# Properties

 Label 5040.2.a.a Level $5040$ Weight $2$ Character orbit 5040.a Self dual yes Analytic conductor $40.245$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5040,2,Mod(1,5040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5040, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5040.a (trivial)

## 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: yes Analytic conductor: $$40.2446026187$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{5} - q^{7}+O(q^{10})$$ q - q^5 - q^7 $$q - q^{5} - q^{7} - 5 q^{11} - 5 q^{13} + 7 q^{17} + 2 q^{19} - 2 q^{23} + q^{25} - 7 q^{29} - 4 q^{31} + q^{35} - 6 q^{37} + 12 q^{41} + 2 q^{43} + q^{47} + q^{49} + 5 q^{55} - 4 q^{59} + 4 q^{61} + 5 q^{65} - 8 q^{67} + 6 q^{73} + 5 q^{77} + 3 q^{79} - 4 q^{83} - 7 q^{85} + 5 q^{91} - 2 q^{95} + 13 q^{97}+O(q^{100})$$ q - q^5 - q^7 - 5 * q^11 - 5 * q^13 + 7 * q^17 + 2 * q^19 - 2 * q^23 + q^25 - 7 * q^29 - 4 * q^31 + q^35 - 6 * q^37 + 12 * q^41 + 2 * q^43 + q^47 + q^49 + 5 * q^55 - 4 * q^59 + 4 * q^61 + 5 * q^65 - 8 * q^67 + 6 * q^73 + 5 * q^77 + 3 * q^79 - 4 * q^83 - 7 * q^85 + 5 * q^91 - 2 * q^95 + 13 * q^97

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 −1.00000 0 −1.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5040.2.a.a 1
3.b odd 2 1 560.2.a.f 1
4.b odd 2 1 2520.2.a.i 1
12.b even 2 1 280.2.a.a 1
15.d odd 2 1 2800.2.a.c 1
15.e even 4 2 2800.2.g.b 2
21.c even 2 1 3920.2.a.c 1
24.f even 2 1 2240.2.a.z 1
24.h odd 2 1 2240.2.a.a 1
60.h even 2 1 1400.2.a.n 1
60.l odd 4 2 1400.2.g.a 2
84.h odd 2 1 1960.2.a.o 1
84.j odd 6 2 1960.2.q.a 2
84.n even 6 2 1960.2.q.o 2
420.o odd 2 1 9800.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.a 1 12.b even 2 1
560.2.a.f 1 3.b odd 2 1
1400.2.a.n 1 60.h even 2 1
1400.2.g.a 2 60.l odd 4 2
1960.2.a.o 1 84.h odd 2 1
1960.2.q.a 2 84.j odd 6 2
1960.2.q.o 2 84.n even 6 2
2240.2.a.a 1 24.h odd 2 1
2240.2.a.z 1 24.f even 2 1
2520.2.a.i 1 4.b odd 2 1
2800.2.a.c 1 15.d odd 2 1
2800.2.g.b 2 15.e even 4 2
3920.2.a.c 1 21.c even 2 1
5040.2.a.a 1 1.a even 1 1 trivial
9800.2.a.a 1 420.o odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5040))$$:

 $$T_{11} + 5$$ T11 + 5 $$T_{13} + 5$$ T13 + 5 $$T_{17} - 7$$ T17 - 7 $$T_{19} - 2$$ T19 - 2 $$T_{29} + 7$$ T29 + 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T + 1$$
$11$ $$T + 5$$
$13$ $$T + 5$$
$17$ $$T - 7$$
$19$ $$T - 2$$
$23$ $$T + 2$$
$29$ $$T + 7$$
$31$ $$T + 4$$
$37$ $$T + 6$$
$41$ $$T - 12$$
$43$ $$T - 2$$
$47$ $$T - 1$$
$53$ $$T$$
$59$ $$T + 4$$
$61$ $$T - 4$$
$67$ $$T + 8$$
$71$ $$T$$
$73$ $$T - 6$$
$79$ $$T - 3$$
$83$ $$T + 4$$
$89$ $$T$$
$97$ $$T - 13$$