# Properties

 Label 256.4.b.e Level $256$ Weight $4$ Character orbit 256.b Analytic conductor $15.104$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,4,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, 4, names="a")

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

chi := DirichletCharacter("256.129");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$15.1044889615$$ 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 32) 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.

 $$f(q)$$ $$=$$ $$q + 4 \beta q^{3} - 5 \beta q^{5} + 16 q^{7} - 37 q^{9} +O(q^{10})$$ q + 4*b * q^3 - 5*b * q^5 + 16 * q^7 - 37 * q^9 $$q + 4 \beta q^{3} - 5 \beta q^{5} + 16 q^{7} - 37 q^{9} + 20 \beta q^{11} + 25 \beta q^{13} + 80 q^{15} - 30 q^{17} + 20 \beta q^{19} + 64 \beta q^{21} + 48 q^{23} + 25 q^{25} - 40 \beta q^{27} + 17 \beta q^{29} - 320 q^{31} - 320 q^{33} - 80 \beta q^{35} + 155 \beta q^{37} - 400 q^{39} - 410 q^{41} - 76 \beta q^{43} + 185 \beta q^{45} + 416 q^{47} - 87 q^{49} - 120 \beta q^{51} - 205 \beta q^{53} + 400 q^{55} - 320 q^{57} + 100 \beta q^{59} - 15 \beta q^{61} - 592 q^{63} + 500 q^{65} + 388 \beta q^{67} + 192 \beta q^{69} + 400 q^{71} + 630 q^{73} + 100 \beta q^{75} + 320 \beta q^{77} + 1120 q^{79} - 359 q^{81} + 276 \beta q^{83} + 150 \beta q^{85} - 272 q^{87} + 326 q^{89} + 400 \beta q^{91} - 1280 \beta q^{93} + 400 q^{95} - 110 q^{97} - 740 \beta q^{99} +O(q^{100})$$ q + 4*b * q^3 - 5*b * q^5 + 16 * q^7 - 37 * q^9 + 20*b * q^11 + 25*b * q^13 + 80 * q^15 - 30 * q^17 + 20*b * q^19 + 64*b * q^21 + 48 * q^23 + 25 * q^25 - 40*b * q^27 + 17*b * q^29 - 320 * q^31 - 320 * q^33 - 80*b * q^35 + 155*b * q^37 - 400 * q^39 - 410 * q^41 - 76*b * q^43 + 185*b * q^45 + 416 * q^47 - 87 * q^49 - 120*b * q^51 - 205*b * q^53 + 400 * q^55 - 320 * q^57 + 100*b * q^59 - 15*b * q^61 - 592 * q^63 + 500 * q^65 + 388*b * q^67 + 192*b * q^69 + 400 * q^71 + 630 * q^73 + 100*b * q^75 + 320*b * q^77 + 1120 * q^79 - 359 * q^81 + 276*b * q^83 + 150*b * q^85 - 272 * q^87 + 326 * q^89 + 400*b * q^91 - 1280*b * q^93 + 400 * q^95 - 110 * q^97 - 740*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 32 q^{7} - 74 q^{9}+O(q^{10})$$ 2 * q + 32 * q^7 - 74 * q^9 $$2 q + 32 q^{7} - 74 q^{9} + 160 q^{15} - 60 q^{17} + 96 q^{23} + 50 q^{25} - 640 q^{31} - 640 q^{33} - 800 q^{39} - 820 q^{41} + 832 q^{47} - 174 q^{49} + 800 q^{55} - 640 q^{57} - 1184 q^{63} + 1000 q^{65} + 800 q^{71} + 1260 q^{73} + 2240 q^{79} - 718 