# Properties

 Label 256.2.b.c Level $256$ Weight $2$ Character orbit 256.b Analytic conductor $2.044$ 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] = [256,2,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, 2, names="a")

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

chi := DirichletCharacter("256.129");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$2.04417029174$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 128) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} - \beta q^{5} + 4 q^{7} - q^{9} +O(q^{10})$$ q + b * q^3 - b * q^5 + 4 * q^7 - q^9 $$q + \beta q^{3} - \beta q^{5} + 4 q^{7} - q^{9} + \beta q^{11} + \beta q^{13} + 4 q^{15} - 2 q^{17} + \beta q^{19} + 4 \beta q^{21} - 4 q^{23} + q^{25} + 2 \beta q^{27} - 3 \beta q^{29} - 4 q^{33} - 4 \beta q^{35} - 5 \beta q^{37} - 4 q^{39} + 6 q^{41} - 3 \beta q^{43} + \beta q^{45} - 8 q^{47} + 9 q^{49} - 2 \beta q^{51} + 3 \beta q^{53} + 4 q^{55} - 4 q^{57} - 7 \beta q^{59} + \beta q^{61} - 4 q^{63} + 4 q^{65} + 5 \beta q^{67} - 4 \beta q^{69} - 12 q^{71} - 14 q^{73} + \beta q^{75} + 4 \beta q^{77} - 8 q^{79} - 11 q^{81} - 3 \beta q^{83} + 2 \beta q^{85} + 12 q^{87} + 2 q^{89} + 4 \beta q^{91} + 4 q^{95} - 2 q^{97} - \beta q^{99} +O(q^{100})$$ q + b * q^3 - b * q^5 + 4 * q^7 - q^9 + b * q^11 + b * q^13 + 4 * q^15 - 2 * q^17 + b * q^19 + 4*b * q^21 - 4 * q^23 + q^25 + 2*b * q^27 - 3*b * q^29 - 4 * q^33 - 4*b * q^35 - 5*b * q^37 - 4 * q^39 + 6 * q^41 - 3*b * q^43 + b * q^45 - 8 * q^47 + 9 * q^49 - 2*b * q^51 + 3*b * q^53 + 4 * q^55 - 4 * q^57 - 7*b * q^59 + b * q^61 - 4 * q^63 + 4 * q^65 + 5*b * q^67 - 4*b * q^69 - 12 * q^71 - 14 * q^73 + b * q^75 + 4*b * q^77 - 8 * q^79 - 11 * q^81 - 3*b * q^83 + 2*b * q^85 + 12 * q^87 + 2 * q^89 + 4*b * q^91 + 4 * q^95 - 2 * q^97 - b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{7} - 2 q^{9}+O(q^{10})$$ 2 * q + 8 * q^7 - 2 * q^9 $$2 q + 8 q^{7} - 2 q^{9} + 8 q^{15} - 4 q^{17} - 8 q^{23} + 2 q^{25} - 8 q^{33} - 8 q^{39} + 12 q^{41} - 16 q^{47} + 18 q^{49} + 8 q^{55} - 8 q^{57} - 8 q^{63} + 8 q^{65} - 24 q^{71} - 28 q^{73} - 16 q^{79} - 22 q^{81} + 24 q^{87} + 4 q^{89} + 8 q^{95} - 4 q^{97}+O(q^{100})$$ 2 * q + 8 * q^7 - 2 * q^9 + 8 * q^15 - 4 * q^17 - 8 * q^23 + 2 * q^25 - 8 * q^33 - 8 * q^39 + 12 * q^41 - 16 * q^47 + 18 * q^49 + 8 * q^55 - 8 * q^57 - 8 * q^63 + 8 * q^65 - 24 * q^71 - 28 * q^73 - 16 * q^79 - 22 * q^81 + 24 * q^87 + 4 * q^89 + 8 * q^95 - 4 * 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 2.00000i 0 2.00000i 0 4.00000 0 −1.00000 0
