# Properties

 Label 256.8.b.c Level $256$ Weight $8$ Character orbit 256.b Analytic conductor $79.971$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 256.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$79.9705665239$$ 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: $$2$$ Twist minimal: no (minimal twist has level 8) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 84 i q^{3} + 82 i q^{5} -456 q^{7} -4869 q^{9} +O(q^{10})$$ $$q + 84 i q^{3} + 82 i q^{5} -456 q^{7} -4869 q^{9} -2524 i q^{11} -10778 i q^{13} -6888 q^{15} -11150 q^{17} -4124 i q^{19} -38304 i q^{21} + 81704 q^{23} + 71401 q^{25} -225288 i q^{27} + 99798 i q^{29} + 40480 q^{31} + 212016 q^{33} -37392 i q^{35} + 419442 i q^{37} + 905352 q^{39} -141402 q^{41} -690428 i q^{43} -399258 i q^{45} + 682032 q^{47} -615607 q^{49} -936600 i q^{51} -1813118 i q^{53} + 206968 q^{55} + 346416 q^{57} -966028 i q^{59} + 1887670 i q^{61} + 2220264 q^{63} + 883796 q^{65} -2965868 i q^{67} + 6863136 i q^{69} -2548232 q^{71} + 1680326 q^{73} + 5997684 i q^{75} + 1150944 i q^{77} -4038064 q^{79} + 8275689 q^{81} + 5385764 i q^{83} -914300 i q^{85} -8383032 q^{87} + 6473046 q^{89} + 4914768 i q^{91} + 3400320 i q^{93} + 338168 q^{95} -6065758 q^{97} + 12289356 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 912q^{7} - 9738q^{9} + O(q^{10})$$ $$2q - 912q^{7} - 9738q^{9} - 13776q^{15} - 22300q^{17} + 163408q^{23} + 142802q^{25} + 80960q^{31} + 424032q^{33} + 1810704q^{39} - 282804q^{41} + 1364064q^{47} - 1231214q^{49} + 413936q^{55} + 692832q^{57} + 4440528q^{63} + 1767592q^{65} - 5096464q^{71} + 3360652q^{73} - 8076128q^{79} + 16551378q^{81} - 16766064q^{87} + 12946092q^{89} + 676336q^{95} - 12131516q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$5$$ $$255$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
129.1
 − 1.00000i 1.00000i
0 84.0000i 0 82.0000i 0 −456.000 0 −4869.00 0
129.2 0 84.0000i 0 82.0000i 0 −456.000 0 −4869.00 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
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 256.8.b.c 2
4.b odd 2 1 256.8.b.e 2
8.b even 2 1 inner 256.8.b.c 2
8.d odd 2 1 256.8.b.e 2
16.e even 4 1 16.8.a.c 1
16.e even 4 1 64.8.a.a 1
16.f odd 4 1 8.8.a.a 1
16.f odd 4 1 64.8.a.g 1
48.i odd 4 1 144.8.a.g 1
48.i odd 4 1 576.8.a.k 1
48.k even 4 1 72.8.a.d 1
48.k even 4 1 576.8.a.j 1
80.i odd 4 1 400.8.c.b 2
80.j even 4 1 200.8.c.a 2
80.k odd 4 1 200.8.a.i 1
80.q even 4 1 400.8.a.b 1
80.s even 4 1 200.8.c.a 2
80.t odd 4 1 400.8.c.b 2
112.j even 4 1 392.8.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.8.a.a 1 16.f odd 4 1
16.8.a.c 1 16.e even 4 1
64.8.a.a 1 16.e even 4 1
64.8.a.g 1 16.f odd 4 1
72.8.a.d 1 48.k even 4 1
144.8.a.g 1 48.i odd 4 1
200.8.a.i 1 80.k odd 4 1
200.8.c.a 2 80.j even 4 1
200.8.c.a 2 80.s even 4 1
256.8.b.c 2 1.a even 1 1 trivial
256.8.b.c 2 8.b even 2 1 inner
256.8.b.e 2 4.b odd 2 1
256.8.b.e 2 8.d odd 2 1
392.8.a.d 1 112.j even 4 1
400.8.a.b 1 80.q even 4 1
400.8.c.b 2 80.i odd 4 1
400.8.c.b 2 80.t odd 4 1
576.8.a.j 1 48.k even 4 1
576.8.a.k 1 48.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{8}^{\mathrm{new}}(256, [\chi])$$:

 $$T_{3}^{2} + 7056$$ $$T_{7} + 456$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$7056 + T^{2}$$
$5$ $$6724 + T^{2}$$
$7$ $$( 456 + T )^{2}$$
$11$ $$6370576 + T^{2}$$
$13$ $$116165284 + T^{2}$$
$17$ $$( 11150 + T )^{2}$$
$19$ $$17007376 + T^{2}$$
$23$ $$( -81704 + T )^{2}$$
$29$ $$9959640804 + T^{2}$$
$31$ $$( -40480 + T )^{2}$$
$37$ $$175931591364 + T^{2}$$
$41$ $$( 141402 + T )^{2}$$
$43$ $$476690823184 + T^{2}$$
$47$ $$( -682032 + T )^{2}$$
$53$ $$3287396881924 + T^{2}$$
$59$ $$933210096784 + T^{2}$$
$61$ $$3563298028900 + T^{2}$$
$67$ $$8796372993424 + T^{2}$$
$71$ $$( 2548232 + T )^{2}$$
$73$ $$( -1680326 + T )^{2}$$
$79$ $$( 4038064 + T )^{2}$$
$83$ $$29006453863696 + T^{2}$$
$89$ $$( -6473046 + T )^{2}$$
$97$ $$( 6065758 + T )^{2}$$