# Properties

 Label 135.1.d.a Level $135$ Weight $1$ Character orbit 135.d Self dual yes Analytic conductor $0.067$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -15 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [135,1,Mod(134,135)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(135, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("135.134");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$135 = 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 135.d (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: yes Analytic conductor: $$0.0673737767055$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.135.1 Artin image: $S_3$ Artin field: Galois closure of 3.1.135.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{5} + q^{8}+O(q^{10})$$ q - q^2 + q^5 + q^8 $$q - q^{2} + q^{5} + q^{8} - q^{10} - q^{16} - q^{17} - q^{19} - q^{23} + q^{25} - q^{31} + q^{34} + q^{38} + q^{40} + q^{46} + 2 q^{47} + q^{49} - q^{50} - q^{53} - q^{61} + q^{62} + q^{64} - q^{79} - q^{80} - q^{83} - q^{85} - 2 q^{94} - q^{95} - q^{98}+O(q^{100})$$ q - q^2 + q^5 + q^8 - q^10 - q^16 - q^17 - q^19 - q^23 + q^25 - q^31 + q^34 + q^38 + q^40 + q^46 + 2 * q^47 + q^49 - q^50 - q^53 - q^61 + q^62 + q^64 - q^79 - q^80 - q^83 - q^85 - 2 * q^94 - q^95 - q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/135\mathbb{Z}\right)^\times$$.

 $$n$$ $$56$$ $$82$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
134.1
 0
−1.00000 0 0 1.00000 0 0 1.00000 0 −1.00000
 $$n$$: 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 $$\Q(\sqrt{-15})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 135.1.d.a 1
3.b odd 2 1 135.1.d.b yes 1
4.b odd 2 1 2160.1.c.b 1
5.b even 2 1 135.1.d.b yes 1
5.c odd 4 2 675.1.c.c 2
9.c even 3 2 405.1.h.b 2
9.d odd 6 2 405.1.h.a 2
12.b even 2 1 2160.1.c.a 1
15.d odd 2 1 CM 135.1.d.a 1
15.e even 4 2 675.1.c.c 2
20.d odd 2 1 2160.1.c.a 1
27.e even 9 6 3645.1.n.d 6
27.f odd 18 6 3645.1.n.e 6
45.h odd 6 2 405.1.h.b 2
45.j even 6 2 405.1.h.a 2
45.k odd 12 4 2025.1.j.c 4
45.l even 12 4 2025.1.j.c 4
60.h even 2 1 2160.1.c.b 1
135.n odd 18 6 3645.1.n.d 6
135.p even 18 6 3645.1.n.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.1.d.a 1 1.a even 1 1 trivial
135.1.d.a 1 15.d odd 2 1 CM
135.1.d.b yes 1 3.b odd 2 1
135.1.d.b yes 1 5.b even 2 1
405.1.h.a 2 9.d odd 6 2
405.1.h.a 2 45.j even 6 2
405.1.h.b 2 9.c even 3 2
405.1.h.b 2 45.h odd 6 2
675.1.c.c 2 5.c odd 4 2
675.1.c.c 2 15.e even 4 2
2025.1.j.c 4 45.k odd 12 4
2025.1.j.c 4 45.l even 12 4
2160.1.c.a 1 12.b even 2 1
2160.1.c.a 1 20.d odd 2 1
2160.1.c.b 1 4.b odd 2 1
2160.1.c.b 1 60.h even 2 1
3645.1.n.d 6 27.e even 9 6
3645.1.n.d 6 135.n odd 18 6
3645.1.n.e 6 27.f odd 18 6
3645.1.n.e 6 135.p even 18 6

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(135, [\chi])$$.

## Hecke characteristic polynomials

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