# Properties

 Label 2200.4.a.b 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 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)

 $$f(q)$$ $$=$$ $$q - 6 q^{3} + 32 q^{7} + 9 q^{9}+O(q^{10})$$ q - 6 * q^3 + 32 * q^7 + 9 * q^9 $$q - 6 q^{3} + 32 q^{7} + 9 q^{9} + 11 q^{11} + 48 q^{13} + 36 q^{17} - 44 q^{19} - 192 q^{21} - 58 q^{23} + 108 q^{27} - 278 q^{29} - 112 q^{31} - 66 q^{33} - 194 q^{37} - 288 q^{39} - 314 q^{41} - 396 q^{43} + 410 q^{47} + 681 q^{49} - 216 q^{51} - 170 q^{53} + 264 q^{57} + 404 q^{59} + 250 q^{61} + 288 q^{63} + 26 q^{67} + 348 q^{69} - 468 q^{71} + 164 q^{73} + 352 q^{77} - 664 q^{79} - 891 q^{81} - 1348 q^{83} + 1668 q^{87} + 534 q^{89} + 1536 q^{91} + 672 q^{93} + 1498 q^{97} + 99 q^{99}+O(q^{100})$$ q - 6 * q^3 + 32 * q^7 + 9 * q^9 + 11 * q^11 + 48 * q^13 + 36 * q^17 - 44 * q^19 - 192 * q^21 - 58 * q^23 + 108 * q^27 - 278 * q^29 - 112 * q^31 - 66 * q^33 - 194 * q^37 - 288 * q^39 - 314 * q^41 - 396 * q^43 + 410 * q^47 + 681 * q^49 - 216 * q^51 - 170 * q^53 + 264 * q^57 + 404 * q^59 + 250 * q^61 + 288 * q^63 + 26 * q^67 + 348 * q^69 - 468 * q^71 + 164 * q^73 + 352 * q^77 - 664 * q^79 - 891 * q^81 - 1348 * q^83 + 1668 * q^87 + 534 * q^89 + 1536 * q^91 + 672 * q^93 + 1498 * q^97 + 99 * 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 −6.00000 0 0 0 32.0000 0 9.00000 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 2200.4.a.b 1
5.b even 2 1 440.4.a.d 1
20.d odd 2 1 880.4.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.4.a.d 1 5.b even 2 1
880.4.a.d 1 20.d odd 2 1
2200.4.a.b 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2200))$$:

 $$T_{3} + 6$$ T3 + 6 $$T_{7} - 32$$ T7 - 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 6$$
$5$ $$T$$
$7$ $$T - 32$$
$11$ $$T - 11$$
$13$ $$T - 48$$
$17$ $$T - 36$$
$19$ $$T + 44$$
$23$ $$T + 58$$
$29$ $$T + 278$$
$31$ $$T + 112$$
$37$ $$T + 194$$
$41$ $$T + 314$$
$43$ $$T + 396$$
$47$ $$T - 410$$
$53$ $$T + 170$$
$59$ $$T - 404$$
$61$ $$T - 250$$
$67$ $$T - 26$$
$71$ $$T + 468$$
$73$ $$T - 164$$
$79$ $$T + 664$$
$83$ $$T + 1348$$
$89$ $$T - 534$$
$97$ $$T - 1498$$