129.2 0 2.00000i 0 2.00000i 0 4.00000 0 −1.00000 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 256.2.b.c 2
3.b odd 2 1 2304.2.d.r 2
4.b odd 2 1 256.2.b.a 2
8.b even 2 1 inner 256.2.b.c 2
8.d odd 2 1 256.2.b.a 2
12.b even 2 1 2304.2.d.b 2
16.e even 4 1 128.2.a.a 1
16.e even 4 1 128.2.a.d yes 1
16.f odd 4 1 128.2.a.b yes 1
16.f odd 4 1 128.2.a.c yes 1
24.f even 2 1 2304.2.d.b 2
24.h odd 2 1 2304.2.d.r 2
32.g even 8 4 1024.2.e.i 4
32.h odd 8 4 1024.2.e.m 4
48.i odd 4 1 1152.2.a.c 1
48.i odd 4 1 1152.2.a.m 1
48.k even 4 1 1152.2.a.h 1
48.k even 4 1 1152.2.a.r 1
80.i odd 4 1 3200.2.c.e 2
80.i odd 4 1 3200.2.c.l 2
80.j even 4 1 3200.2.c.f 2
80.j even 4 1 3200.2.c.k 2
80.k odd 4 1 3200.2.a.e 1
80.k odd 4 1 3200.2.a.u 1
80.q even 4 1 3200.2.a.h 1
80.q even 4 1 3200.2.a.x 1
80.s even 4 1 3200.2.c.f 2
80.s even 4 1 3200.2.c.k 2
80.t odd 4 1 3200.2.c.e 2
80.t odd 4 1 3200.2.c.l 2
112.j even 4 1 6272.2.a.b 1
112.j even 4 1 6272.2.a.g 1
112.l odd 4 1 6272.2.a.a 1
112.l odd 4 1 6272.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.a.a 1 16.e even 4 1
128.2.a.b yes 1 16.f odd 4 1
128.2.a.c yes 1 16.f odd 4 1
128.2.a.d yes 1 16.e even 4 1
256.2.b.a 2 4.b odd 2 1
256.2.b.a 2 8.d odd 2 1
256.2.b.c 2 1.a even 1 1 trivial
256.2.b.c 2 8.b even 2 1 inner
1024.2.e.i 4 32.g even 8 4
1024.2.e.m 4 32.h odd 8 4
1152.2.a.c 1 48.i odd 4 1
1152.2.a.h 1 48.k even 4 1
1152.2.a.m 1 48.i odd 4 1
1152.2.a.r 1 48.k even 4 1
2304.2.d.b 2 12.b even 2 1
2304.2.d.b 2 24.f even 2 1
2304.2.d.r 2 3.b odd 2 1
2304.2.d.r 2 24.h odd 2 1
3200.2.a.e 1 80.k odd 4 1
3200.2.a.h 1 80.q even 4 1
3200.2.a.u 1 80.k odd 4 1
3200.2.a.x 1 80.q even 4 1
3200.2.c.e 2 80.i odd 4 1
3200.2.c.e 2 80.t odd 4 1
3200.2.c.f 2 80.j even 4 1
3200.2.c.f 2 80.s even 4 1
3200.2.c.k 2 80.j even 4 1
3200.2.c.k 2 80.s even 4 1
3200.2.c.l 2 80.i odd 4 1
3200.2.c.l 2 80.t odd 4 1
6272.2.a.a 1 112.l odd 4 1
6272.2.a.b 1 112.j even 4 1
6272.2.a.g 1 112.j even 4 1
6272.2.a.h 1 112.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(256, [\chi])$$:

 $$T_{3}^{2} + 4$$ T3^2 + 4 $$T_{7} - 4$$ T7 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2} + 4$$
$7$ $$(T - 4)^{2}$$
$11$ $$T^{2} + 4$$
$13$ $$T^{2} + 4$$
$17$ $$(T + 2)^{2}$$
$19$ $$T^{2} + 4$$
$23$ $$(T + 4)^{2}$$
$29$ $$T^{2} + 36$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$(T - 6)^{2}$$
$43$ $$T^{2} + 36$$
$47$ $$(T + 8)^{2}$$
$53$ $$T^{2} + 36$$
$59$ $$T^{2} + 196$$
$61$ $$T^{2} + 4$$
$67$ $$T^{2} + 100$$
$71$ $$(T + 12)^{2}$$
$73$ $$(T + 14)^{2}$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T - 2)^{2}$$
$97$ $$(T + 2)^{2}$$