# Properties

 Label 1024.2.b.b Level $1024$ Weight $2$ Character orbit 1024.b Analytic conductor $8.177$ 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] = [1024,2,Mod(513,1024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1024, 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("1024.513");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1024 = 2^{10}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1024.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: $$8.17668116698$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 16) 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 $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} - \beta q^{5} - 2 q^{7} + q^{9} +O(q^{10})$$ q + b * q^3 - b * q^5 - 2 * q^7 + q^9 $$q + \beta q^{3} - \beta q^{5} - 2 q^{7} + q^{9} + \beta q^{11} + \beta q^{13} + 2 q^{15} + 2 q^{17} + 3 \beta q^{19} - 2 \beta q^{21} - 6 q^{23} + 3 q^{25} + 4 \beta q^{27} + 3 \beta q^{29} + 8 q^{31} - 2 q^{33} + 2 \beta q^{35} + 3 \beta q^{37} - 2 q^{39} + 5 \beta q^{43} - \beta q^{45} + 8 q^{47} - 3 q^{49} + 2 \beta q^{51} + 5 \beta q^{53} + 2 q^{55} - 6 q^{57} + 3 \beta q^{59} - 9 \beta q^{61} - 2 q^{63} + 2 q^{65} + 5 \beta q^{67} - 6 \beta q^{69} - 10 q^{71} - 4 q^{73} + 3 \beta q^{75} - 2 \beta q^{77} - 5 q^{81} - \beta q^{83} - 2 \beta q^{85} - 6 q^{87} - 4 q^{89} - 2 \beta q^{91} + 8 \beta q^{93} + 6 q^{95} - 2 q^{97} + \beta q^{99} +O(q^{100})$$ q + b * q^3 - b * q^5 - 2 * q^7 + q^9 + b * q^11 + b * q^13 + 2 * q^15 + 2 * q^17 + 3*b * q^19 - 2*b * q^21 - 6 * q^23 + 3 * q^25 + 4*b * q^27 + 3*b * q^29 + 8 * q^31 - 2 * q^33 + 2*b * q^35 + 3*b * q^37 - 2 * q^39 + 5*b * q^43 - b * q^45 + 8 * q^47 - 3 * q^49 + 2*b * q^51 + 5*b * q^53 + 2 * q^55 - 6 * q^57 + 3*b * q^59 - 9*b * q^61 - 2 * q^63 + 2 * q^65 + 5*b * q^67 - 6*b * q^69 - 10 * q^71 - 4 * q^73 + 3*b * q^75 - 2*b * q^77 - 5 * q^81 - b * q^83 - 2*b * q^85 - 6 * q^87 - 4 * q^89 - 2*b * q^91 + 8*b * q^93 + 6 * q^95 - 2 * q^97 + b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 4 * q^7 + 2 * q^9 $$2 q - 4 q^{7} + 2 q^{9} + 4 q^{15} + 4 q^{17} - 12 q^{23} + 6 q^{25} + 16 q^{31} - 4 q^{33} - 4 q^{39} + 16 q^{47} - 6 q^{49} + 4 q^{55} - 12 q^{57} - 4 q^{63} + 4 q^{65} - 20 q^{71} - 8 q^{73} - 10 q^{81} - 12 q^{87} - 8 q^{89} + 12 q^{95} - 4 q^{97}+O(q^{100})$$ 2 * q - 4 * q^7 + 2 * q^9 + 4 * q^15 + 4 * q^17 - 12 * q^23 + 6 * q^25 + 16 * q^31 - 4 * q^33 - 4 * q^39 + 16 * q^47 - 6 * q^49 + 4 * q^55 - 12 * q^57 - 4 * q^63 + 4 * q^65 - 20 * q^71 - 8 * q^73 - 10 * q^81 - 12 * q^87 - 8 * q^89 + 12 * q^95 - 4 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$1023$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
513.1
 − 1.41421i 1.41421i
0 1.41421i 0 1.41421i 0 −2.00000 0 1.00000 0
513.2 0 1.41421i 0 1.41421i 0 −2.00000 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 1024.2.b.b 2
4.b odd 2 1 1024.2.b.e 2
8.b even 2 1 inner 1024.2.b.b 2
8.d odd 2 1 1024.2.b.e 2
16.e even 4 2 1024.2.a.e 2
16.f odd 4 2 1024.2.a.b 2
32.g even 8 2 64.2.e.a 2
32.g even 8 2 128.2.e.a 2
32.h odd 8 2 16.2.e.a 2
32.h odd 8 2 128.2.e.b 2
48.i odd 4 2 9216.2.a.s 2
48.k even 4 2 9216.2.a.d 2
96.o even 8 2 144.2.k.a 2
96.o even 8 2 1152.2.k.b 2
96.p odd 8 2 576.2.k.a 2
96.p odd 8 2 1152.2.k.a 2
160.u even 8 2 400.2.q.b 2
160.v odd 8 2 1600.2.q.b 2
160.y odd 8 2 400.2.l.c 2
160.z even 8 2 1600.2.l.a 2
160.ba even 8 2 400.2.q.a 2
160.bb odd 8 2 1600.2.q.a 2
224.x even 8 2 784.2.m.b 2
224.be even 24 4 784.2.x.c 4
224.bf odd 24 4 784.2.x.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.2.e.a 2 32.h odd 8 2
64.2.e.a 2 32.g even 8 2
128.2.e.a 2 32.g even 8 2
128.2.e.b 2 32.h odd 8 2
144.2.k.a 2 96.o even 8 2
400.2.l.c 2 160.y odd 8 2
400.2.q.a 2 160.ba even 8 2
400.2.q.b 2 160.u even 8 2
576.2.k.a 2 96.p odd 8 2
784.2.m.b 2 224.x even 8 2
784.2.x.c 4 224.be even 24 4
784.2.x.f 4 224.bf odd 24 4
1024.2.a.b 2 16.f odd 4 2
1024.2.a.e 2 16.e even 4 2
1024.2.b.b 2 1.a even 1 1 trivial
1024.2.b.b 2 8.b even 2 1 inner
1024.2.b.e 2 4.b odd 2 1
1024.2.b.e 2 8.d odd 2 1
1152.2.k.a 2 96.p odd 8 2
1152.2.k.b 2 96.o even 8 2
1600.2.l.a 2 160.z even 8 2
1600.2.q.a 2 160.bb odd 8 2
1600.2.q.b 2 160.v odd 8 2
9216.2.a.d 2 48.k even 4 2
9216.2.a.s 2 48.i odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1024, [\chi])$$:

 $$T_{3}^{2} + 2$$ T3^2 + 2 $$T_{5}^{2} + 2$$ T5^2 + 2 $$T_{7} + 2$$ T7 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2$$
$5$ $$T^{2} + 2$$
$7$ $$(T + 2)^{2}$$
$11$ $$T^{2} + 2$$
$13$ $$T^{2} + 2$$
$17$ $$(T - 2)^{2}$$
$19$ $$T^{2} + 18$$
$23$ $$(T + 6)^{2}$$
$29$ $$T^{2} + 18$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} + 18$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 50$$
$47$ $$(T - 8)^{2}$$
$53$ $$T^{2} + 50$$
$59$ $$T^{2} + 18$$
$61$ $$T^{2} + 162$$
$67$ $$T^{2} + 50$$
$71$ $$(T + 10)^{2}$$
$73$ $$(T + 4)^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 2$$
$89$ $$(T + 4)^{2}$$
$97$ $$(T + 2)^{2}$$