# Properties

 Label 256.6.b.b Level $256$ Weight $6$ Character orbit 256.b Analytic conductor $41.058$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 256.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$41.0582578721$$ 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 32) 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.

 $$f(q)$$ $$=$$ $$q + 8 i q^{3} -14 i q^{5} -208 q^{7} + 179 q^{9} +O(q^{10})$$ $$q + 8 i q^{3} -14 i q^{5} -208 q^{7} + 179 q^{9} -536 i q^{11} + 694 i q^{13} + 112 q^{15} -1278 q^{17} -1112 i q^{19} -1664 i q^{21} + 3216 q^{23} + 2929 q^{25} + 3376 i q^{27} + 2918 i q^{29} + 2624 q^{31} + 4288 q^{33} + 2912 i q^{35} + 9458 i q^{37} -5552 q^{39} -170 q^{41} -19928 i q^{43} -2506 i q^{45} -32 q^{47} + 26457 q^{49} -10224 i q^{51} + 22178 i q^{53} -7504 q^{55} + 8896 q^{57} + 41480 i q^{59} + 15462 i q^{61} -37232 q^{63} + 9716 q^{65} + 20744 i q^{67} + 25728 i q^{69} + 28592 q^{71} + 53670 q^{73} + 23432 i q^{75} + 111488 i q^{77} + 69152 q^{79} + 16489 q^{81} + 37800 i q^{83} + 17892 i q^{85} -23344 q^{87} + 126806 q^{89} -144352 i q^{91} + 20992 i q^{93} -15568 q^{95} + 62290 q^{97} -95944 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 416q^{7} + 358q^{9} + O(q^{10})$$ $$2q - 416q^{7} + 358q^{9} + 224q^{15} - 2556q^{17} + 6432q^{23} + 5858q^{25} + 5248q^{31} + 8576q^{33} - 11104q^{39} - 340q^{41} - 64q^{47} + 52914q^{49} - 15008q^{55} + 17792q^{57} - 74464q^{63} + 19432q^{65} + 57184q^{71} + 107340q^{73} + 138304q^{79} + 32978q^{81} - 46688q^{87} + 253612q^{89} - 31136q^{95} + 124580q^{97} + O(q^{100})$$

## 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 8.00000i 0 14.0000i 0 −208.000 0 179.000 0
129.2 0 8.00000i 0 14.0000i 0 −208.000 0 179.000 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 256.6.b.b 2
4.b odd 2 1 256.6.b.h 2
8.b even 2 1 inner 256.6.b.b 2
8.d odd 2 1 256.6.b.h 2
16.e even 4 1 32.6.a.c yes 1
16.e even 4 1 64.6.a.c 1
16.f odd 4 1 32.6.a.a 1
16.f odd 4 1 64.6.a.e 1
48.i odd 4 1 288.6.a.e 1
48.i odd 4 1 576.6.a.v 1
48.k even 4 1 288.6.a.d 1
48.k even 4 1 576.6.a.u 1
80.i odd 4 1 800.6.c.b 2
80.j even 4 1 800.6.c.a 2
80.k odd 4 1 800.6.a.e 1
80.q even 4 1 800.6.a.a 1
80.s even 4 1 800.6.c.a 2
80.t odd 4 1 800.6.c.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.6.a.a 1 16.f odd 4 1
32.6.a.c yes 1 16.e even 4 1
64.6.a.c 1 16.e even 4 1
64.6.a.e 1 16.f odd 4 1
256.6.b.b 2 1.a even 1 1 trivial
256.6.b.b 2 8.b even 2 1 inner
256.6.b.h 2 4.b odd 2 1
256.6.b.h 2 8.d odd 2 1
288.6.a.d 1 48.k even 4 1
288.6.a.e 1 48.i odd 4 1
576.6.a.u 1 48.k even 4 1
576.6.a.v 1 48.i odd 4 1
800.6.a.a 1 80.q even 4 1
800.6.a.e 1 80.k odd 4 1
800.6.c.a 2 80.j even 4 1
800.6.c.a 2 80.s even 4 1
800.6.c.b 2 80.i odd 4 1
800.6.c.b 2 80.t odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(256, [\chi])$$:

 $$T_{3}^{2} + 64$$ $$T_{7} + 208$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$64 + T^{2}$$
$5$ $$196 + T^{2}$$
$7$ $$( 208 + T )^{2}$$
$11$ $$287296 + T^{2}$$
$13$ $$481636 + T^{2}$$
$17$ $$( 1278 + T )^{2}$$
$19$ $$1236544 + T^{2}$$
$23$ $$( -3216 + T )^{2}$$
$29$ $$8514724 + T^{2}$$
$31$ $$( -2624 + T )^{2}$$
$37$ $$89453764 + T^{2}$$
$41$ $$( 170 + T )^{2}$$
$43$ $$397125184 + T^{2}$$
$47$ $$( 32 + T )^{2}$$
$53$ $$491863684 + T^{2}$$
$59$ $$1720590400 + T^{2}$$
$61$ $$239073444 + T^{2}$$
$67$ $$430313536 + T^{2}$$
$71$ $$( -28592 + T )^{2}$$
$73$ $$( -53670 + T )^{2}$$
$79$ $$( -69152 + T )^{2}$$
$83$ $$1428840000 + T^{2}$$
$89$ $$( -126806 + T )^{2}$$
$97$ $$( -62290 + T )^{2}$$