# Properties

 Label 135.2.b.a Level $135$ Weight $2$ Character orbit 135.b Analytic conductor $1.078$ Analytic rank $0$ Dimension $4$ CM discriminant -15 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [135,2,Mod(109,135)]

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([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("135.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$135 = 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 135.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: $$1.07798042729$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$3^{2}$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{3} - 2) q^{4} + (\beta_{2} - \beta_1) q^{5} + ( - \beta_{2} + 3 \beta_1) q^{8}+O(q^{10})$$ q - b1 * q^2 + (b3 - 2) * q^4 + (b2 - b1) * q^5 + (-b2 + 3*b1) * q^8 $$q - \beta_1 q^{2} + (\beta_{3} - 2) q^{4} + (\beta_{2} - \beta_1) q^{5} + ( - \beta_{2} + 3 \beta_1) q^{8} + (\beta_{3} - 3) q^{10} + ( - \beta_{3} + 7) q^{16} + ( - 3 \beta_{2} - \beta_1) q^{17} + ( - 2 \beta_{3} + 3) q^{19} + (\beta_{2} + 4 \beta_1) q^{20} + (3 \beta_{2} - \beta_1) q^{23} - 5 q^{25} + ( - 2 \beta_{3} - 3) q^{31} + ( - \beta_{2} - 4 \beta_1) q^{32} + (\beta_{3} - 7) q^{34} + (2 \beta_{2} - 9 \beta_1) q^{38} + ( - 2 \beta_{3} + 11) q^{40} + (\beta_{3} - 1) q^{46} + ( - 4 \beta_{2} + 4 \beta_1) q^{47} + 7 q^{49} + 5 \beta_1 q^{50} + (3 \beta_{2} + 5 \beta_1) q^{53} + (4 \beta_{3} - 3) q^{61} + (2 \beta_{2} - 3 \beta_1) q^{62} + (2 \beta_{3} - 3) q^{64} + ( - 7 \beta_{2} + 8 \beta_1) q^{68} + (5 \beta_{3} - 28) q^{76} + ( - 2 \beta_{3} + 9) q^{79} + (4 \beta_{2} - 9 \beta_1) q^{80} + ( - 3 \beta_{2} + 5 \beta_1) q^{83} + (4 \beta_{3} + 3) q^{85} + (5 \beta_{2} + 2 \beta_1) q^{92} + ( - 4 \beta_{3} + 12) q^{94} + ( - 3 \beta_{2} - 7 \beta_1) q^{95} - 7 \beta_1 q^{98}+O(q^{100})$$ q - b1 * q^2 + (b3 - 2) * q^4 + (b2 - b1) * q^5 + (-b2 + 3*b1) * q^8 + (b3 - 3) * q^10 + (-b3 + 7) * q^16 + (-3*b2 - b1) * q^17 + (-2*b3 + 3) * q^19 + (b2 + 4*b1) * q^20 + (3*b2 - b1) * q^23 - 5 * q^25 + (-2*b3 - 3) * q^31 + (-b2 - 4*b1) * q^32 + (b3 - 7) * q^34 + (2*b2 - 9*b1) * q^38 + (-2*b3 + 11) * q^40 + (b3 - 1) * q^46 + (-4*b2 + 4*b1) * q^47 + 7 * q^49 + 5*b1 * q^50 + (3*b2 + 5*b1) * q^53 + (4*b3 - 3) * q^61 + (2*b2 - 3*b1) * q^62 + (2*b3 - 3) * q^64 + (-7*b2 + 8*b1) * q^68 + (5*b3 - 28) * q^76 + (-2*b3 + 9) * q^79 + (4*b2 - 9*b1) * q^80 + (-3*b2 + 5*b1) * q^83 + (4*b3 + 3) * q^85 + (5*b2 + 2*b1) * q^92 + (-4*b3 + 12) * q^94 + (-3*b2 - 7*b1) * q^95 - 7*b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{4}+O(q^{10})$$ 4 * q - 6 * q^4 $$4 q - 6 q^{4} - 10 q^{10} + 26 q^{16} + 8 q^{19} - 20 q^{25} - 16 q^{31} - 26 q^{34} + 40 q^{40} - 2 q^{46} + 28 q^{49} - 4 q^{61} - 8 q^{64} - 102 q^{76} + 32 q^{79} + 20 q^{85} + 40 q^{94}+O(q^{100})$$ 4 * q - 6 * q^4 - 10 * q^10 + 26 * q^16 + 8 * q^19 - 20 * q^25 - 16 * q^31 - 26 * q^34 + 40 * q^40 - 2 * q^46 + 28 * q^49 - 4 * q^61 - 8 * q^64 - 102 * q^76 + 32 * q^79 + 20 * q^85 + 40 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} + \nu$$ v^3 + v $$\beta_{2}$$ $$=$$ $$2\nu^{3} + 5\nu$$ 2*v^3 + 5*v $$\beta_{3}$$ $$=$$ $$3\nu^{2} + 5$$ 3*v^2 + 5
 $$\nu$$ $$=$$ $$( \beta_{2} - 2\beta_1 ) / 3$$ (b2 - 2*b1) / 3 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 5 ) / 3$$ (b3 - 5) / 3 $$\nu^{3}$$ $$=$$ $$( -\beta_{2} + 5\beta_1 ) / 3$$ (-b2 + 5*b1) / 3

