# Properties

 Label 225.2.a.f Level $225$ Weight $2$ Character orbit 225.a Self dual yes Analytic conductor $1.797$ Analytic rank $0$ Dimension $2$ CM discriminant -15 Inner twists $4$

# 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 45) 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + 3 q^{4} - \beta q^{8} +O(q^{10})$$ q - b * q^2 + 3 * q^4 - b * q^8 $$q - \beta q^{2} + 3 q^{4} - \beta q^{8} - q^{16} + 2 \beta q^{17} + 4 q^{19} + 4 \beta q^{23} + 8 q^{31} + 3 \beta q^{32} - 10 q^{34} - 4 \beta q^{38} - 20 q^{46} - 4 \beta q^{47} - 7 q^{49} - 2 \beta q^{53} + 2 q^{61} - 8 \beta q^{62} - 13 q^{64} + 6 \beta q^{68} + 12 q^{76} + 16 q^{79} - 8 \beta q^{83} + 12 \beta q^{92} + 20 q^{94} + 7 \beta q^{98} +O(q^{100})$$ q - b * q^2 + 3 * q^4 - b * q^8 - q^16 + 2*b * q^17 + 4 * q^19 + 4*b * q^23 + 8 * q^31 + 3*b * q^32 - 10 * q^34 - 4*b * q^38 - 20 * q^46 - 4*b * q^47 - 7 * q^49 - 2*b * q^53 + 2 * q^61 - 8*b * q^62 - 13 * q^64 + 6*b * q^68 + 12 * q^76 + 16 * q^79 - 8*b * q^83 + 12*b * q^92 + 20 * q^94 + 7*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{4}+O(q^{10})$$ 2 * q + 6 * q^4 $$2 q + 6 q^{4} - 2 q^{16} + 8 q^{19} + 16 q^{31} - 20 q^{34} - 40 q^{46} - 14 q^{49} + 4 q^{61} - 26 q^{64} + 24 q^{76} + 32 q^{79} + 40 q^{94}+O(q^{100})$$ 2 * q + 6 * q^4 - 2 * q^16 + 8 * q^19 + 16 * q^31 - 20 * q^34 - 40 * q^46 - 14 * q^49 + 4 * q^61 - 26 * q^64 + 24 * q^76 + 32 * q^79 + 40 * q^94

## 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
 1.61803 −0.618034
−2.23607 0 3.00000 0 0 0 −2.23607 0 0
1.2 2.23607 0 3.00000 0 0 0 2.23607 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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
3.b odd 2 1 inner
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.2.a.f 2
3.b odd 2 1 inner 225.2.a.f 2
4.b odd 2 1 3600.2.a.bs 2
5.b even 2 1 inner 225.2.a.f 2
5.c odd 4 2 45.2.b.a 2
12.b even 2 1 3600.2.a.bs 2
15.d odd 2 1 CM 225.2.a.f 2
15.e even 4 2 45.2.b.a 2
20.d odd 2 1 3600.2.a.bs 2
20.e even 4 2 720.2.f.d 2
35.f even 4 2 2205.2.d.a 2
40.i odd 4 2 2880.2.f.k 2
40.k even 4 2 2880.2.f.j 2
45.k odd 12 4 405.2.j.c 4
45.l even 12 4 405.2.j.c 4
60.h even 2 1 3600.2.a.bs 2
60.l odd 4 2 720.2.f.d 2
105.k odd 4 2 2205.2.d.a 2
120.q odd 4 2 2880.2.f.j 2
120.w even 4 2 2880.2.f.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.b.a 2 5.c odd 4 2
45.2.b.a 2 15.e even 4 2
225.2.a.f 2 1.a even 1 1 trivial
225.2.a.f 2 3.b odd 2 1 inner
225.2.a.f 2 5.b even 2 1 inner
225.2.a.f 2 15.d odd 2 1 CM
405.2.j.c 4 45.k odd 12 4
405.2.j.c 4 45.l even 12 4
720.2.f.d 2 20.e even 4 2
720.2.f.d 2 60.l odd 4 2
2205.2.d.a 2 35.f even 4 2
2205.2.d.a 2 105.k odd 4 2
2880.2.f.j 2 40.k even 4 2
2880.2.f.j 2 120.q odd 4 2
2880.2.f.k 2 40.i odd 4 2
2880.2.f.k 2 120.w even 4 2
3600.2.a.bs 2 4.b odd 2 1
3600.2.a.bs 2 12.b even 2 1
3600.2.a.bs 2 20.d odd 2 1
3600.2.a.bs 2 60.h even 2 1

## Hecke kernels

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

 $$T_{2}^{2} - 5$$ T2^2 - 5 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 5$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 20$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2} - 80$$
$29$ $$T^{2}$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2} - 80$$
$53$ $$T^{2} - 20$$
$59$ $$T^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$(T - 16)^{2}$$
$83$ $$T^{2} - 320$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$