# Properties

 Label 2200.4.a.h 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)

 $$f(q)$$ $$=$$ $$q + 4 q^{3} - 8 q^{7} - 11 q^{9}+O(q^{10})$$ q + 4 * q^3 - 8 * q^7 - 11 * q^9 $$q + 4 q^{3} - 8 q^{7} - 11 q^{9} + 11 q^{11} + 58 q^{13} - 114 q^{17} - 4 q^{19} - 32 q^{21} + 152 q^{23} - 152 q^{27} - 138 q^{29} + 208 q^{31} + 44 q^{33} + 226 q^{37} + 232 q^{39} - 294 q^{41} - 276 q^{43} + 240 q^{47} - 279 q^{49} - 456 q^{51} + 370 q^{53} - 16 q^{57} - 716 q^{59} - 650 q^{61} + 88 q^{63} - 124 q^{67} + 608 q^{69} + 232 q^{71} + 454 q^{73} - 88 q^{77} - 144 q^{79} - 311 q^{81} + 692 q^{83} - 552 q^{87} - 1206 q^{89} - 464 q^{91} + 832 q^{93} + 1438 q^{97} - 121 q^{99}+O(q^{100})$$ q + 4 * q^3 - 8 * q^7 - 11 * q^9 + 11 * q^11 + 58 * q^13 - 114 * q^17 - 4 * q^19 - 32 * q^21 + 152 * q^23 - 152 * q^27 - 138 * q^29 + 208 * q^31 + 44 * q^33 + 226 * q^37 + 232 * q^39 - 294 * q^41 - 276 * q^43 + 240 * q^47 - 279 * q^49 - 456 * q^51 + 370 * q^53 - 16 * q^57 - 716 * q^59 - 650 * q^61 + 88 * q^63 - 124 * q^67 + 608 * q^69 + 232 * q^71 + 454 * q^73 - 88 * q^77 - 144 * q^79 - 311 * q^81 + 692 * q^83 - 552 * q^87 - 1206 * q^89 - 464 * q^91 + 832 * q^93 + 1438 * q^97 - 121 * 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 4.00000 0 0 0 −8.00000 0 −11.0000 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.h 1
5.b even 2 1 440.4.a.b 1
20.d odd 2 1 880.4.a.l 1

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

 $$T_{3} - 4$$ T3 - 4 $$T_{7} + 8$$ T7 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 4$$
$5$ $$T$$
$7$ $$T + 8$$
$11$ $$T - 11$$
$13$ $$T - 58$$
$17$ $$T + 114$$
$19$ $$T + 4$$
$23$ $$T - 152$$
$29$ $$T + 138$$
$31$ $$T - 208$$
$37$ $$T - 226$$
$41$ $$T + 294$$
$43$ $$T + 276$$
$47$ $$T - 240$$
$53$ $$T - 370$$
$59$ $$T + 716$$
$61$ $$T + 650$$
$67$ $$T + 124$$
$71$ $$T - 232$$
$73$ $$T - 454$$
$79$ $$T + 144$$
$83$ $$T - 692$$
$89$ $$T + 1206$$
$97$ $$T - 1438$$