_{16}^{4} ) q^{93} + ( -2 \zeta_{16} + 2 \zeta_{16}^{7} ) 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})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{5} + O(q^{10})$$ $$8q - 8q^{5} + 8q^{13} - 8q^{29} - 32q^{33} + 40q^{37} - 40q^{45} - 8q^{49} + 8q^{53} + 40q^{61} + 16q^{65} - 64q^{69} + 64q^{77} + 24q^{81} - 32q^{85} - 64q^{93} + 64q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$5$$ $$511$$ $$\chi(n)$$ $$-\zeta_{16}^{2}$$ $$1$$

## 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
 −0.382683 + 0.923880i 0.923880 + 0.382683i −0.923880 − 0.382683i 0.382683 − 0.923880i −0.382683 − 0.923880i 0.923880 − 0.382683i −0.923880 + 0.382683i 0.382683 + 0.923880i
0 −1.84776 1.84776i 0 −2.41421 + 2.41421i 0 1.53073i 0 3.82843i 0
129.2 0 −0.765367 0.765367i 0 0.414214 0.414214i 0 3.69552i 0 1.82843i 0
129.3 0 0.765367 + 0.765367i 0 0.414214 0.414214i 0 3.69552i 0 1.82843i 0
129.4 0 1.84776 + 1.84776i 0 −2.41421 + 2.41421i 0 1.53073i 0 3.82843i 0
385.1 0 −1.84776 + 1.84776i 0 −2.41421 2.41421i 0 1.53073i 0 3.82843i 0
385.2 0 −0.765367 + 0.765367i 0 0.414214 + 0.414214i 0 3.69552i 0 1.82843i 0
385.3 0 0.765367 0.765367i 0 0.414214 + 0.414214i 0 3.69552i 0 1.82843i 0
385.4 0 1.84776 1.84776i 0 −2.41421 2.41421i 0 1.53073i 0 3.82843i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 385.4 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
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 512.2.e.i 8
3.b odd 2 1 4608.2.k.bi 8
4.b odd 2 1 inner 512.2.e.i 8
8.b even 2 1 512.2.e.j yes 8
8.d odd 2 1 512.2.e.j yes 8
12.b even 2 1 4608.2.k.bi 8
16.e even 4 1 inner 512.2.e.i 8
16.e even 4 1 512.2.e.j yes 8
16.f odd 4 1 inner 512.2.e.i 8
16.f odd 4 1 512.2.e.j yes 8
24.f even 2 1 4608.2.k.bd 8
24.h odd 2 1 4608.2.k.bd 8
32.g even 8 1 1024.2.a.h 4
32.g even 8 1 1024.2.a.i 4
32.g even 8 2 1024.2.b.g 8
32.h odd 8 1 1024.2.a.h 4
32.h odd 8 1 1024.2.a.i 4
32.h odd 8 2 1024.2.b.g 8
48.i odd 4 1 4608.2.k.bd 8
48.i odd 4 1 4608.2.k.bi 8
48.k even 4 1 4608.2.k.bd 8
48.k even 4 1 4608.2.k.bi 8
96.o even 8 1 9216.2.a.w 4
96.o even 8 1 9216.2.a.bp 4
96.p odd 8 1 9216.2.a.w 4
96.p odd 8 1 9216.2.a.bp 4

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

## Hecke kernels

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

 $$T_{3}^{8} + 48 T_{3}^{4} + 64$$ $$T_{5}^{4} + 4 T_{5}^{3} + 8 T_{5}^{2} - 8 T_{5} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$64 + 48 T^{4} + T^{8}$$
$5$ $$( 4 - 8 T + 8 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$7$ $$( 32 + 16 T^{2} + T^{4} )^{2}$$
$11$ $$153664 + 816 T^{4} + T^{8}$$
$13$ $$( 4 + 8 T + 8 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$17$ $$( -8 + T^{2} )^{4}$$
$19$ $$64 + 1584 T^{4} + T^{8}$$
$23$ $$( 1568 + 80 T^{2} + T^{4} )^{2}$$
$29$ $$( 1156 - 136 T + 8 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$31$ $$( 512 - 64 T^{2} + T^{4} )^{2}$$
$37$ $$( 2116 - 920 T + 200 T^{2} - 20 T^{3} + T^{4} )^{2}$$
$41$ $$( 16 + T^{2} )^{4}$$
$43$ $$64 + 48 T^{4} + T^{8}$$
$47$ $$( 512 - 64 T^{2} + T^{4} )^{2}$$
$53$ $$( 1156 + 136 T + 8 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$59$ $$64 + 48 T^{4} + T^{8}$$
$61$ $$( 196 - 280 T + 200 T^{2} - 20 T^{3} + T^{4} )^{2}$$
$67$ $$153664 + 10032 T^{4} + T^{8}$$
$71$ $$( 9248 + 208 T^{2} + T^{4} )^{2}$$
$73$ $$( 4624 + 152 T^{2} + T^{4} )^{2}$$
$79$ $$( 8192 - 256 T^{2} + T^{4} )^{2}$$
$83$ $$5345344 + 35376 T^{4} + T^{8}$$
$89$ $$( 16 + 24 T^{2} + T^{4} )^{2}$$
$97$ $$( 56 - 16 T + T^{2} )^{4}$$