# Properties

 Label 225.1.g.a Level $225$ Weight $1$ Character orbit 225.g Analytic conductor $0.112$ Analytic rank $0$ Dimension $2$ Projective image $D_{2}$ CM/RM discs -3, -15, 5 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,1,Mod(82,225)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("225.82");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 225.g (of order $$4$$, degree $$2$$, 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: $$0.112289627842$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{-3}, \sqrt{5})$$ Artin image: $\OD_{16}$ Artin field: Galois closure of 8.4.56953125.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + i q^{4}+O(q^{10})$$ q + z * q^4 $$q + i q^{4} - q^{16} - 2 i q^{19} - 2 q^{31} + i q^{49} + 2 q^{61} - i q^{64} + 2 q^{76} + 2 i q^{79} +O(q^{100})$$ q + z * q^4 - q^16 - 2*z * q^19 - 2 * q^31 + z * q^49 + 2 * q^61 - z * q^64 + 2 * q^76 + 2*z * q^79 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 2 q^{16} - 4 q^{31} + 4 q^{61} + 4 q^{76}+O(q^{100})$$ 2 * q - 2 * q^16 - 4 * q^31 + 4 * q^61 + 4 * q^76

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-i$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
82.1
 − 1.00000i 1.00000i
0 0 1.00000i 0 0 0 0 0 0
118.1 0 0 1.00000i 0 0 0 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 CM by $$\Q(\sqrt{-3})$$
5.b even 2 1 RM by $$\Q(\sqrt{5})$$
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
5.c odd 4 2 inner
15.e even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.1.g.a 2
3.b odd 2 1 CM 225.1.g.a 2
4.b odd 2 1 3600.1.bh.a 2
5.b even 2 1 RM 225.1.g.a 2
5.c odd 4 2 inner 225.1.g.a 2
9.c even 3 2 2025.1.p.a 4
9.d odd 6 2 2025.1.p.a 4
12.b even 2 1 3600.1.bh.a 2
15.d odd 2 1 CM 225.1.g.a 2
15.e even 4 2 inner 225.1.g.a 2
20.d odd 2 1 3600.1.bh.a 2
20.e even 4 2 3600.1.bh.a 2
45.h odd 6 2 2025.1.p.a 4
45.j even 6 2 2025.1.p.a 4
45.k odd 12 4 2025.1.p.a 4
45.l even 12 4 2025.1.p.a 4
60.h even 2 1 3600.1.bh.a 2
60.l odd 4 2 3600.1.bh.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.1.g.a 2 1.a even 1 1 trivial
225.1.g.a 2 3.b odd 2 1 CM
225.1.g.a 2 5.b even 2 1 RM
225.1.g.a 2 5.c odd 4 2 inner
225.1.g.a 2 15.d odd 2 1 CM
225.1.g.a 2 15.e even 4 2 inner
2025.1.p.a 4 9.c even 3 2
2025.1.p.a 4 9.d odd 6 2
2025.1.p.a 4 45.h odd 6 2
2025.1.p.a 4 45.j even 6 2
2025.1.p.a 4 45.k odd 12 4
2025.1.p.a 4 45.l even 12 4
3600.1.bh.a 2 4.b odd 2 1
3600.1.bh.a 2 12.b even 2 1
3600.1.bh.a 2 20.d odd 2 1
3600.1.bh.a 2 20.e even 4 2
3600.1.bh.a 2 60.h even 2 1
3600.1.bh.a 2 60.l odd 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(225, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 4$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$(T + 2)^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2} + 4$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$