# Properties

 Label 2200.4.a.a Level $2200$ Weight $4$ Character orbit 2200.a Self dual yes Analytic conductor $129.804$ Analytic rank $1$ 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] = [2200,4,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, 4, names="a")

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

chi := DirichletCharacter("2200.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2200 = 2^{3} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$129.804202013$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 88) 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)

 $$f(q)$$ $$=$$ $$q - 7 q^{3} - 2 q^{7} + 22 q^{9}+O(q^{10})$$ q - 7 * q^3 - 2 * q^7 + 22 * q^9 $$q - 7 q^{3} - 2 q^{7} + 22 q^{9} - 11 q^{11} + 38 q^{17} + 44 q^{19} + 14 q^{21} - 175 q^{23} + 35 q^{27} - 264 q^{29} + 159 q^{31} + 77 q^{33} + 173 q^{37} - 220 q^{41} + 542 q^{43} + 264 q^{47} - 339 q^{49} - 266 q^{51} - 682 q^{53} - 308 q^{57} + 421 q^{59} + 308 q^{61} - 44 q^{63} - 177 q^{67} + 1225 q^{69} + 365 q^{71} + 528 q^{73} + 22 q^{77} + 686 q^{79} - 839 q^{81} - 698 q^{83} + 1848 q^{87} + 967 q^{89} - 1113 q^{93} + 1127 q^{97} - 242 q^{99}+O(q^{100})$$ q - 7 * q^3 - 2 * q^7 + 22 * q^9 - 11 * q^11 + 38 * q^17 + 44 * q^19 + 14 * q^21 - 175 * q^23 + 35 * q^27 - 264 * q^29 + 159 * q^31 + 77 * q^33 + 173 * q^37 - 220 * q^41 + 542 * q^43 + 264 * q^47 - 339 * q^49 - 266 * q^51 - 682 * q^53 - 308 * q^57 + 421 * q^59 + 308 * q^61 - 44 * q^63 - 177 * q^67 + 1225 * q^69 + 365 * q^71 + 528 * q^73 + 22 * q^77 + 686 * q^79 - 839 * q^81 - 698 * q^83 + 1848 * q^87 + 967 * q^89 - 1113 * q^93 + 1127 * q^97 - 242 * 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 −7.00000 0 0 0 −2.00000 0 22.0000 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 2200.4.a.a 1
5.b even 2 1 88.4.a.b 1
15.d odd 2 1 792.4.a.b 1
20.d odd 2 1 176.4.a.a 1
40.e odd 2 1 704.4.a.k 1
40.f even 2 1 704.4.a.a 1
55.d odd 2 1 968.4.a.e 1
60.h even 2 1 1584.4.a.g 1
220.g even 2 1 1936.4.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.4.a.b 1 5.b even 2 1
176.4.a.a 1 20.d odd 2 1
704.4.a.a 1 40.f even 2 1
704.4.a.k 1 40.e odd 2 1
792.4.a.b 1 15.d odd 2 1
968.4.a.e 1 55.d odd 2 1
1584.4.a.g 1 60.h even 2 1
1936.4.a.b 1 220.g even 2 1
2200.4.a.a 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2200))$$:

 $$T_{3} + 7$$ T3 + 7 $$T_{7} + 2$$ T7 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 7$$
$5$ $$T$$
$7$ $$T + 2$$
$11$ $$T + 11$$
$13$ $$T$$
$17$ $$T - 38$$
$19$ $$T - 44$$
$23$ $$T + 175$$
$29$ $$T + 264$$
$31$ $$T - 159$$
$37$ $$T - 173$$
$41$ $$T + 220$$
$43$ $$T - 542$$
$47$ $$T - 264$$
$53$ $$T + 682$$
$59$ $$T - 421$$
$61$ $$T - 308$$
$67$ $$T + 177$$
$71$ $$T - 365$$
$73$ $$T - 528$$
$79$ $$T - 686$$
$83$ $$T + 698$$
$89$ $$T - 967$$
$97$ $$T - 1127$$