# Properties

 Label 1800.1.l.a Level $1800$ Weight $1$ Character orbit 1800.l Analytic conductor $0.898$ Analytic rank $0$ Dimension $2$ Projective image $S_{4}$ CM/RM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,1,Mod(1601,1800)]

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, 1, 0]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1800.1601");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1800.l (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$0.898317022739$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$S_{4}$$ Projective field: Galois closure of 4.2.10800.2 Artin image: $\GL(2,3)$ Artin field: Galois closure of 8.2.1399680000.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{7}+O(q^{10})$$ q - q^7 $$q - q^{7} - \beta q^{11} + q^{13} + \beta q^{17} + q^{19} - \beta q^{23} - \beta q^{29} + q^{31} + q^{43} - \beta q^{47} + \beta q^{59} + q^{61} - q^{67} + \beta q^{77} + \beta q^{83} - q^{91} + q^{97} +O(q^{100})$$ q - q^7 - b * q^11 + q^13 + b * q^17 + q^19 - b * q^23 - b * q^29 + q^31 + q^43 - b * q^47 + b * q^59 + q^61 - q^67 + b * q^77 + b * q^83 - q^91 + q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{7}+O(q^{10})$$ 2 * q - 2 * q^7 $$2 q - 2 q^{7} + 2 q^{13} + 2 q^{19} + 2 q^{31} + 2 q^{43} + 2 q^{61} - 2 q^{67} - 2 q^{91} + 2 q^{97}+O(q^{100})$$ 2 * q - 2 * q^7 + 2 * q^13 + 2 * q^19 + 2 * q^31 + 2 * q^43 + 2 * q^61 - 2 * q^67 - 2 * q^91 + 2 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1001$$ $$1351$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1601.1
 1.41421i − 1.41421i
0 0 0 0 0 −1.00000 0 0 0
1601.2 0 0 0 0 0 −1.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1800.1.l.a 2
3.b odd 2 1 inner 1800.1.l.a 2
4.b odd 2 1 3600.1.l.b 2
5.b even 2 1 1800.1.l.b yes 2
5.c odd 4 2 1800.1.c.a 4
12.b even 2 1 3600.1.l.b 2
15.d odd 2 1 1800.1.l.b yes 2
15.e even 4 2 1800.1.c.a 4
20.d odd 2 1 3600.1.l.a 2
20.e even 4 2 3600.1.c.a 4
60.h even 2 1 3600.1.l.a 2
60.l odd 4 2 3600.1.c.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1800.1.c.a 4 5.c odd 4 2
1800.1.c.a 4 15.e even 4 2
1800.1.l.a 2 1.a even 1 1 trivial
1800.1.l.a 2 3.b odd 2 1 inner
1800.1.l.b yes 2 5.b even 2 1
1800.1.l.b yes 2 15.d odd 2 1
3600.1.c.a 4 20.e even 4 2
3600.1.c.a 4 60.l odd 4 2
3600.1.l.a 2 20.d odd 2 1
3600.1.l.a 2 60.h even 2 1
3600.1.l.b 2 4.b odd 2 1
3600.1.l.b 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7} + 1$$ acting on $$S_{1}^{\mathrm{new}}(1800, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} + 2$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} + 2$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} + 2$$
$29$ $$T^{2} + 2$$
$31$ $$(T - 1)^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$(T - 1)^{2}$$
$47$ $$T^{2} + 2$$
$53$ $$T^{2}$$
$59$ $$T^{2} + 2$$
$61$ $$(T - 1)^{2}$$
$67$ $$(T + 1)^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 2$$
$89$ $$T^{2}$$
$97$ $$(T - 1)^{2}$$