# Properties

 Label 135.2.a.d Level $135$ Weight $2$ Character orbit 135.a Self dual yes Analytic conductor $1.078$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [135,2,Mod(1,135)]

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, 0]))

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

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

chi := DirichletCharacter("135.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$135 = 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 135.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: $$1.07798042729$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + (\beta + 1) q^{4} - q^{5} + ( - 2 \beta + 2) q^{7} + 3 q^{8}+O(q^{10})$$ q + b * q^2 + (b + 1) * q^4 - q^5 + (-2*b + 2) * q^7 + 3 * q^8 $$q + \beta q^{2} + (\beta + 1) q^{4} - q^{5} + ( - 2 \beta + 2) q^{7} + 3 q^{8} - \beta q^{10} - 2 \beta q^{11} + (2 \beta + 2) q^{13} - 6 q^{14} + (\beta - 2) q^{16} + ( - 2 \beta + 3) q^{17} + (2 \beta - 1) q^{19} + ( - \beta - 1) q^{20} + ( - 2 \beta - 6) q^{22} + 3 q^{23} + q^{25} + (4 \beta + 6) q^{26} + ( - 2 \beta - 4) q^{28} + (2 \beta - 6) q^{29} + ( - 2 \beta - 1) q^{31} + ( - \beta - 3) q^{32} + (\beta - 6) q^{34} + (2 \beta - 2) q^{35} + 2 q^{37} + (\beta + 6) q^{38} - 3 q^{40} + 2 \beta q^{41} + (2 \beta - 4) q^{43} + ( - 4 \beta - 6) q^{44} + 3 \beta q^{46} + 4 \beta q^{47} + ( - 4 \beta + 9) q^{49} + \beta q^{50} + (6 \beta + 8) q^{52} + ( - 2 \beta + 3) q^{53} + 2 \beta q^{55} + ( - 6 \beta + 6) q^{56} + ( - 4 \beta + 6) q^{58} + ( - 2 \beta + 6) q^{59} + ( - 4 \beta + 5) q^{61} + ( - 3 \beta - 6) q^{62} + ( - 6 \beta + 1) q^{64} + ( - 2 \beta - 2) q^{65} + (4 \beta - 10) q^{67} + ( - \beta - 3) q^{68} + 6 q^{70} + (2 \beta - 12) q^{71} + (2 \beta + 8) q^{73} + 2 \beta q^{74} + (3 \beta + 5) q^{76} + 12 q^{77} + ( - 2 \beta - 7) q^{79} + ( - \beta + 2) q^{80} + (2 \beta + 6) q^{82} - 3 q^{83} + (2 \beta - 3) q^{85} + ( - 2 \beta + 6) q^{86} - 6 \beta q^{88} + 6 \beta q^{89} + ( - 4 \beta - 8) q^{91} + (3 \beta + 3) q^{92} + (4 \beta + 12) q^{94} + ( - 2 \beta + 1) q^{95} + 8 q^{97} + (5 \beta - 12) q^{98} +O(q^{100})$$ q + b * q^2 + (b + 1) * q^4 - q^5 + (-2*b + 2) * q^7 + 3 * q^8 - b * q^10 - 2*b * q^11 + (2*b + 2) * q^13 - 6 * q^14 + (b - 2) * q^16 + (-2*b + 3) * q^17 + (2*b - 1) * q^19 + (-b - 1) * q^20 + (-2*b - 6) * q^22 + 3 * q^23 + q^25 + (4*b + 6) * q^26 + (-2*b - 4) * q^28 + (2*b - 6) * q^29 + (-2*b - 1) * q^31 + (-b - 3) * q^32 + (b - 6) * q^34 + (2*b - 2) * q^35 + 2 * q^37 + (b + 6) * q^38 - 3 * q^40 + 2*b * q^41 + (2*b - 4) * q^43 + (-4*b - 6) * q^44 + 3*b * q^46 + 4*b * q^47 + (-4*b + 9) * q^49 + b * q^50 + (6*b + 8) * q^52 + (-2*b + 3) * q^53 + 2*b * q^55 + (-6*b + 6) * q^56 + (-4*b + 6) * q^58 + (-2*b + 6) * q^59 + (-4*b + 5) * q^61 + (-3*b - 6) * q^62 + (-6*b + 1) * q^64 + (-2*b - 2) * q^65 + (4*b - 10) * q^67 + (-b - 3) * q^68 + 6 * q^70 + (2*b - 12) * q^71 + (2*b + 8) * q^73 + 2*b * q^74 + (3*b + 5) * q^76 + 12 * q^77 + (-2*b - 7) * q^79 + (-b + 2) * q^80 + (2*b + 6) * q^82 - 3 * q^83 + (2*b - 3) * q^85 + (-2*b + 6) * q^86 - 6*b * q^88 + 6*b * q^89 + (-4*b - 8) * q^91 + (3*b + 3) * q^92 + (4*b + 12) * q^94 + (-2*b + 1) * q^95 + 8 * q^97 + (5*b - 12) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 3 q^{4} - 2 q^{5} + 2 q^{7} + 6 q^{8}+O(q^{10})$$ 2 * q + q^2 + 3 * q^4 - 2 * q^5 + 2 * q^7 + 6 * q^8 $$2 q + q^{2} + 3 q^{4} - 2 q^{5} + 2 q^{7} + 6 q^{8} - q^{10} - 2 q^{11} + 6 q^{13} - 12 q^{14} - 3 q^{16} + 4 q^{17} - 3 q^{20} - 14 q^{22} + 6 q^{23} + 2 q^{25} + 16 q^{26} - 10 q^{28} - 10 q^{29} - 4 q^{31} - 7 q^{32} - 11 q^{34} - 2 q^{35} + 4 q^{37} + 13 q^{38} - 6 q^{40} + 2 q^{41} - 6 q^{43} - 16 q^{44} + 3 q^{46} + 4 q^{47} + 14 q^{49} + q^{50} + 22 q^{52} + 4 q^{53} + 2 q^{55} + 6 q^{56} + 8 q^{58} + 10 q^{59} + 6 q^{61} - 15 q^{62} - 4 q^{64} - 6 q^{65} - 16 q^{67} - 7 q^{68} + 12 q^{70} - 22 q^{71} + 18 q^{73} + 2 q^{74} + 13 q^{76} + 24 q^{77} - 16 q^{79} + 3 q^{80} + 14 q^{82} - 6 q^{83} - 4 q^{85} + 10 q^{86} - 6 q^{88} + 6 q^{89} - 20 q^{91} + 9 q^{92} + 28 q^{94} + 16 q^{97} - 19 q^{98}+O(q^{100})$$ 2 * q + q^2 + 3 * q^4 - 2 * q^5 + 2 * q^7 + 6 * q^8 - q^10 - 2 * q^11 + 6 * q^13 - 12 * q^14 - 3 * q^16 + 4 * q^17 - 3 * q^20 - 14 * q^22 + 6 * q^23 + 2 * q^25 + 16 * q^26 - 10 * q^28 - 10 * q^29 - 4 * q^31 - 7 * q^32 - 11 * q^34 - 2 * q^35 + 4 * q^37 + 13 * q^38 - 6 * q^40 + 2 * q^41 - 6 * q^43 - 16 * q^44 + 3 * q^46 + 4 * q^47 + 14 * q^49 + q^50 + 22 * q^52 + 4 * q^53 + 2 * q^55 + 6 * q^56 + 8 * q^58 + 10 * q^59 + 6 * q^61 - 15 * q^62 - 4 * q^64 - 6 * q^65 - 16 * q^67 - 7 * q^68 + 12 * q^70 - 22 * q^71 + 18 * q^73 + 2 * q^74 + 13 * q^76 + 24 * q^77 - 16 * q^79 + 3 * q^80 + 14 * q^82 - 6 * q^83 - 4 * q^85 + 10 * q^86 - 6 * q^88 + 6 * q^89 - 20 * q^91 + 9 * q^92 + 28 * q^94 + 16 * q^97 - 19 * q^98

