# Properties

 Label 1800.4.a.a 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 360) 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)

 $$f(q)$$ $$=$$ $$q - 34 q^{7}+O(q^{10})$$ q - 34 * q^7 $$q - 34 q^{7} - 18 q^{11} - 12 q^{13} - 106 q^{17} - 44 q^{19} + 56 q^{23} - 270 q^{29} + 204 q^{31} - 120 q^{37} - 80 q^{41} - 536 q^{43} - 536 q^{47} + 813 q^{49} + 542 q^{53} + 174 q^{59} + 186 q^{61} - 332 q^{67} + 132 q^{71} + 602 q^{73} + 612 q^{77} - 548 q^{79} - 492 q^{83} + 1052 q^{89} + 408 q^{91} - 482 q^{97}+O(q^{100})$$ q - 34 * q^7 - 18 * q^11 - 12 * q^13 - 106 * q^17 - 44 * q^19 + 56 * q^23 - 270 * q^29 + 204 * q^31 - 120 * q^37 - 80 * q^41 - 536 * q^43 - 536 * q^47 + 813 * q^49 + 542 * q^53 + 174 * q^59 + 186 * q^61 - 332 * q^67 + 132 * q^71 + 602 * q^73 + 612 * q^77 - 548 * q^79 - 492 * q^83 + 1052 * q^89 + 408 * q^91 - 482 * 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 −34.0000 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.a 1
3.b odd 2 1 1800.4.a.b 1
5.b even 2 1 360.4.a.o yes 1
5.c odd 4 2 1800.4.f.i 2
15.d odd 2 1 360.4.a.g 1
15.e even 4 2 1800.4.f.o 2
20.d odd 2 1 720.4.a.p 1
60.h even 2 1 720.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.4.a.g 1 15.d odd 2 1
360.4.a.o yes 1 5.b even 2 1
720.4.a.a 1 60.h even 2 1
720.4.a.p 1 20.d odd 2 1
1800.4.a.a 1 1.a even 1 1 trivial
1800.4.a.b 1 3.b odd 2 1
1800.4.f.i 2 5.c odd 4 2
1800.4.f.o 2 15.e even 4 2

## Hecke kernels

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

 $$T_{7} + 34$$ T7 + 34 $$T_{11} + 18$$ T11 + 18 $$T_{17} + 106$$ T17 + 106

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 34$$
$11$ $$T + 18$$
$13$ $$T + 12$$
$17$ $$T + 106$$
$19$ $$T + 44$$
$23$ $$T - 56$$
$29$ $$T + 270$$
$31$ $$T - 204$$
$37$ $$T + 120$$
$41$ $$T + 80$$
$43$ $$T + 536$$
$47$ $$T + 536$$
$53$ $$T - 542$$
$59$ $$T - 174$$
$61$ $$T - 186$$
$67$ $$T + 332$$
$71$ $$T - 132$$
$73$ $$T - 602$$
$79$ $$T + 548$$
$83$ $$T + 492$$
$89$ $$T - 1052$$
$97$ $$T + 482$$