# Properties

 Label 2200.2.a.d Level $2200$ Weight $2$ Character orbit 2200.a Self dual yes Analytic conductor $17.567$ 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] = [2200,2,Mod(1,2200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2200 = 2^{3} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2200.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: $$17.5670884447$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 440) 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)

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{7} - 2 q^{9}+O(q^{10})$$ q - q^3 + q^7 - 2 * q^9 $$q - q^{3} + q^{7} - 2 q^{9} - q^{11} + q^{17} + q^{19} - q^{21} + 5 q^{27} - q^{29} - q^{31} + q^{33} + q^{37} + 6 q^{43} + 8 q^{47} - 6 q^{49} - q^{51} + 9 q^{53} - q^{57} + 4 q^{59} - 7 q^{61} - 2 q^{63} - 4 q^{67} + 5 q^{71} + 14 q^{73} - q^{77} + 4 q^{79} + q^{81} + 16 q^{83} + q^{87} - 7 q^{89} + q^{93} - 16 q^{97} + 2 q^{99}+O(q^{100})$$ q - q^3 + q^7 - 2 * q^9 - q^11 + q^17 + q^19 - q^21 + 5 * q^27 - q^29 - q^31 + q^33 + q^37 + 6 * q^43 + 8 * q^47 - 6 * q^49 - q^51 + 9 * q^53 - q^57 + 4 * q^59 - 7 * q^61 - 2 * q^63 - 4 * q^67 + 5 * q^71 + 14 * q^73 - q^77 + 4 * q^79 + q^81 + 16 * q^83 + q^87 - 7 * q^89 + q^93 - 16 * q^97 + 2 * q^99

## 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 −1.00000 0 0 0 1.00000 0 −2.00000 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
$$2$$ $$+1$$
$$5$$ $$-1$$
$$11$$ $$+1$$

## 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 2200.2.a.d 1
4.b odd 2 1 4400.2.a.u 1
5.b even 2 1 2200.2.a.h 1
5.c odd 4 2 440.2.b.c 2
15.e even 4 2 3960.2.d.a 2
20.d odd 2 1 4400.2.a.j 1
20.e even 4 2 880.2.b.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.b.c 2 5.c odd 4 2
880.2.b.g 2 20.e even 4 2
2200.2.a.d 1 1.a even 1 1 trivial
2200.2.a.h 1 5.b even 2 1
3960.2.d.a 2 15.e even 4 2
4400.2.a.j 1 20.d odd 2 1
4400.2.a.u 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2200))$$:

 $$T_{3} + 1$$ T3 + 1 $$T_{7} - 1$$ T7 - 1 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T - 1$$
$11$ $$T + 1$$
$13$ $$T$$
$17$ $$T - 1$$
$19$ $$T - 1$$
$23$ $$T$$
$29$ $$T + 1$$
$31$ $$T + 1$$
$37$ $$T - 1$$
$41$ $$T$$
$43$ $$T - 6$$
$47$ $$T - 8$$
$53$ $$T - 9$$
$59$ $$T - 4$$
$61$ $$T + 7$$
$67$ $$T + 4$$
$71$ $$T - 5$$
$73$ $$T - 14$$
$79$ $$T - 4$$
$83$ $$T - 16$$
$89$ $$T + 7$$
$97$ $$T + 16$$