# Properties

 Label 45.2.b.a Level $45$ Weight $2$ Character orbit 45.b Analytic conductor $0.359$ 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] = [45,2,Mod(19,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("45.19");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 45.b (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: $$0.359326809096$$ 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: $$x^{2} + 5$$ x^2 + 5 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - 3 q^{4} - \beta q^{5} - \beta q^{8} +O(q^{10})$$ q + b * q^2 - 3 * q^4 - b * q^5 - b * q^8 $$q + \beta q^{2} - 3 q^{4} - \beta q^{5} - \beta q^{8} + 5 q^{10} - q^{16} - 2 \beta q^{17} - 4 q^{19} + 3 \beta q^{20} + 4 \beta q^{23} - 5 q^{25} + 8 q^{31} - 3 \beta q^{32} + 10 q^{34} - 4 \beta q^{38} - 5 q^{40} - 20 q^{46} + 4 \beta q^{47} + 7 q^{49} - 5 \beta q^{50} - 2 \beta q^{53} + 2 q^{61} + 8 \beta q^{62} + 13 q^{64} + 6 \beta q^{68} + 12 q^{76} - 16 q^{79} + \beta q^{80} - 8 \beta q^{83} - 10 q^{85} - 12 \beta q^{92} - 20 q^{94} + 4 \beta q^{95} + 7 \beta q^{98} +O(q^{100})$$ q + b * q^2 - 3 * q^4 - b * q^5 - b * q^8 + 5 * q^10 - q^16 - 2*b * q^17 - 4 * q^19 + 3*b * q^20 + 4*b * q^23 - 5 * q^25 + 8 * q^31 - 3*b * q^32 + 10 * q^34 - 4*b * q^38 - 5 * q^40 - 20 * q^46 + 4*b * q^47 + 7 * q^49 - 5*b * q^50 - 2*b * q^53 + 2 * q^61 + 8*b * q^62 + 13 * q^64 + 6*b * q^68 + 12 * q^76 - 16 * q^79 + b * q^80 - 8*b * q^83 - 10 * q^85 - 12*b * q^92 - 20 * q^94 + 4*b * q^95 + 7*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{4}+O(q^{10})$$ 2 * q - 6 * q^4 $$2 q - 6 q^{4} + 10 q^{10} - 2 q^{16} - 8 q^{19} - 10 q^{25} + 16 q^{31} + 20 q^{34} - 10 q^{40} - 40 q^{46} + 14 q^{49} + 4 q^{61} + 26 q^{64} + 24 q^{76} - 32 q^{79} - 20 q^{85} - 40 q^{94}+O(q^{100})$$ 2 * q - 6 * q^4 + 10 * q^10 - 2 * q^16 - 8 * q^19 - 10 * q^25 + 16 * q^31 + 20 * q^34 - 10 * q^40 - 40 * q^46 + 14 * q^49 + 4 * q^61 + 26 * q^64 + 24 * q^76 - 32 * q^79 - 20 * q^85 - 40 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$37$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 − 2.23607i 2.23607i
2.23607i 0 −3.00000 2.23607i 0 0 2.23607i 0 5.00000
19.2 2.23607i 0 −3.00000 2.23607i 0 0 2.23607i 0 5.00000
 $$n$$: 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
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 45.2.b.a 2
3.b odd 2 1 inner 45.2.b.a 2
4.b odd 2 1 720.2.f.d 2
5.b even 2 1 inner 45.2.b.a 2
5.c odd 4 2 225.2.a.f 2
7.b odd 2 1 2205.2.d.a 2
8.b even 2 1 2880.2.f.k 2
8.d odd 2 1 2880.2.f.j 2
9.c even 3 2 405.2.j.c 4
9.d odd 6 2 405.2.j.c 4
12.b even 2 1 720.2.f.d 2
15.d odd 2 1 CM 45.2.b.a 2
15.e even 4 2 225.2.a.f 2
20.d odd 2 1 720.2.f.d 2
20.e even 4 2 3600.2.a.bs 2
21.c even 2 1 2205.2.d.a 2
24.f even 2 1 2880.2.f.j 2
24.h odd 2 1 2880.2.f.k 2
35.c odd 2 1 2205.2.d.a 2
40.e odd 2 1 2880.2.f.j 2
40.f even 2 1 2880.2.f.k 2
45.h odd 6 2 405.2.j.c 4
45.j even 6 2 405.2.j.c 4
60.h even 2 1 720.2.f.d 2
60.l odd 4 2 3600.2.a.bs 2
105.g even 2 1 2205.2.d.a 2
120.i odd 2 1 2880.2.f.k 2
120.m even 2 1 2880.2.f.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.b.a 2 1.a even 1 1 trivial
45.2.b.a 2 3.b odd 2 1 inner
45.2.b.a 2 5.b even 2 1 inner
45.2.b.a 2 15.d odd 2 1 CM
225.2.a.f 2 5.c odd 4 2
225.2.a.f 2 15.e even 4 2
405.2.j.c 4 9.c even 3 2
405.2.j.c 4 9.d odd 6 2
405.2.j.c 4 45.h odd 6 2
405.2.j.c 4 45.j even 6 2
720.2.f.d 2 4.b odd 2 1
720.2.f.d 2 12.b even 2 1
720.2.f.d 2 20.d odd 2 1
720.2.f.d 2 60.h even 2 1
2205.2.d.a 2 7.b odd 2 1
2205.2.d.a 2 21.c even 2 1
2205.2.d.a 2 35.c odd 2 1
2205.2.d.a 2 105.g even 2 1
2880.2.f.j 2 8.d odd 2 1
2880.2.f.j 2 24.f even 2 1
2880.2.f.j 2 40.e odd 2 1
2880.2.f.j 2 120.m even 2 1
2880.2.f.k 2 8.b even 2 1
2880.2.f.k 2 24.h odd 2 1
2880.2.f.k 2 40.f even 2 1
2880.2.f.k 2 120.i odd 2 1
3600.2.a.bs 2 20.e even 4 2
3600.2.a.bs 2 60.l odd 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(45, [\chi])$$.

## Hecke characteristic polynomials

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