# Properties

 Label 1800.4.a.m 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 600) 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)

 $$f(q)$$ $$=$$ $$q - 5 q^{7}+O(q^{10})$$ q - 5 * q^7 $$q - 5 q^{7} - 14 q^{11} - q^{13} + 46 q^{17} + 19 q^{19} - 46 q^{23} - 14 q^{29} + 133 q^{31} - 258 q^{37} - 84 q^{41} + 167 q^{43} + 410 q^{47} - 318 q^{49} + 456 q^{53} + 194 q^{59} - 17 q^{61} - 653 q^{67} - 828 q^{71} - 570 q^{73} + 70 q^{77} - 552 q^{79} + 142 q^{83} + 1104 q^{89} + 5 q^{91} - 841 q^{97}+O(q^{100})$$ q - 5 * q^7 - 14 * q^11 - q^13 + 46 * q^17 + 19 * q^19 - 46 * q^23 - 14 * q^29 + 133 * q^31 - 258 * q^37 - 84 * q^41 + 167 * q^43 + 410 * q^47 - 318 * q^49 + 456 * q^53 + 194 * q^59 - 17 * q^61 - 653 * q^67 - 828 * q^71 - 570 * q^73 + 70 * q^77 - 552 * q^79 + 142 * q^83 + 1104 * q^89 + 5 * q^91 - 841 * 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 −5.00000 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.m 1
3.b odd 2 1 600.4.a.e 1
5.b even 2 1 1800.4.a.v 1
5.c odd 4 2 1800.4.f.l 2
12.b even 2 1 1200.4.a.bd 1
15.d odd 2 1 600.4.a.n yes 1
15.e even 4 2 600.4.f.e 2
60.h even 2 1 1200.4.a.g 1
60.l odd 4 2 1200.4.f.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.4.a.e 1 3.b odd 2 1
600.4.a.n yes 1 15.d odd 2 1
600.4.f.e 2 15.e even 4 2
1200.4.a.g 1 60.h even 2 1
1200.4.a.bd 1 12.b even 2 1
1200.4.f.i 2 60.l odd 4 2
1800.4.a.m 1 1.a even 1 1 trivial
1800.4.a.v 1 5.b even 2 1
1800.4.f.l 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1800))$$:

 $$T_{7} + 5$$ T7 + 5 $$T_{11} + 14$$ T11 + 14 $$T_{17} - 46$$ T17 - 46

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 5$$
$11$ $$T + 14$$
$13$ $$T + 1$$
$17$ $$T - 46$$
$19$ $$T - 19$$
$23$ $$T + 46$$
$29$ $$T + 14$$
$31$ $$T - 133$$
$37$ $$T + 258$$
$41$ $$T + 84$$
$43$ $$T - 167$$
$47$ $$T - 410$$
$53$ $$T - 456$$
$59$ $$T - 194$$
$61$ $$T + 17$$
$67$ $$T + 653$$
$71$ $$T + 828$$
$73$ $$T + 570$$
$79$ $$T + 552$$
$83$ $$T - 142$$
$89$ $$T - 1104$$
$97$ $$T + 841$$