# Properties

 Label 135.4.a.a Level $135$ Weight $4$ Character orbit 135.a Self dual yes Analytic conductor $7.965$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [135,4,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, 4, names="a")

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

chi := DirichletCharacter("135.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$135 = 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$7.96525785077$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} - 4 q^{4} + 5 q^{5} + 24 q^{8}+O(q^{10})$$ q - 2 * q^2 - 4 * q^4 + 5 * q^5 + 24 * q^8 $$q - 2 q^{2} - 4 q^{4} + 5 q^{5} + 24 q^{8} - 10 q^{10} + 10 q^{11} - 80 q^{13} - 16 q^{16} + 7 q^{17} - 113 q^{19} - 20 q^{20} - 20 q^{22} - 81 q^{23} + 25 q^{25} + 160 q^{26} - 220 q^{29} - 189 q^{31} - 160 q^{32} - 14 q^{34} + 170 q^{37} + 226 q^{38} + 120 q^{40} - 130 q^{41} + 10 q^{43} - 40 q^{44} + 162 q^{46} + 160 q^{47} - 343 q^{49} - 50 q^{50} + 320 q^{52} + 631 q^{53} + 50 q^{55} + 440 q^{58} - 560 q^{59} + 229 q^{61} + 378 q^{62} + 448 q^{64} - 400 q^{65} + 750 q^{67} - 28 q^{68} + 890 q^{71} - 890 q^{73} - 340 q^{74} + 452 q^{76} - 27 q^{79} - 80 q^{80} + 260 q^{82} + 429 q^{83} + 35 q^{85} - 20 q^{86} + 240 q^{88} - 750 q^{89} + 324 q^{92} - 320 q^{94} - 565 q^{95} - 1480 q^{97} + 686 q^{98}+O(q^{100})$$ q - 2 * q^2 - 4 * q^4 + 5 * q^5 + 24 * q^8 - 10 * q^10 + 10 * q^11 - 80 * q^13 - 16 * q^16 + 7 * q^17 - 113 * q^19 - 20 * q^20 - 20 * q^22 - 81 * q^23 + 25 * q^25 + 160 * q^26 - 220 * q^29 - 189 * q^31 - 160 * q^32 - 14 * q^34 + 170 * q^37 + 226 * q^38 + 120 * q^40 - 130 * q^41 + 10 * q^43 - 40 * q^44 + 162 * q^46 + 160 * q^47 - 343 * q^49 - 50 * q^50 + 320 * q^52 + 631 * q^53 + 50 * q^55 + 440 * q^58 - 560 * q^59 + 229 * q^61 + 378 * q^62 + 448 * q^64 - 400 * q^65 + 750 * q^67 - 28 * q^68 + 890 * q^71 - 890 * q^73 - 340 * q^74 + 452 * q^76 - 27 * q^79 - 80 * q^80 + 260 * q^82 + 429 * q^83 + 35 * q^85 - 20 * q^86 + 240 * q^88 - 750 * q^89 + 324 * q^92 - 320 * q^94 - 565 * q^95 - 1480 * q^97 + 686 * 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
 0
−2.00000 0 −4.00000 5.00000 0 0 24.0000 0 −10.0000
 $$n$$: 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 135.4.a.a 1
3.b odd 2 1 135.4.a.d yes 1
4.b odd 2 1 2160.4.a.n 1
5.b even 2 1 675.4.a.i 1
5.c odd 4 2 675.4.b.d 2
9.c even 3 2 405.4.e.j 2
9.d odd 6 2 405.4.e.e 2
12.b even 2 1 2160.4.a.d 1
15.d odd 2 1 675.4.a.b 1
15.e even 4 2 675.4.b.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.a 1 1.a even 1 1 trivial
135.4.a.d yes 1 3.b odd 2 1
405.4.e.e 2 9.d odd 6 2
405.4.e.j 2 9.c even 3 2
675.4.a.b 1 15.d odd 2 1
675.4.a.i 1 5.b even 2 1
675.4.b.c 2 15.e even 4 2
675.4.b.d 2 5.c odd 4 2
2160.4.a.d 1 12.b even 2 1
2160.4.a.n 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 2$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(135))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T$$
$5$ $$T - 5$$
$7$ $$T$$
$11$ $$T - 10$$
$13$ $$T + 80$$
$17$ $$T - 7$$
$19$ $$T + 113$$
$23$ $$T + 81$$
$29$ $$T + 220$$
$31$ $$T + 189$$
$37$ $$T - 170$$
$41$ $$T + 130$$
$43$ $$T - 10$$
$47$ $$T - 160$$
$53$ $$T - 631$$
$59$ $$T + 560$$
$61$ $$T - 229$$
$67$ $$T - 750$$
$71$ $$T - 890$$
$73$ $$T + 890$$
$79$ $$T + 27$$
$83$ $$T - 429$$
$89$ $$T + 750$$
$97$ $$T + 1480$$