Properties

 Label 27.4.a.b Level $27$ Weight $4$ Character orbit 27.a Self dual yes Analytic conductor $1.593$ Analytic rank $0$ 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] = [27,4,Mod(1,27)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(27, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("27.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$27 = 3^{3}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 27.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: $$1.59305157015$$ Analytic rank: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{2} + q^{4} + 15 q^{5} - 25 q^{7} - 21 q^{8}+O(q^{10})$$ q + 3 * q^2 + q^4 + 15 * q^5 - 25 * q^7 - 21 * q^8 $$q + 3 q^{2} + q^{4} + 15 q^{5} - 25 q^{7} - 21 q^{8} + 45 q^{10} - 15 q^{11} + 20 q^{13} - 75 q^{14} - 71 q^{16} + 72 q^{17} + 2 q^{19} + 15 q^{20} - 45 q^{22} + 114 q^{23} + 100 q^{25} + 60 q^{26} - 25 q^{28} + 30 q^{29} + 101 q^{31} - 45 q^{32} + 216 q^{34} - 375 q^{35} - 430 q^{37} + 6 q^{38} - 315 q^{40} - 30 q^{41} + 110 q^{43} - 15 q^{44} + 342 q^{46} - 330 q^{47} + 282 q^{49} + 300 q^{50} + 20 q^{52} + 621 q^{53} - 225 q^{55} + 525 q^{56} + 90 q^{58} - 660 q^{59} - 376 q^{61} + 303 q^{62} + 433 q^{64} + 300 q^{65} - 250 q^{67} + 72 q^{68} - 1125 q^{70} - 360 q^{71} + 785 q^{73} - 1290 q^{74} + 2 q^{76} + 375 q^{77} + 488 q^{79} - 1065 q^{80} - 90 q^{82} + 489 q^{83} + 1080 q^{85} + 330 q^{86} + 315 q^{88} - 450 q^{89} - 500 q^{91} + 114 q^{92} - 990 q^{94} + 30 q^{95} - 1105 q^{97} + 846 q^{98}+O(q^{100})$$ q + 3 * q^2 + q^4 + 15 * q^5 - 25 * q^7 - 21 * q^8 + 45 * q^10 - 15 * q^11 + 20 * q^13 - 75 * q^14 - 71 * q^16 + 72 * q^17 + 2 * q^19 + 15 * q^20 - 45 * q^22 + 114 * q^23 + 100 * q^25 + 60 * q^26 - 25 * q^28 + 30 * q^29 + 101 * q^31 - 45 * q^32 + 216 * q^34 - 375 * q^35 - 430 * q^37 + 6 * q^38 - 315 * q^40 - 30 * q^41 + 110 * q^43 - 15 * q^44 + 342 * q^46 - 330 * q^47 + 282 * q^49 + 300 * q^50 + 20 * q^52 + 621 * q^53 - 225 * q^55 + 525 * q^56 + 90 * q^58 - 660 * q^59 - 376 * q^61 + 303 * q^62 + 433 * q^64 + 300 * q^65 - 250 * q^67 + 72 * q^68 - 1125 * q^70 - 360 * q^71 + 785 * q^73 - 1290 * q^74 + 2 * q^76 + 375 * q^77 + 488 * q^79 - 1065 * q^80 - 90 * q^82 + 489 * q^83 + 1080 * q^85 + 330 * q^86 + 315 * q^88 - 450 * q^89 - 500 * q^91 + 114 * q^92 - 990 * q^94 + 30 * q^95 - 1105 * q^97 + 846 * 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
3.00000 0 1.00000 15.0000 0 −25.0000 −21.0000 0 45.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$+1$$

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 27.4.a.b yes 1
3.b odd 2 1 27.4.a.a 1
4.b odd 2 1 432.4.a.n 1
5.b even 2 1 675.4.a.a 1
5.c odd 4 2 675.4.b.a 2
7.b odd 2 1 1323.4.a.k 1
8.b even 2 1 1728.4.a.c 1
8.d odd 2 1 1728.4.a.d 1
9.c even 3 2 81.4.c.a 2
9.d odd 6 2 81.4.c.c 2
12.b even 2 1 432.4.a.a 1
15.d odd 2 1 675.4.a.j 1
15.e even 4 2 675.4.b.b 2
21.c even 2 1 1323.4.a.d 1
24.f even 2 1 1728.4.a.bd 1
24.h odd 2 1 1728.4.a.bc 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.4.a.a 1 3.b odd 2 1
27.4.a.b yes 1 1.a even 1 1 trivial
81.4.c.a 2 9.c even 3 2
81.4.c.c 2 9.d odd 6 2
432.4.a.a 1 12.b even 2 1
432.4.a.n 1 4.b odd 2 1
675.4.a.a 1 5.b even 2 1
675.4.a.j 1 15.d odd 2 1
675.4.b.a 2 5.c odd 4 2
675.4.b.b 2 15.e even 4 2
1323.4.a.d 1 21.c even 2 1
1323.4.a.k 1 7.b odd 2 1
1728.4.a.c 1 8.b even 2 1
1728.4.a.d 1 8.d odd 2 1
1728.4.a.bc 1 24.h odd 2 1
1728.4.a.bd 1 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 3$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(27))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 3$$
$3$ $$T$$
$5$ $$T - 15$$
$7$ $$T + 25$$
$11$ $$T + 15$$
$13$ $$T - 20$$
$17$ $$T - 72$$
$19$ $$T - 2$$
$23$ $$T - 114$$
$29$ $$T - 30$$
$31$ $$T - 101$$
$37$ $$T + 430$$
$41$ $$T + 30$$
$43$ $$T - 110$$
$47$ $$T + 330$$
$53$ $$T - 621$$
$59$ $$T + 660$$
$61$ $$T + 376$$
$67$ $$T + 250$$
$71$ $$T + 360$$
$73$ $$T - 785$$
$79$ $$T - 488$$
$83$ $$T - 489$$
$89$ $$T + 450$$
$97$ $$T + 1105$$