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 − 1.61803i 0.618034i − 0.618034i 1.61803i
2.61803i 0 −4.85410 2.23607i 0 0 7.47214i 0 −5.85410
109.2 0.381966i 0 1.85410 2.23607i 0 0 1.47214i 0 0.854102
109.3 0.381966i 0 1.85410 2.23607i 0 0 1.47214i 0 0.854102
109.4 2.61803i 0 −4.85410 2.23607i 0 0 7.47214i 0 −5.85410
 $$n$$: 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 135.2.b.a 4
3.b odd 2 1 inner 135.2.b.a 4
4.b odd 2 1 2160.2.f.j 4
5.b even 2 1 inner 135.2.b.a 4
5.c odd 4 1 675.2.a.j 2
5.c odd 4 1 675.2.a.q 2
9.c even 3 2 405.2.j.h 8
9.d odd 6 2 405.2.j.h 8
12.b even 2 1 2160.2.f.j 4
15.d odd 2 1 CM 135.2.b.a 4
15.e even 4 1 675.2.a.j 2
15.e even 4 1 675.2.a.q 2
20.d odd 2 1 2160.2.f.j 4
45.h odd 6 2 405.2.j.h 8
45.j even 6 2 405.2.j.h 8
60.h even 2 1 2160.2.f.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.b.a 4 1.a even 1 1 trivial
135.2.b.a 4 3.b odd 2 1 inner
135.2.b.a 4 5.b even 2 1 inner
135.2.b.a 4 15.d odd 2 1 CM
405.2.j.h 8 9.c even 3 2
405.2.j.h 8 9.d odd 6 2
405.2.j.h 8 45.h odd 6 2
405.2.j.h 8 45.j even 6 2
675.2.a.j 2 5.c odd 4 1
675.2.a.j 2 15.e even 4 1
675.2.a.q 2 5.c odd 4 1
675.2.a.q 2 15.e even 4 1
2160.2.f.j 4 4.b odd 2 1
2160.2.f.j 4 12.b even 2 1
2160.2.f.j 4 20.d odd 2 1
2160.2.f.j 4 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 7T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(135, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 7T^{2} + 1$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 5)^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4} + 82T^{2} + 961$$
$19$ $$(T^{2} - 4 T - 41)^{2}$$
$23$ $$T^{4} + 58T^{2} + 121$$
$29$ $$T^{4}$$
$31$ $$(T^{2} + 8 T - 29)^{2}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$(T^{2} + 80)^{2}$$
$53$ $$T^{4} + 298 T^{2} + 19321$$
$59$ $$T^{4}$$
$61$ $$(T^{2} + 2 T - 179)^{2}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$(T^{2} - 16 T + 19)^{2}$$
$83$ $$T^{4} + 178T^{2} + 5041$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$