# Properties

 Label 2200.4.a.d 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: yes 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{3} + 8 q^{7} - 23 q^{9}+O(q^{10})$$ q - 2 * q^3 + 8 * q^7 - 23 * q^9 $$q - 2 q^{3} + 8 q^{7} - 23 q^{9} - 11 q^{11} + 25 q^{13} - 52 q^{17} + 19 q^{19} - 16 q^{21} - 75 q^{23} + 100 q^{27} + 241 q^{29} + 189 q^{31} + 22 q^{33} + 68 q^{37} - 50 q^{39} + 240 q^{41} - 183 q^{43} - 56 q^{47} - 279 q^{49} + 104 q^{51} - 142 q^{53} - 38 q^{57} - 584 q^{59} - 622 q^{61} - 184 q^{63} - 142 q^{67} + 150 q^{69} + 435 q^{71} + 628 q^{73} - 88 q^{77} - 824 q^{79} + 421 q^{81} + 837 q^{83} - 482 q^{87} - 53 q^{89} + 200 q^{91} - 378 q^{93} + 1677 q^{97} + 253 q^{99}+O(q^{100})$$ q - 2 * q^3 + 8 * q^7 - 23 * q^9 - 11 * q^11 + 25 * q^13 - 52 * q^17 + 19 * q^19 - 16 * q^21 - 75 * q^23 + 100 * q^27 + 241 * q^29 + 189 * q^31 + 22 * q^33 + 68 * q^37 - 50 * q^39 + 240 * q^41 - 183 * q^43 - 56 * q^47 - 279 * q^49 + 104 * q^51 - 142 * q^53 - 38 * q^57 - 584 * q^59 - 622 * q^61 - 184 * q^63 - 142 * q^67 + 150 * q^69 + 435 * q^71 + 628 * q^73 - 88 * q^77 - 824 * q^79 + 421 * q^81 + 837 * q^83 - 482 * q^87 - 53 * q^89 + 200 * q^91 - 378 * q^93 + 1677 * q^97 + 253 * 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 −2.00000 0 0 0 8.00000 0 −23.0000 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.d 1
5.b even 2 1 2200.4.a.g yes 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2200.4.a.d 1 1.a even 1 1 trivial
2200.4.a.g yes 1 5.b even 2 1

## Hecke kernels

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

 $$T_{3} + 2$$ T3 + 2 $$T_{7} - 8$$ T7 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 2$$
$5$ $$T$$
$7$ $$T - 8$$
$11$ $$T + 11$$
$13$ $$T - 25$$
$17$ $$T + 52$$
$19$ $$T - 19$$
$23$ $$T + 75$$
$29$ $$T - 241$$
$31$ $$T - 189$$
$37$ $$T - 68$$
$41$ $$T - 240$$
$43$ $$T + 183$$
$47$ $$T + 56$$
$53$ $$T + 142$$
$59$ $$T + 584$$
$61$ $$T + 622$$
$67$ $$T + 142$$
$71$ $$T - 435$$
$73$ $$T - 628$$
$79$ $$T + 824$$
$83$ $$T - 837$$
$89$ $$T + 53$$
$97$ $$T - 1677$$