# Properties

 Label 1024.2.e.m Level $1024$ Weight $2$ Character orbit 1024.e Analytic conductor $8.177$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1024,2,Mod(257,1024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1024, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1024.257");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1024 = 2^{10}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1024.e (of order $$4$$, degree $$2$$, 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: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 128) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - \beta_{3} q^{5} - 4 \beta_{2} q^{7} + \beta_{2} q^{9}+O(q^{10})$$ q + b1 * q^3 - b3 * q^5 - 4*b2 * q^7 + b2 * q^9 $$q + \beta_1 q^{3} - \beta_{3} q^{5} - 4 \beta_{2} q^{7} + \beta_{2} q^{9} + \beta_{3} q^{11} + \beta_1 q^{13} + 4 q^{15} + 2 q^{17} - \beta_1 q^{19} - 4 \beta_{3} q^{21} - 4 \beta_{2} q^{23} + \beta_{2} q^{25} - 2 \beta_{3} q^{27} + 3 \beta_1 q^{29} - 4 q^{33} - 4 \beta_1 q^{35} - 5 \beta_{3} q^{37} + 4 \beta_{2} q^{39} - 6 \beta_{2} q^{41} - 3 \beta_{3} q^{43} + \beta_1 q^{45} - 8 q^{47} - 9 q^{49} + 2 \beta_1 q^{51} - 3 \beta_{3} q^{53} + 4 \beta_{2} q^{55} - 4 \beta_{2} q^{57} + 7 \beta_{3} q^{59} - \beta_1 q^{61} + 4 q^{63} + 4 q^{65} + 5 \beta_1 q^{67} - 4 \beta_{3} q^{69} + 12 \beta_{2} q^{71} + 14 \beta_{2} q^{73} + \beta_{3} q^{75} + 4 \beta_1 q^{77} - 8 q^{79} + 11 q^{81} + 3 \beta_1 q^{83} - 2 \beta_{3} q^{85} + 12 \beta_{2} q^{87} + 2 \beta_{2} q^{89} - 4 \beta_{3} q^{91} - 4 q^{95} - 2 q^{97} - \beta_1 q^{99}+O(q^{100})$$ q + b1 * q^3 - b3 * q^5 - 4*b2 * q^7 + b2 * q^9 + b3 * q^11 + b1 * q^13 + 4 * q^15 + 2 * q^17 - b1 * q^19 - 4*b3 * q^21 - 4*b2 * q^23 + b2 * q^25 - 2*b3 * q^27 + 3*b1 * q^29 - 4 * q^33 - 4*b1 * q^35 - 5*b3 * q^37 + 4*b2 * q^39 - 6*b2 * q^41 - 3*b3 * q^43 + b1 * q^45 - 8 * q^47 - 9 * q^49 + 2*b1 * q^51 - 3*b3 * q^53 + 4*b2 * q^55 - 4*b2 * q^57 + 7*b3 * q^59 - b1 * q^61 + 4 * q^63 + 4 * q^65 + 5*b1 * q^67 - 4*b3 * q^69 + 12*b2 * q^71 + 14*b2 * q^73 + b3 * q^75 + 4*b1 * q^77 - 8 * q^79 + 11 * q^81 + 3*b1 * q^83 - 2*b3 * q^85 + 12*b2 * q^87 + 2*b2 * q^89 - 4*b3 * q^91 - 4 * q^95 - 2 * q^97 - b1 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 16 q^{15} + 8 q^{17} - 16 q^{33} - 32 q^{47} - 36 q^{49} + 16 q^{63} + 16 q^{65} - 32 q^{79} + 44 q^{81} - 16 q^{95} - 8 q^{97}+O(q^{100})$$ 4 * q + 16 * q^15 + 8 * q^17 - 16 * q^33 - 32 * q^47 - 36 * q^49 + 16 * q^63 + 16 * q^65 - 32 * q^79 + 44 * q^81 - 16 * q^95 - 8 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{8}$$ 2*v $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{3}$$ $$=$$ $$2\zeta_{8}^{3}$$ 2*v^3
 $$\zeta_{8}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{8}^{3}$$ $$=$$ $$( \beta_{3} ) / 2$$ (b3) / 2

