# Properties

 Label 256.2.e.b Level $256$ Weight $2$ Character orbit 256.e Analytic conductor $2.044$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 256.e (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.04417029174$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{8}$$ 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 $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{3} + ( 1 + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{5} + ( 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{7} + ( 2 - 4 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{3} + ( 1 + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{5} + ( 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{7} + ( 2 - 4 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{9} + ( 3 \zeta_{24} + \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{11} + ( 3 - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} + 3 \zeta_{24}^{6} ) q^{13} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{15} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{17} + ( -\zeta_{24} - 3 \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{19} + ( 2 - 2 \zeta_{24}^{6} ) q^{21} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{23} -\zeta_{24}^{6} q^{25} + ( -4 \zeta_{24} + 4 \zeta_{24}^{5} ) q^{27} + ( -1 + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{29} + ( 4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} ) q^{31} + ( -4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{33} + ( 2 \zeta_{24} + 6 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{35} + ( -3 - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + 3 \zeta_{24}^{6} ) q^{37} + ( -8 \zeta_{24} + 8 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{39} + ( -4 + 8 \zeta_{24}^{4} ) q^{41} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{43} + ( -7 + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} - 7 \zeta_{24}^{6} ) q^{45} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{47} + ( -1 - 8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{49} + ( 2 \zeta_{24} - 6 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{51} + ( -7 - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + 7 \zeta_{24}^{6} ) q^{53} + ( -6 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{55} + ( -2 + 4 \zeta_{24}^{4} ) q^{57} + ( -9 \zeta_{24} + \zeta_{24}^{3} + 9 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{59} + ( -1 - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{61} + ( 6 \zeta_{24} + 6 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{63} + ( -6 + 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{65} + ( 3 \zeta_{24} + \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{67} + ( 2 - 8 \zeta_{24}^{2} + 8 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{69} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{71} + ( -2 + 4 \zeta_{24}^{4} - 6 \zeta_{24}^{6} ) q^{73} + ( \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{75} + ( 2 + 8 \zeta_{24}^{2} + 8 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{77} + ( 8 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{79} + ( -1 - 4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{81} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{83} + ( 6 - 6 \zeta_{24}^{6} ) q^{85} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{87} + ( 2 - 4 \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{89} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{91} + ( 8 - 8 \zeta_{24}^{2} - 8 \zeta_{24}^{4} + 8 \zeta_{24}^{6} ) q^{93} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{7} ) q^{95} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{97} + ( -5 \zeta_{24} - 3 \zeta_{24}^{3} - 5 \zeta_{24}^{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 16q^{13} + 16q^{21} - 16q^{37} - 48q^{45} - 8q^{49} - 48q^{53} - 16q^{61} - 48q^{65} + 48q^{69} + 48q^{77} - 8q^{81} + 48q^{85} + 32q^{93} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$255$$ $$\chi(n)$$ $$-\zeta_{24}$$ $$1$$

