# Properties

 Label 512.2.g.a Level $512$ Weight $2$ Character orbit 512.g Analytic conductor $4.088$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$512 = 2^{9}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 512.g (of order $$8$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.08834058349$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 32) Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \zeta_{8}^{3} ) q^{3} + ( -2 - 2 \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{5} + ( -1 - \zeta_{8}^{2} ) q^{7} + ( 1 - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \zeta_{8}^{3} ) q^{3} + ( -2 - 2 \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{5} + ( -1 - \zeta_{8}^{2} ) q^{7} + ( 1 - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{9} + ( -1 + \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{11} + ( -\zeta_{8} + \zeta_{8}^{2} ) q^{13} + ( 3 \zeta_{8} - 2 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{15} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{17} + ( -3 + 2 \zeta_{8} - 2 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{19} + ( 1 - \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{21} + ( -3 + 3 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{23} + ( -1 + 5 \zeta_{8} - \zeta_{8}^{2} ) q^{25} + ( -1 - \zeta_{8} - 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{27} + ( -2 - \zeta_{8} - \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{29} -4 q^{31} + ( 2 + \zeta_{8} - \zeta_{8}^{3} ) q^{33} + ( 3 + \zeta_{8} + \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{35} + ( -\zeta_{8}^{2} + \zeta_{8}^{3} ) q^{37} + ( -1 + 2 \zeta_{8} - \zeta_{8}^{2} ) q^{39} + ( 3 - 3 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{41} + ( -3 + 3 \zeta_{8} - 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{43} + ( -3 - 2 \zeta_{8} + 2 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{45} + ( -4 \zeta_{8} + 6 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{47} -5 \zeta_{8}^{2} q^{49} + ( 2 - 2 \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{51} + ( -4 + 4 \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{53} + ( 5 - 5 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{55} + ( 5 - 4 \zeta_{8} + 5 \zeta_{8}^{2} ) q^{57} + ( 1 + \zeta_{8} + 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{59} + ( \zeta_{8} + \zeta_{8}^{2} ) q^{61} + ( -2 - \zeta_{8} + \zeta_{8}^{3} ) q^{63} + ( -2 + 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{65} + ( -3 - 2 \zeta_{8} - 2 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{67} + ( 3 + 3 \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{69} + ( 3 + 4 \zeta_{8} + 3 \zeta_{8}^{2} ) q^{71} + ( -7 + 7 \zeta_{8}^{2} ) q^{73} + ( 6 - 6 \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{75} + ( 3 + \zeta_{8} - \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{77} -6 \zeta_{8}^{2} q^{79} + ( -5 \zeta_{8} + 3 \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{81} + ( -1 - 4 \zeta_{8} + 4 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{83} + ( 6 - 6 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{85} + ( 1 - \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{87} + ( -3 - 8 \zeta_{8} - 3 \zeta_{8}^{2} ) q^{89} + ( 1 + \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{91} + ( 4 + 4 \zeta_{8}^{3} ) q^{93} + ( 16 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{95} + ( -10 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{97} + ( 2 + 5 \zeta_{8} + 5 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} - 8 q^{5} - 4 q^{7} + 4 q^{9} + O(q^{10})$$ $$4 q - 4 q^{3} - 8 q^{5} - 4 q^{7} + 4 q^{9} - 4 q^{11} - 12 q^{19} + 4 q^{21} - 12 q^{23} - 4 q^{25} - 4 q^{27} - 8 q^{29} - 16 q^{31} + 8 q^{33} + 12 q^{35} - 4 q^{39} + 12 q^{41} - 12 q^{43} - 12 q^{45} + 8 q^{51} - 16 q^{53} + 20 q^{55} + 20 q^{57} + 4 q^{59} - 8 q^{63} - 8 q^{65} - 12 q^{67} + 12 q^{69} + 12 q^{71} - 28 q^{73} + 24 q^{75} + 12 q^{77} - 4 q^{83} + 24 q^{85} + 4 q^{87} - 12 q^{89} + 4 q^{91} + 16 q^{93} + 64 q^{95} - 40 q^{97} + 8 q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$5$$ $$511$$ $$\chi(n)$$ $$\zeta_{8}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
0 −0.292893 + 0.707107i 0 −2.70711 + 1.12132i 0 −1.00000 + 1.00000i 0 1.70711 + 1.70711i 0
193.1 0 −1.70711 + 0.707107i 0 −1.29289 + 3.12132i 0 −1.00000 1.00000i 0 0.292893 0.292893i 0
321.1 0 −1.70711 0.707107i 0 −1.29289 3.12132i 0 −1.00000 + 1.00000i 0 0.292893 + 0.292893i 0
449.1 0 −0.292893 0.707107i 0 −2.70711 1.12132i 0 −1.00000 1.00000i 0 1.70711 1.70711i 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
32.g even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 512.2.g.a 4
4.b odd 2 1 512.2.g.c 4
8.b even 2 1 512.2.g.d 4
8.d odd 2 1 512.2.g.b 4
16.e even 4 1 32.2.g.a 4
16.e even 4 1 256.2.g.b 4
16.f odd 4 1 128.2.g.a 4
16.f odd 4 1 256.2.g.a 4
32.g even 8 1 32.2.g.a 4
32.g even 8 1 256.2.g.b 4
32.g even 8 1 inner 512.2.g.a 4
32.g even 8 1 512.2.g.d 4
32.h odd 8 1 128.2.g.a 4
32.h odd 8 1 256.2.g.a 4
32.h odd 8 1 512.2.g.b 4
32.h odd 8 1 512.2.g.c 4
48.i odd 4 1 288.2.v.a 4
48.k even 4 1 1152.2.v.a 4
64.i even 16 2 4096.2.a.e 4
64.j odd 16 2 4096.2.a.f 4
80.i odd 4 1 800.2.ba.b 4
80.q even 4 1 800.2.y.a 4
80.t odd 4 1 800.2.ba.a 4
96.o even 8 1 1152.2.v.a 4
96.p odd 8 1 288.2.v.a 4
160.v odd 8 1 800.2.ba.a 4
160.z even 8 1 800.2.y.a 4
160.bb odd 8 1 800.2.ba.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.g.a 4 16.e even 4 1
32.2.g.a 4 32.g even 8 1
128.2.g.a 4 16.f odd 4 1
128.2.g.a 4 32.h odd 8 1
256.2.g.a 4 16.f odd 4 1
256.2.g.a 4 32.h odd 8 1
256.2.g.b 4 16.e even 4 1
256.2.g.b 4 32.g even 8 1
288.2.v.a 4 48.i odd 4 1
288.2.v.a 4 96.p odd 8 1
512.2.g.a 4 1.a even 1 1 trivial
512.2.g.a 4 32.g even 8 1 inner
512.2.g.b 4 8.d odd 2 1
512.2.g.b 4 32.h odd 8 1
512.2.g.c 4 4.b odd 2 1
512.2.g.c 4 32.h odd 8 1
512.2.g.d 4 8.b even 2 1
512.2.g.d 4 32.g even 8 1
800.2.y.a 4 80.q even 4 1
800.2.y.a 4 160.z even 8 1
800.2.ba.a 4 80.t odd 4 1
800.2.ba.a 4 160.v odd 8 1
800.2.ba.b 4 80.i odd 4 1
800.2.ba.b 4 160.bb odd 8 1
1152.2.v.a 4 48.k even 4 1
1152.2.v.a 4 96.o even 8 1
4096.2.a.e 4 64.i even 16 2
4096.2.a.f 4 64.j odd 16 2

