# Properties

 Label 1800.4.a.bd 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 40) 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)

 $$f(q)$$ $$=$$ $$q + 18 q^{7}+O(q^{10})$$ q + 18 * q^7 $$q + 18 q^{7} + 16 q^{11} + 6 q^{13} - 6 q^{17} - 124 q^{19} + 42 q^{23} - 142 q^{29} - 188 q^{31} - 202 q^{37} - 54 q^{41} - 66 q^{43} + 38 q^{47} - 19 q^{49} + 738 q^{53} - 564 q^{59} - 262 q^{61} + 554 q^{67} - 140 q^{71} - 882 q^{73} + 288 q^{77} - 1160 q^{79} + 642 q^{83} + 854 q^{89} + 108 q^{91} + 478 q^{97}+O(q^{100})$$ q + 18 * q^7 + 16 * q^11 + 6 * q^13 - 6 * q^17 - 124 * q^19 + 42 * q^23 - 142 * q^29 - 188 * q^31 - 202 * q^37 - 54 * q^41 - 66 * q^43 + 38 * q^47 - 19 * q^49 + 738 * q^53 - 564 * q^59 - 262 * q^61 + 554 * q^67 - 140 * q^71 - 882 * q^73 + 288 * q^77 - 1160 * q^79 + 642 * q^83 + 854 * q^89 + 108 * q^91 + 478 * 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 18.0000 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.bd 1
3.b odd 2 1 200.4.a.a 1
5.b even 2 1 360.4.a.i 1
5.c odd 4 2 1800.4.f.n 2
12.b even 2 1 400.4.a.u 1
15.d odd 2 1 40.4.a.c 1
15.e even 4 2 200.4.c.a 2
20.d odd 2 1 720.4.a.ba 1
24.f even 2 1 1600.4.a.a 1
24.h odd 2 1 1600.4.a.ca 1
60.h even 2 1 80.4.a.a 1
60.l odd 4 2 400.4.c.a 2
105.g even 2 1 1960.4.a.a 1
120.i odd 2 1 320.4.a.a 1
120.m even 2 1 320.4.a.n 1
240.t even 4 2 1280.4.d.b 2
240.bm odd 4 2 1280.4.d.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.a.c 1 15.d odd 2 1
80.4.a.a 1 60.h even 2 1
200.4.a.a 1 3.b odd 2 1
200.4.c.a 2 15.e even 4 2
320.4.a.a 1 120.i odd 2 1
320.4.a.n 1 120.m even 2 1
360.4.a.i 1 5.b even 2 1
400.4.a.u 1 12.b even 2 1
400.4.c.a 2 60.l odd 4 2
720.4.a.ba 1 20.d odd 2 1
1280.4.d.b 2 240.t even 4 2
1280.4.d.o 2 240.bm odd 4 2
1600.4.a.a 1 24.f even 2 1
1600.4.a.ca 1 24.h odd 2 1
1800.4.a.bd 1 1.a even 1 1 trivial
1800.4.f.n 2 5.c odd 4 2
1960.4.a.a 1 105.g even 2 1

## Hecke kernels

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

 $$T_{7} - 18$$ T7 - 18 $$T_{11} - 16$$ T11 - 16 $$T_{17} + 6$$ T17 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 18$$
$11$ $$T - 16$$
$13$ $$T - 6$$
$17$ $$T + 6$$
$19$ $$T + 124$$
$23$ $$T - 42$$
$29$ $$T + 142$$
$31$ $$T + 188$$
$37$ $$T + 202$$
$41$ $$T + 54$$
$43$ $$T + 66$$
$47$ $$T - 38$$
$53$ $$T - 738$$
$59$ $$T + 564$$
$61$ $$T + 262$$
$67$ $$T - 554$$
$71$ $$T + 140$$
$73$ $$T + 882$$
$79$ $$T + 1160$$
$83$ $$T - 642$$
$89$ $$T - 854$$
$97$ $$T - 478$$