# Properties

 Label 54.2.a.b Level $54$ Weight $2$ Character orbit 54.a Self dual yes Analytic conductor $0.431$ 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] = [54,2,Mod(1,54)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(54, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("54.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$54 = 2 \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 54.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: $$0.431192170915$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - 3 q^{5} - q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 - 3 * q^5 - q^7 + q^8 $$q + q^{2} + q^{4} - 3 q^{5} - q^{7} + q^{8} - 3 q^{10} + 3 q^{11} - 4 q^{13} - q^{14} + q^{16} + 2 q^{19} - 3 q^{20} + 3 q^{22} + 6 q^{23} + 4 q^{25} - 4 q^{26} - q^{28} - 6 q^{29} + 5 q^{31} + q^{32} + 3 q^{35} + 2 q^{37} + 2 q^{38} - 3 q^{40} + 6 q^{41} - 10 q^{43} + 3 q^{44} + 6 q^{46} - 6 q^{47} - 6 q^{49} + 4 q^{50} - 4 q^{52} - 9 q^{53} - 9 q^{55} - q^{56} - 6 q^{58} - 12 q^{59} + 8 q^{61} + 5 q^{62} + q^{64} + 12 q^{65} + 14 q^{67} + 3 q^{70} - 7 q^{73} + 2 q^{74} + 2 q^{76} - 3 q^{77} + 8 q^{79} - 3 q^{80} + 6 q^{82} + 3 q^{83} - 10 q^{86} + 3 q^{88} + 18 q^{89} + 4 q^{91} + 6 q^{92} - 6 q^{94} - 6 q^{95} - q^{97} - 6 q^{98}+O(q^{100})$$ q + q^2 + q^4 - 3 * q^5 - q^7 + q^8 - 3 * q^10 + 3 * q^11 - 4 * q^13 - q^14 + q^16 + 2 * q^19 - 3 * q^20 + 3 * q^22 + 6 * q^23 + 4 * q^25 - 4 * q^26 - q^28 - 6 * q^29 + 5 * q^31 + q^32 + 3 * q^35 + 2 * q^37 + 2 * q^38 - 3 * q^40 + 6 * q^41 - 10 * q^43 + 3 * q^44 + 6 * q^46 - 6 * q^47 - 6 * q^49 + 4 * q^50 - 4 * q^52 - 9 * q^53 - 9 * q^55 - q^56 - 6 * q^58 - 12 * q^59 + 8 * q^61 + 5 * q^62 + q^64 + 12 * q^65 + 14 * q^67 + 3 * q^70 - 7 * q^73 + 2 * q^74 + 2 * q^76 - 3 * q^77 + 8 * q^79 - 3 * q^80 + 6 * q^82 + 3 * q^83 - 10 * q^86 + 3 * q^88 + 18 * q^89 + 4 * q^91 + 6 * q^92 - 6 * q^94 - 6 * q^95 - q^97 - 6 * 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
1.00000 0 1.00000 −3.00000 0 −1.00000 1.00000 0 −3.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$

## 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 54.2.a.b yes 1
3.b odd 2 1 54.2.a.a 1
4.b odd 2 1 432.2.a.b 1
5.b even 2 1 1350.2.a.h 1
5.c odd 4 2 1350.2.c.k 2
7.b odd 2 1 2646.2.a.bd 1
8.b even 2 1 1728.2.a.y 1
8.d odd 2 1 1728.2.a.z 1
9.c even 3 2 162.2.c.b 2
9.d odd 6 2 162.2.c.c 2
11.b odd 2 1 6534.2.a.b 1
12.b even 2 1 432.2.a.g 1
13.b even 2 1 9126.2.a.r 1
15.d odd 2 1 1350.2.a.r 1
15.e even 4 2 1350.2.c.b 2
21.c even 2 1 2646.2.a.a 1
24.f even 2 1 1728.2.a.d 1
24.h odd 2 1 1728.2.a.c 1
33.d even 2 1 6534.2.a.bc 1
36.f odd 6 2 1296.2.i.o 2
36.h even 6 2 1296.2.i.c 2
39.d odd 2 1 9126.2.a.u 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.2.a.a 1 3.b odd 2 1
54.2.a.b yes 1 1.a even 1 1 trivial
162.2.c.b 2 9.c even 3 2
162.2.c.c 2 9.d odd 6 2
432.2.a.b 1 4.b odd 2 1
432.2.a.g 1 12.b even 2 1
1296.2.i.c 2 36.h even 6 2
1296.2.i.o 2 36.f odd 6 2
1350.2.a.h 1 5.b even 2 1
1350.2.a.r 1 15.d odd 2 1
1350.2.c.b 2 15.e even 4 2
1350.2.c.k 2 5.c odd 4 2
1728.2.a.c 1 24.h odd 2 1
1728.2.a.d 1 24.f even 2 1
1728.2.a.y 1 8.b even 2 1
1728.2.a.z 1 8.d odd 2 1
2646.2.a.a 1 21.c even 2 1
2646.2.a.bd 1 7.b odd 2 1
6534.2.a.b 1 11.b odd 2 1
6534.2.a.bc 1 33.d even 2 1
9126.2.a.r 1 13.b even 2 1
9126.2.a.u 1 39.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 3$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(54))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T + 3$$
$7$ $$T + 1$$
$11$ $$T - 3$$
$13$ $$T + 4$$
$17$ $$T$$
$19$ $$T - 2$$
$23$ $$T - 6$$
$29$ $$T + 6$$
$31$ $$T - 5$$
$37$ $$T - 2$$
$41$ $$T - 6$$
$43$ $$T + 10$$
$47$ $$T + 6$$
$53$ $$T + 9$$
$59$ $$T + 12$$
$61$ $$T - 8$$
$67$ $$T - 14$$
$71$ $$T$$
$73$ $$T + 7$$
$79$ $$T - 8$$
$83$ $$T - 3$$
$89$ $$T - 18$$
$97$ $$T + 1$$