## 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
 −1.30278 2.30278
−1.30278 0 −0.302776 −1.00000 0 4.60555 3.00000 0 1.30278
1.2 2.30278 0 3.30278 −1.00000 0 −2.60555 3.00000 0 −2.30278
 $$n$$: 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 135.2.a.d yes 2
3.b odd 2 1 135.2.a.c 2
4.b odd 2 1 2160.2.a.y 2
5.b even 2 1 675.2.a.k 2
5.c odd 4 2 675.2.b.h 4
7.b odd 2 1 6615.2.a.v 2
8.b even 2 1 8640.2.a.df 2
8.d odd 2 1 8640.2.a.cy 2
9.c even 3 2 405.2.e.j 4
9.d odd 6 2 405.2.e.k 4
12.b even 2 1 2160.2.a.ba 2
15.d odd 2 1 675.2.a.p 2
15.e even 4 2 675.2.b.i 4
21.c even 2 1 6615.2.a.p 2
24.f even 2 1 8640.2.a.ck 2
24.h odd 2 1 8640.2.a.cr 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.a.c 2 3.b odd 2 1
135.2.a.d yes 2 1.a even 1 1 trivial
405.2.e.j 4 9.c even 3 2
405.2.e.k 4 9.d odd 6 2
675.2.a.k 2 5.b even 2 1
675.2.a.p 2 15.d odd 2 1
675.2.b.h 4 5.c odd 4 2
675.2.b.i 4 15.e even 4 2
2160.2.a.y 2 4.b odd 2 1
2160.2.a.ba 2 12.b even 2 1
6615.2.a.p 2 21.c even 2 1
6615.2.a.v 2 7.b odd 2 1
8640.2.a.ck 2 24.f even 2 1
8640.2.a.cr 2 24.h odd 2 1
8640.2.a.cy 2 8.d odd 2 1
8640.2.a.df 2 8.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - T_{2} - 3$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(135))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 3$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - 2T - 12$$
$11$ $$T^{2} + 2T - 12$$
$13$ $$T^{2} - 6T - 4$$
$17$ $$T^{2} - 4T - 9$$
$19$ $$T^{2} - 13$$
$23$ $$(T - 3)^{2}$$
$29$ $$T^{2} + 10T + 12$$
$31$ $$T^{2} + 4T - 9$$
$37$ $$(T - 2)^{2}$$
$41$ $$T^{2} - 2T - 12$$
$43$ $$T^{2} + 6T - 4$$
$47$ $$T^{2} - 4T - 48$$
$53$ $$T^{2} - 4T - 9$$
$59$ $$T^{2} - 10T + 12$$
$61$ $$T^{2} - 6T - 43$$
$67$ $$T^{2} + 16T + 12$$
$71$ $$T^{2} + 22T + 108$$
$73$ $$T^{2} - 18T + 68$$
$79$ $$T^{2} + 16T + 51$$
$83$ $$(T + 3)^{2}$$
$89$ $$T^{2} - 6T - 108$$
$97$ $$(T - 8)^{2}$$