# Properties

 Label 1800.4.a.o Level $1800$ Weight $4$ Character orbit 1800.a Self dual yes Analytic conductor $106.203$ Analytic rank $0$ 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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q - 4 q^{7}+O(q^{10})$$ q - 4 * q^7 $$q - 4 q^{7} + 28 q^{11} + 16 q^{13} + 108 q^{17} + 32 q^{19} - 28 q^{23} + 238 q^{29} - 180 q^{31} + 40 q^{37} - 422 q^{41} - 276 q^{43} + 60 q^{47} - 327 q^{49} + 220 q^{53} + 804 q^{59} - 358 q^{61} + 884 q^{67} + 64 q^{71} + 152 q^{73} - 112 q^{77} - 932 q^{79} - 1292 q^{83} + 1146 q^{89} - 64 q^{91} - 824 q^{97}+O(q^{100})$$ q - 4 * q^7 + 28 * q^11 + 16 * q^13 + 108 * q^17 + 32 * q^19 - 28 * q^23 + 238 * q^29 - 180 * q^31 + 40 * q^37 - 422 * q^41 - 276 * q^43 + 60 * q^47 - 327 * q^49 + 220 * q^53 + 804 * q^59 - 358 * q^61 + 884 * q^67 + 64 * q^71 + 152 * q^73 - 112 * q^77 - 932 * q^79 - 1292 * q^83 + 1146 * q^89 - 64 * q^91 - 824 * 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 −4.00000 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.o 1
3.b odd 2 1 600.4.a.k 1
5.b even 2 1 1800.4.a.u 1
5.c odd 4 2 360.4.f.a 2
12.b even 2 1 1200.4.a.l 1
15.d odd 2 1 600.4.a.f 1
15.e even 4 2 120.4.f.c 2
20.e even 4 2 720.4.f.b 2
60.h even 2 1 1200.4.a.z 1
60.l odd 4 2 240.4.f.e 2
120.q odd 4 2 960.4.f.a 2
120.w even 4 2 960.4.f.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.f.c 2 15.e even 4 2
240.4.f.e 2 60.l odd 4 2
360.4.f.a 2 5.c odd 4 2
600.4.a.f 1 15.d odd 2 1
600.4.a.k 1 3.b odd 2 1
720.4.f.b 2 20.e even 4 2
960.4.f.a 2 120.q odd 4 2
960.4.f.b 2 120.w even 4 2
1200.4.a.l 1 12.b even 2 1
1200.4.a.z 1 60.h even 2 1
1800.4.a.o 1 1.a even 1 1 trivial
1800.4.a.u 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1800))$$:

 $$T_{7} + 4$$ T7 + 4 $$T_{11} - 28$$ T11 - 28 $$T_{17} - 108$$ T17 - 108

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 4$$
$11$ $$T - 28$$
$13$ $$T - 16$$
$17$ $$T - 108$$
$19$ $$T - 32$$
$23$ $$T + 28$$
$29$ $$T - 238$$
$31$ $$T + 180$$
$37$ $$T - 40$$
$41$ $$T + 422$$
$43$ $$T + 276$$
$47$ $$T - 60$$
$53$ $$T - 220$$
$59$ $$T - 804$$
$61$ $$T + 358$$
$67$ $$T - 884$$
$71$ $$T - 64$$
$73$ $$T - 152$$
$79$ $$T + 932$$
$83$ $$T + 1292$$
$89$ $$T - 1146$$
$97$ $$T + 824$$