# Properties

 Label 256.8.b.c Level $256$ Weight $8$ Character orbit 256.b Analytic conductor $79.971$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,8,Mod(129,256)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(256, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("256.129");

S:= CuspForms(chi, 8);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$79.9705665239$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 42 \beta q^{3} + 41 \beta q^{5} - 456 q^{7} - 4869 q^{9}+O(q^{10})$$ q + 42*b * q^3 + 41*b * q^5 - 456 * q^7 - 4869 * q^9 $$q + 42 \beta q^{3} + 41 \beta q^{5} - 456 q^{7} - 4869 q^{9} - 1262 \beta q^{11} - 5389 \beta q^{13} - 6888 q^{15} - 11150 q^{17} - 2062 \beta q^{19} - 19152 \beta q^{21} + 81704 q^{23} + 71401 q^{25} - 112644 \beta q^{27} + 49899 \beta q^{29} + 40480 q^{31} + 212016 q^{33} - 18696 \beta q^{35} + 209721 \beta q^{37} + 905352 q^{39} - 141402 q^{41} - 345214 \beta q^{43} - 199629 \beta q^{45} + 682032 q^{47} - 615607 q^{49} - 468300 \beta q^{51} - 906559 \beta q^{53} + 206968 q^{55} + 346416 q^{57} - 483014 \beta q^{59} + 943835 \beta q^{61} + 2220264 q^{63} + 883796 q^{65} - 1482934 \beta q^{67} + 3431568 \beta q^{69} - 2548232 q^{71} + 1680326 q^{73} + 2998842 \beta q^{75} + 575472 \beta q^{77} - 4038064 q^{79} + 8275689 q^{81} + 2692882 \beta q^{83} - 457150 \beta q^{85} - 8383032 q^{87} + 6473046 q^{89} + 2457384 \beta q^{91} + 1700160 \beta q^{93} + 338168 q^{95} - 6065758 q^{97} + 6144678 \beta q^{99} +O(q^{100})$$ q + 42*b * q^3 + 41*b * q^5 - 456 * q^7 - 4869 * q^9 - 1262*b * q^11 - 5389*b * q^13 - 6888 * q^15 - 11150 * q^17 - 2062*b * q^19 - 19152*b * q^21 + 81704 * q^23 + 71401 * q^25 - 112644*b * q^27 + 49899*b * q^29 + 40480 * q^31 + 212016 * q^33 - 18696*b * q^35 + 209721*b * q^37 + 905352 * q^39 - 141402 * q^41 - 345214*b * q^43 - 199629*b * q^45 + 682032 * q^47 - 615607 * q^49 - 468300*b * q^51 - 906559*b * q^53 + 206968 * q^55 + 346416 * q^57 - 483014*b * q^59 + 943835*b * q^61 + 2220264 * q^63 + 883796 * q^65 - 1482934*b * q^67 + 3431568*b * q^69 - 2548232 * q^71 + 1680326 * q^73 + 2998842*b * q^75 + 575472*b * q^77 - 4038064 * q^79 + 8275689 * q^81 + 2692882*b * q^83 - 457150*b * q^85 - 8383032 * q^87 + 6473046 * q^89 + 2457384*b * q^91 + 1700160*b * q^93 + 338168 * q^95 - 6065758 * q^97 + 6144678*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 912 q^{7} - 9738 q^{9}+O(q^{10})$$ 2 * q - 912 * q^7 - 9738 * q^9 $$2 q - 912 q^{7} - 9738 q^{9} - 13776 q^{15} - 22300 q^{17} + 163408 q^{23} + 142802 q^{25} + 80960 q^{31} + 424032 q^{33} + 1810704 q^{39} - 282804 q^{41} + 1364064 q^{47} - 1231214 q^{49} + 413936 q^{55} + 692832 q^{57} + 4440528 q^{63} + 1767592 q^{65} - 5096464 q^{71} + 3360652 q^{73} - 8076128 q^{79} + 16551378 q^{81} - 16766064 q^{87} + 12946092 q^{89} + 676336 q^{95} - 12131516 q^{97}+O(q^{100})$$ 2 * q - 912 * q^7 - 9738 * q^9 - 13776 * q^15 - 22300 * q^17 + 163408 * q^23 + 142802 * q^25 + 80960 * q^31 + 424032 * q^33 + 1810704 * q^39 - 282804 * q^41 + 1364064 * q^47 - 1231214 * q^49 + 413936 * q^55 + 692832 * q^57 + 4440528 * q^63 + 1767592 * q^65 - 5096464 * q^71 + 3360652 * q^73 - 8076128 * q^79 + 16551378 * q^81 - 16766064 * q^87 + 12946092 * q^89 + 676336 * q^95 - 12131516 * q^97

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 $$S_{8}^{\mathrm{new}}(256, [\chi])$$:

 $$T_{3}^{2} + 7056$$ T3^2 + 7056 $$T_{7} + 456$$ T7 + 456

## Hecke characteristic polynomials

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