q^{81} - 544 q^{87} + 652 q^{89} + 800 q^{95} - 220 q^{97}+O(q^{100})$$ 2 * q + 32 * q^7 - 74 * q^9 + 160 * q^15 - 60 * q^17 + 96 * q^23 + 50 * q^25 - 640 * q^31 - 640 * q^33 - 800 * q^39 - 820 * q^41 + 832 * q^47 - 174 * q^49 + 800 * q^55 - 640 * q^57 - 1184 * q^63 + 1000 * q^65 + 800 * q^71 + 1260 * q^73 + 2240 * q^79 - 718 * q^81 - 544 * q^87 + 652 * q^89 + 800 * q^95 - 220 * 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 8.00000i 0 10.0000i 0 16.0000 0 −37.0000 0
129.2 0 8.00000i 0 10.0000i 0 16.0000 0 −37.0000 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 256.4.b.e 2
4.b odd 2 1 256.4.b.c 2
8.b even 2 1 inner 256.4.b.e 2
8.d odd 2 1 256.4.b.c 2
16.e even 4 1 32.4.a.a 1
16.e even 4 1 64.4.a.e 1
16.f odd 4 1 32.4.a.c yes 1
16.f odd 4 1 64.4.a.a 1
48.i odd 4 1 288.4.a.h 1
48.i odd 4 1 576.4.a.g 1
48.k even 4 1 288.4.a.i 1
48.k even 4 1 576.4.a.h 1
80.i odd 4 1 800.4.c.b 2
80.j even 4 1 800.4.c.a 2
80.k odd 4 1 800.4.a.a 1
80.k odd 4 1 1600.4.a.bw 1
80.q even 4 1 800.4.a.k 1
80.q even 4 1 1600.4.a.e 1
80.s even 4 1 800.4.c.a 2
80.t odd 4 1 800.4.c.b 2
112.j even 4 1 1568.4.a.c 1
112.l odd 4 1 1568.4.a.o 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.4.a.a 1 16.e even 4 1
32.4.a.c yes 1 16.f odd 4 1
64.4.a.a 1 16.f odd 4 1
64.4.a.e 1 16.e even 4 1
256.4.b.c 2 4.b odd 2 1
256.4.b.c 2 8.d odd 2 1
256.4.b.e 2 1.a even 1 1 trivial
256.4.b.e 2 8.b even 2 1 inner
288.4.a.h 1 48.i odd 4 1
288.4.a.i 1 48.k even 4 1
576.4.a.g 1 48.i odd 4 1
576.4.a.h 1 48.k even 4 1
800.4.a.a 1 80.k odd 4 1
800.4.a.k 1 80.q even 4 1
800.4.c.a 2 80.j even 4 1
800.4.c.a 2 80.s even 4 1
800.4.c.b 2 80.i odd 4 1
800.4.c.b 2 80.t odd 4 1
1568.4.a.c 1 112.j even 4 1
1568.4.a.o 1 112.l odd 4 1
1600.4.a.e 1 80.q even 4 1
1600.4.a.bw 1 80.k odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(256, [\chi])$$:

 $$T_{3}^{2} + 64$$ T3^2 + 64 $$T_{7} - 16$$ T7 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 64$$
$5$ $$T^{2} + 100$$
$7$ $$(T - 16)^{2}$$
$11$ $$T^{2} + 1600$$
$13$ $$T^{2} + 2500$$
$17$ $$(T + 30)^{2}$$
$19$ $$T^{2} + 1600$$
$23$ $$(T - 48)^{2}$$
$29$ $$T^{2} + 1156$$
$31$ $$(T + 320)^{2}$$
$37$ $$T^{2} + 96100$$
$41$ $$(T + 410)^{2}$$
$43$ $$T^{2} + 23104$$
$47$ $$(T - 416)^{2}$$
$53$ $$T^{2} + 168100$$
$59$ $$T^{2} + 40000$$
$61$ $$T^{2} + 900$$
$67$ $$T^{2} + 602176$$
$71$ $$(T - 400)^{2}$$
$73$ $$(T - 630)^{2}$$
$79$ $$(T - 1120)^{2}$$
$83$ $$T^{2} + 304704$$
$89$ $$(T - 326)^{2}$$
$97$ $$(T + 110)^{2}$$