# Properties

 Label 225.2.a.c Level $225$ Weight $2$ Character orbit 225.a Self dual yes Analytic conductor $1.797$ Analytic rank $1$ Dimension $1$ CM discriminant -3 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,2,Mod(1,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 225.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: $$1.79663404548$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 2 q^{4} - 5 q^{7}+O(q^{10})$$ q - 2 * q^4 - 5 * q^7 $$q - 2 q^{4} - 5 q^{7} - 5 q^{13} + 4 q^{16} - q^{19} + 10 q^{28} - 7 q^{31} + 10 q^{37} - 5 q^{43} + 18 q^{49} + 10 q^{52} - 13 q^{61} - 8 q^{64} - 5 q^{67} + 10 q^{73} + 2 q^{76} - 4 q^{79} + 25 q^{91} - 5 q^{97}+O(q^{100})$$ q - 2 * q^4 - 5 * q^7 - 5 * q^13 + 4 * q^16 - q^19 + 10 * q^28 - 7 * q^31 + 10 * q^37 - 5 * q^43 + 18 * q^49 + 10 * q^52 - 13 * q^61 - 8 * q^64 - 5 * q^67 + 10 * q^73 + 2 * q^76 - 4 * q^79 + 25 * q^91 - 5 * 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 −2.00000 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
$$3$$ $$+1$$
$$5$$ $$+1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.2.a.c 1
3.b odd 2 1 CM 225.2.a.c 1
4.b odd 2 1 3600.2.a.br 1
5.b even 2 1 225.2.a.d yes 1
5.c odd 4 2 225.2.b.c 2
12.b even 2 1 3600.2.a.br 1
15.d odd 2 1 225.2.a.d yes 1
15.e even 4 2 225.2.b.c 2
20.d odd 2 1 3600.2.a.b 1
20.e even 4 2 3600.2.f.k 2
60.h even 2 1 3600.2.a.b 1
60.l odd 4 2 3600.2.f.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.a.c 1 1.a even 1 1 trivial
225.2.a.c 1 3.b odd 2 1 CM
225.2.a.d yes 1 5.b even 2 1
225.2.a.d yes 1 15.d odd 2 1
225.2.b.c 2 5.c odd 4 2
225.2.b.c 2 15.e even 4 2
3600.2.a.b 1 20.d odd 2 1
3600.2.a.b 1 60.h even 2 1
3600.2.a.br 1 4.b odd 2 1
3600.2.a.br 1 12.b even 2 1
3600.2.f.k 2 20.e even 4 2
3600.2.f.k 2 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(225))$$:

 $$T_{2}$$ T2 $$T_{7} + 5$$ T7 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 5$$
$11$ $$T$$
$13$ $$T + 5$$
$17$ $$T$$
$19$ $$T + 1$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T + 7$$
$37$ $$T - 10$$
$41$ $$T$$
$43$ $$T + 5$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T + 13$$
$67$ $$T + 5$$
$71$ $$T$$
$73$ $$T - 10$$
$79$ $$T + 4$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T + 5$$