## Hecke kernels

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

 $$T_{3}^{4} + 4 T_{3}^{3} + 6 T_{3}^{2} + 4 T_{3} + 2$$ $$T_{5}^{4} + 8 T_{5}^{3} + 34 T_{5}^{2} + 84 T_{5} + 98$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$2 + 4 T + 6 T^{2} + 4 T^{3} + T^{4}$$
$5$ $$98 + 84 T + 34 T^{2} + 8 T^{3} + T^{4}$$
$7$ $$( 2 + 2 T + T^{2} )^{2}$$
$11$ $$2 + 12 T + 22 T^{2} + 4 T^{3} + T^{4}$$
$13$ $$2 + 4 T + 2 T^{2} + T^{4}$$
$17$ $$( 8 + T^{2} )^{2}$$
$19$ $$578 + 340 T + 86 T^{2} + 12 T^{3} + T^{4}$$
$23$ $$4 + 24 T + 72 T^{2} + 12 T^{3} + T^{4}$$
$29$ $$98 + 84 T + 34 T^{2} + 8 T^{3} + T^{4}$$
$31$ $$( 4 + T )^{4}$$
$37$ $$2 + 4 T + 2 T^{2} + T^{4}$$
$41$ $$4 - 24 T + 72 T^{2} - 12 T^{3} + T^{4}$$
$43$ $$1922 + 124 T + 38 T^{2} + 12 T^{3} + T^{4}$$
$47$ $$16 + 136 T^{2} + T^{4}$$
$53$ $$98 + 84 T + 82 T^{2} + 16 T^{3} + T^{4}$$
$59$ $$1058 - 276 T + 22 T^{2} - 4 T^{3} + T^{4}$$
$61$ $$2 + 4 T + 2 T^{2} + T^{4}$$
$67$ $$578 + 340 T + 86 T^{2} + 12 T^{3} + T^{4}$$
$71$ $$4 - 24 T + 72 T^{2} - 12 T^{3} + T^{4}$$
$73$ $$( 98 + 14 T + T^{2} )^{2}$$
$79$ $$( 36 + T^{2} )^{2}$$
$83$ $$1058 + 276 T + 22 T^{2} + 4 T^{3} + T^{4}$$
$89$ $$2116 - 552 T + 72 T^{2} + 12 T^{3} + T^{4}$$
$97$ $$( 28 + 20 T + T^{2} )^{2}$$