## 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
 0.258819 + 0.965926i 0.965926 + 0.258819i −0.965926 − 0.258819i −0.258819 − 0.965926i 0.258819 − 0.965926i 0.965926 − 0.258819i −0.965926 + 0.258819i −0.258819 + 0.965926i
0 −1.93185 1.93185i 0 −1.73205 + 1.73205i 0 1.03528i 0 4.46410i 0
65.2 0 −0.517638 0.517638i 0 1.73205 1.73205i 0 3.86370i 0 2.46410i 0
65.3 0 0.517638 + 0.517638i 0 1.73205 1.73205i 0 3.86370i 0 2.46410i 0
65.4 0 1.93185 + 1.93185i 0 −1.73205 + 1.73205i 0 1.03528i 0 4.46410i 0
193.1 0 −1.93185 + 1.93185i 0 −1.73205 1.73205i 0 1.03528i 0 4.46410i 0
193.2 0 −0.517638 + 0.517638i 0 1.73205 + 1.73205i 0 3.86370i 0 2.46410i 0
193.3 0 0.517638 0.517638i 0 1.73205 + 1.73205i 0 3.86370i 0 2.46410i 0
193.4 0 1.93185 1.93185i 0 −1.73205 1.73205i 0 1.03528i 0 4.46410i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 193.4 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 256.2.e.b yes 8
3.b odd 2 1 2304.2.k.k 8
4.b odd 2 1 inner 256.2.e.b yes 8
8.b even 2 1 256.2.e.a 8
8.d odd 2 1 256.2.e.a 8
12.b even 2 1 2304.2.k.k 8
16.e even 4 1 256.2.e.a 8
16.e even 4 1 inner 256.2.e.b yes 8
16.f odd 4 1 256.2.e.a 8
16.f odd 4 1 inner 256.2.e.b yes 8
24.f even 2 1 2304.2.k.f 8
24.h odd 2 1 2304.2.k.f 8
32.g even 8 1 1024.2.a.g 4
32.g even 8 1 1024.2.a.j 4
32.g even 8 2 1024.2.b.h 8
32.h odd 8 1 1024.2.a.g 4
32.h odd 8 1 1024.2.a.j 4
32.h odd 8 2 1024.2.b.h 8
48.i odd 4 1 2304.2.k.f 8
48.i odd 4 1 2304.2.k.k 8
48.k even 4 1 2304.2.k.f 8
48.k even 4 1 2304.2.k.k 8
96.o even 8 1 9216.2.a.bb 4
96.o even 8 1 9216.2.a.bk 4
96.p odd 8 1 9216.2.a.bb 4
96.p odd 8 1 9216.2.a.bk 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
256.2.e.a 8 8.b even 2 1
256.2.e.a 8 8.d odd 2 1
256.2.e.a 8 16.e even 4 1
256.2.e.a 8 16.f odd 4 1
256.2.e.b yes 8 1.a even 1 1 trivial
256.2.e.b yes 8 4.b odd 2 1 inner
256.2.e.b yes 8 16.e even 4 1 inner
256.2.e.b yes 8 16.f odd 4 1 inner
1024.2.a.g 4 32.g even 8 1
1024.2.a.g 4 32.h odd 8 1
1024.2.a.j 4 32.g even 8 1
1024.2.a.j 4 32.h odd 8 1
1024.2.b.h 8 32.g even 8 2
1024.2.b.h 8 32.h odd 8 2
2304.2.k.f 8 24.f even 2 1
2304.2.k.f 8 24.h odd 2 1
2304.2.k.f 8 48.i odd 4 1
2304.2.k.f 8 48.k even 4 1
2304.2.k.k 8 3.b odd 2 1
2304.2.k.k 8 12.b even 2 1
2304.2.k.k 8 48.i odd 4 1
2304.2.k.k 8 48.k even 4 1
9216.2.a.bb 4 96.o even 8 1
9216.2.a.bb 4 96.p odd 8 1
9216.2.a.bk 4 96.o even 8 1
9216.2.a.bk 4 96.p odd 8 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{4} - 8 T_{13}^{3} + 32 T_{13}^{2} - 16 T_{13} + 4$$ acting on $$S_{2}^{\mathrm{new}}(256, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$16 + 56 T^{4} + T^{8}$$
$5$ $$( 36 + T^{4} )^{2}$$
$7$ $$( 16 + 16 T^{2} + T^{4} )^{2}$$
$11$ $$1296 + 504 T^{4} + T^{8}$$
$13$ $$( 4 - 16 T + 32 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$17$ $$( -12 + T^{2} )^{4}$$
$19$ $$1296 + 504 T^{4} + T^{8}$$
$23$ $$( 144 + 48 T^{2} + T^{4} )^{2}$$
$29$ $$( 36 + T^{4} )^{2}$$
$31$ $$( -32 + T^{2} )^{4}$$
$37$ $$( 4 + 16 T + 32 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$41$ $$( 48 + T^{2} )^{4}$$
$43$ $$104976 + 4536 T^{4} + T^{8}$$
$47$ $$( -96 + T^{2} )^{4}$$
$53$ $$( 4356 + 1584 T + 288 T^{2} + 24 T^{3} + T^{4} )^{2}$$
$59$ $$37015056 + 16056 T^{4} + T^{8}$$
$61$ $$( 4 + 16 T + 32 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$67$ $$456976 + 1784 T^{4} + T^{8}$$
$71$ $$( 144 + 48 T^{2} + T^{4} )^{2}$$
$73$ $$( 576 + 96 T^{2} + T^{4} )^{2}$$
$79$ $$( 4096 - 256 T^{2} + T^{4} )^{2}$$
$83$ $$104976 + 4536 T^{4} + T^{8}$$
$89$ $$( 576 + 96 T^{2} + T^{4} )^{2}$$
$97$ $$( -12 + T^{2} )^{4}$$