# Properties

 Label 2200.4.a.j 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)

 $$f(q)$$ $$=$$ $$q + 5 q^{3} + 13 q^{7} - 2 q^{9}+O(q^{10})$$ q + 5 * q^3 + 13 * q^7 - 2 * q^9 $$q + 5 q^{3} + 13 q^{7} - 2 q^{9} - 11 q^{11} - 12 q^{13} - 67 q^{17} + 151 q^{19} + 65 q^{21} + 12 q^{23} - 145 q^{27} - 143 q^{29} - 337 q^{31} - 55 q^{33} - 125 q^{37} - 60 q^{39} - 240 q^{41} + 270 q^{43} + 448 q^{47} - 174 q^{49} - 335 q^{51} - 45 q^{53} + 755 q^{57} + 704 q^{59} - 217 q^{61} - 26 q^{63} + 284 q^{67} + 60 q^{69} - 515 q^{71} - 1162 q^{73} - 143 q^{77} - 944 q^{79} - 671 q^{81} - 124 q^{83} - 715 q^{87} + 361 q^{89} - 156 q^{91} - 1685 q^{93} - 916 q^{97} + 22 q^{99}+O(q^{100})$$ q + 5 * q^3 + 13 * q^7 - 2 * q^9 - 11 * q^11 - 12 * q^13 - 67 * q^17 + 151 * q^19 + 65 * q^21 + 12 * q^23 - 145 * q^27 - 143 * q^29 - 337 * q^31 - 55 * q^33 - 125 * q^37 - 60 * q^39 - 240 * q^41 + 270 * q^43 + 448 * q^47 - 174 * q^49 - 335 * q^51 - 45 * q^53 + 755 * q^57 + 704 * q^59 - 217 * q^61 - 26 * q^63 + 284 * q^67 + 60 * q^69 - 515 * q^71 - 1162 * q^73 - 143 * q^77 - 944 * q^79 - 671 * q^81 - 124 * q^83 - 715 * q^87 + 361 * q^89 - 156 * q^91 - 1685 * q^93 - 916 * q^97 + 22 * 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 5.00000 0 0 0 13.0000 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 2200.4.a.j 1
5.b even 2 1 2200.4.a.c 1
5.c odd 4 2 440.4.b.a 2
20.e even 4 2 880.4.b.c 2

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

 $$T_{3} - 5$$ T3 - 5 $$T_{7} - 13$$ T7 - 13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 5$$
$5$ $$T$$
$7$ $$T - 13$$
$11$ $$T + 11$$
$13$ $$T + 12$$
$17$ $$T + 67$$
$19$ $$T - 151$$
$23$ $$T - 12$$
$29$ $$T + 143$$
$31$ $$T + 337$$
$37$ $$T + 125$$
$41$ $$T + 240$$
$43$ $$T - 270$$
$47$ $$T - 448$$
$53$ $$T + 45$$
$59$ $$T - 704$$
$61$ $$T + 217$$
$67$ $$T - 284$$
$71$ $$T + 515$$
$73$ $$T + 1162$$
$79$ $$T + 944$$
$83$ $$T + 124$$
$89$ $$T - 361$$
$97$ $$T + 916$$