## Character values

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

 $$n$$ $$5$$ $$1023$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
0 −1.41421 1.41421i 0 −1.41421 + 1.41421i 0 4.00000i 0 1.00000i 0
257.2 0 1.41421 + 1.41421i 0 1.41421 1.41421i 0 4.00000i 0 1.00000i 0
769.1 0 −1.41421 + 1.41421i 0 −1.41421 1.41421i 0 4.00000i 0 1.00000i 0
769.2 0 1.41421 1.41421i 0 1.41421 + 1.41421i 0 4.00000i 0 1.00000i 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
16.e even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1024.2.e.m 4
4.b odd 2 1 1024.2.e.i 4
8.b even 2 1 inner 1024.2.e.m 4
8.d odd 2 1 1024.2.e.i 4
16.e even 4 2 inner 1024.2.e.m 4
16.f odd 4 2 1024.2.e.i 4
32.g even 8 1 128.2.a.b yes 1
32.g even 8 1 128.2.a.c yes 1
32.g even 8 2 256.2.b.a 2
32.h odd 8 1 128.2.a.a 1
32.h odd 8 1 128.2.a.d yes 1
32.h odd 8 2 256.2.b.c 2
96.o even 8 1 1152.2.a.c 1
96.o even 8 1 1152.2.a.m 1
96.o even 8 2 2304.2.d.r 2
96.p odd 8 1 1152.2.a.h 1
96.p odd 8 1 1152.2.a.r 1
96.p odd 8 2 2304.2.d.b 2
160.u even 8 1 3200.2.c.e 2
160.u even 8 1 3200.2.c.l 2
160.v odd 8 1 3200.2.c.f 2
160.v odd 8 1 3200.2.c.k 2
160.y odd 8 1 3200.2.a.h 1
160.y odd 8 1 3200.2.a.x 1
160.z even 8 1 3200.2.a.e 1
160.z even 8 1 3200.2.a.u 1
160.ba even 8 1 3200.2.c.e 2
160.ba even 8 1 3200.2.c.l 2
160.bb odd 8 1 3200.2.c.f 2
160.bb odd 8 1 3200.2.c.k 2
224.v odd 8 1 6272.2.a.b 1
224.v odd 8 1 6272.2.a.g 1
224.x even 8 1 6272.2.a.a 1
224.x even 8 1 6272.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.a.a 1 32.h odd 8 1
128.2.a.b yes 1 32.g even 8 1
128.2.a.c yes 1 32.g even 8 1
128.2.a.d yes 1 32.h odd 8 1
256.2.b.a 2 32.g even 8 2
256.2.b.c 2 32.h odd 8 2
1024.2.e.i 4 4.b odd 2 1
1024.2.e.i 4 8.d odd 2 1
1024.2.e.i 4 16.f odd 4 2
1024.2.e.m 4 1.a even 1 1 trivial
1024.2.e.m 4 8.b even 2 1 inner
1024.2.e.m 4 16.e even 4 2 inner
1152.2.a.c 1 96.o even 8 1
1152.2.a.h 1 96.p odd 8 1
1152.2.a.m 1 96.o even 8 1
1152.2.a.r 1 96.p odd 8 1
2304.2.d.b 2 96.p odd 8 2
2304.2.d.r 2 96.o even 8 2
3200.2.a.e 1 160.z even 8 1
3200.2.a.h 1 160.y odd 8 1
3200.2.a.u 1 160.z even 8 1
3200.2.a.x 1 160.y odd 8 1
3200.2.c.e 2 160.u even 8 1
3200.2.c.e 2 160.ba even 8 1
3200.2.c.f 2 160.v odd 8 1
3200.2.c.f 2 160.bb odd 8 1
3200.2.c.k 2 160.v odd 8 1
3200.2.c.k 2 160.bb odd 8 1
3200.2.c.l 2 160.u even 8 1
3200.2.c.l 2 160.ba even 8 1
6272.2.a.a 1 224.x even 8 1
6272.2.a.b 1 224.v odd 8 1
6272.2.a.g 1 224.v odd 8 1
6272.2.a.h 1 224.x even 8 1

## Hecke kernels

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

 $$T_{3}^{4} + 16$$ T3^4 + 16 $$T_{5}^{4} + 16$$ T5^4 + 16 $$T_{47} + 8$$ T47 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 16$$
$5$ $$T^{4} + 16$$
$7$ $$(T^{2} + 16)^{2}$$
$11$ $$T^{4} + 16$$
$13$ $$T^{4} + 16$$
$17$ $$(T - 2)^{4}$$
$19$ $$T^{4} + 16$$
$23$ $$(T^{2} + 16)^{2}$$
$29$ $$T^{4} + 1296$$
$31$ $$T^{4}$$
$37$ $$T^{4} + 10000$$
$41$ $$(T^{2} + 36)^{2}$$
$43$ $$T^{4} + 1296$$
$47$ $$(T + 8)^{4}$$
$53$ $$T^{4} + 1296$$
$59$ $$T^{4} + 38416$$
$61$ $$T^{4} + 16$$
$67$ $$T^{4} + 10000$$
$71$ $$(T^{2} + 144)^{2}$$
$73$ $$(T^{2} + 196)^{2}$$
$79$ $$(T + 8)^{4}$$
$83$ $$T^{4} + 1296$$
$89$ $$(T^{2} + 4)^{2}$$
$97$ $$(T + 2)^{4}$$