# Properties

 Label 1800.4.a.k Level $1800$ Weight $4$ Character orbit 1800.a Self dual yes Analytic conductor $106.203$ 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] = [1800,4,Mod(1,1800)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, 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("1800.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1800.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: $$106.203438010$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 120) 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)

 $$f(q)$$ $$=$$ $$q - 8 q^{7}+O(q^{10})$$ q - 8 * q^7 $$q - 8 q^{7} - 20 q^{11} - 22 q^{13} - 14 q^{17} + 76 q^{19} + 56 q^{23} + 154 q^{29} + 160 q^{31} + 162 q^{37} + 390 q^{41} - 388 q^{43} - 544 q^{47} - 279 q^{49} - 210 q^{53} + 380 q^{59} - 794 q^{61} + 148 q^{67} + 840 q^{71} - 858 q^{73} + 160 q^{77} + 144 q^{79} + 316 q^{83} - 1098 q^{89} + 176 q^{91} - 994 q^{97}+O(q^{100})$$ q - 8 * q^7 - 20 * q^11 - 22 * q^13 - 14 * q^17 + 76 * q^19 + 56 * q^23 + 154 * q^29 + 160 * q^31 + 162 * q^37 + 390 * q^41 - 388 * q^43 - 544 * q^47 - 279 * q^49 - 210 * q^53 + 380 * q^59 - 794 * q^61 + 148 * q^67 + 840 * q^71 - 858 * q^73 + 160 * q^77 + 144 * q^79 + 316 * q^83 - 1098 * q^89 + 176 * q^91 - 994 * q^97

## 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 0 0 0 0 −8.00000 0 0 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
$$2$$ $$+1$$
$$3$$ $$-1$$
$$5$$ $$+1$$

## 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 1800.4.a.k 1
3.b odd 2 1 600.4.a.d 1
5.b even 2 1 360.4.a.e 1
5.c odd 4 2 1800.4.f.h 2
12.b even 2 1 1200.4.a.bf 1
15.d odd 2 1 120.4.a.f 1
15.e even 4 2 600.4.f.g 2
20.d odd 2 1 720.4.a.f 1
60.h even 2 1 240.4.a.d 1
60.l odd 4 2 1200.4.f.g 2
120.i odd 2 1 960.4.a.g 1
120.m even 2 1 960.4.a.v 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.a.f 1 15.d odd 2 1
240.4.a.d 1 60.h even 2 1
360.4.a.e 1 5.b even 2 1
600.4.a.d 1 3.b odd 2 1
600.4.f.g 2 15.e even 4 2
720.4.a.f 1 20.d odd 2 1
960.4.a.g 1 120.i odd 2 1
960.4.a.v 1 120.m even 2 1
1200.4.a.bf 1 12.b even 2 1
1200.4.f.g 2 60.l odd 4 2
1800.4.a.k 1 1.a even 1 1 trivial
1800.4.f.h 2 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1800))$$:

 $$T_{7} + 8$$ T7 + 8 $$T_{11} + 20$$ T11 + 20 $$T_{17} + 14$$ T17 + 14

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 8$$
$11$ $$T + 20$$
$13$ $$T + 22$$
$17$ $$T + 14$$
$19$ $$T - 76$$
$23$ $$T - 56$$
$29$ $$T - 154$$
$31$ $$T - 160$$
$37$ $$T - 162$$
$41$ $$T - 390$$
$43$ $$T + 388$$
$47$ $$T + 544$$
$53$ $$T + 210$$
$59$ $$T - 380$$
$61$ $$T + 794$$
$67$ $$T - 148$$
$71$ $$T - 840$$
$73$ $$T + 858$$
$79$ $$T - 144$$
$83$ $$T - 316$$
$89$ $$T + 1098$$
$97$ $$T + 994$$