# Properties

 Label 1458.2.a.d Level $1458$ Weight $2$ Character orbit 1458.a Self dual yes Analytic conductor $11.642$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1458,2,Mod(1,1458)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1458, 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("1458.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1458 = 2 \cdot 3^{6}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1458.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: $$11.6421886147$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 3x - 1$$ x^3 - 3*x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 54) 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + (\beta_1 - 1) q^{5} + ( - \beta_{2} - 2) q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + (b1 - 1) * q^5 + (-b2 - 2) * q^7 + q^8 $$q + q^{2} + q^{4} + (\beta_1 - 1) q^{5} + ( - \beta_{2} - 2) q^{7} + q^{8} + (\beta_1 - 1) q^{10} + (2 \beta_{2} - 3 \beta_1 - 1) q^{11} + (2 \beta_{2} - 2 \beta_1 - 3) q^{13} + ( - \beta_{2} - 2) q^{14} + q^{16} + ( - 4 \beta_{2} + \beta_1 - 2) q^{17} + (\beta_{2} + 3 \beta_1 - 3) q^{19} + (\beta_1 - 1) q^{20} + (2 \beta_{2} - 3 \beta_1 - 1) q^{22} + ( - 2 \beta_{2} + 1) q^{23} + (\beta_{2} - 2 \beta_1 - 2) q^{25} + (2 \beta_{2} - 2 \beta_1 - 3) q^{26} + ( - \beta_{2} - 2) q^{28} + (3 \beta_{2} - 2 \beta_1 - 1) q^{29} + (3 \beta_1 - 4) q^{31} + q^{32} + ( - 4 \beta_{2} + \beta_1 - 2) q^{34} + (\beta_{2} - 3 \beta_1 + 1) q^{35} + ( - 3 \beta_{2} + \beta_1 - 5) q^{37} + (\beta_{2} + 3 \beta_1 - 3) q^{38} + (\beta_1 - 1) q^{40} + (2 \beta_{2} + 3 \beta_1 - 1) q^{41} + ( - 2 \beta_{2} - 2 \beta_1 - 4) q^{43} + (2 \beta_{2} - 3 \beta_1 - 1) q^{44} + ( - 2 \beta_{2} + 1) q^{46} + (\beta_{2} + \beta_1 + 3) q^{47} + (3 \beta_{2} + \beta_1 - 1) q^{49} + (\beta_{2} - 2 \beta_1 - 2) q^{50} + (2 \beta_{2} - 2 \beta_1 - 3) q^{52} + ( - 3 \beta_{2} + \beta_1 + 2) q^{53} + ( - 5 \beta_{2} + 4 \beta_1 - 3) q^{55} + ( - \beta_{2} - 2) q^{56} + (3 \beta_{2} - 2 \beta_1 - 1) q^{58} + ( - \beta_{2} + 4 \beta_1 + 1) q^{59} + (4 \beta_{2} - 5 \beta_1 + 2) q^{61} + (3 \beta_1 - 4) q^{62} + q^{64} + ( - 4 \beta_{2} + \beta_1 + 1) q^{65} + (3 \beta_{2} - 1) q^{67} + ( - 4 \beta_{2} + \beta_1 - 2) q^{68} + (\beta_{2} - 3 \beta_1 + 1) q^{70} + (5 \beta_{2} + \beta_1 + 4) q^{71} + ( - 7 \beta_{2} + 5 \beta_1 - 1) q^{73} + ( - 3 \beta_{2} + \beta_1 - 5) q^{74} + (\beta_{2} + 3 \beta_1 - 3) q^{76} + ( - \beta_{2} + 7 \beta_1 + 1) q^{77} + ( - 8 \beta_{2} + 3 \beta_1) q^{79} + (\beta_1 - 1) q^{80} + (2 \beta_{2} + 3 \beta_1 - 1) q^{82} + ( - 2 \beta_{2} + 4 \beta_1) q^{83} + (5 \beta_{2} - 7 \beta_1) q^{85} + ( - 2 \beta_{2} - 2 \beta_1 - 4) q^{86} + (2 \beta_{2} - 3 \beta_1 - 1) q^{88} + ( - 4 \beta_{2} + \beta_1 - 5) q^{89} + (\beta_{2} + 4 \beta_1 + 4) q^{91} + ( - 2 \beta_{2} + 1) q^{92} + (\beta_{2} + \beta_1 + 3) q^{94} + (2 \beta_{2} - 5 \beta_1 + 10) q^{95} + ( - \beta_{2} - 2 \beta_1 - 6) q^{97} + (3 \beta_{2} + \beta_1 - 1) q^{98}+O(q^{100})$$ q + q^2 + q^4 + (b1 - 1) * q^5 + (-b2 - 2) * q^7 + q^8 + (b1 - 1) * q^10 + (2*b2 - 3*b1 - 1) * q^11 + (2*b2 - 2*b1 - 3) * q^13 + (-b2 - 2) * q^14 + q^16 + (-4*b2 + b1 - 2) * q^17 + (b2 + 3*b1 - 3) * q^19 + (b1 - 1) * q^20 + (2*b2 - 3*b1 - 1) * q^22 + (-2*b2 + 1) * q^23 + (b2 - 2*b1 - 2) * q^25 + (2*b2 - 2*b1 - 3) * q^26 + (-b2 - 2) * q^28 + (3*b2 - 2*b1 - 1) * q^29 + (3*b1 - 4) * q^31 + q^32 + (-4*b2 + b1 - 2) * q^34 + (b2 - 3*b1 + 1) * q^35 + (-3*b2 + b1 - 5) * q^37 + (b2 + 3*b1 - 3) * q^38 + (b1 - 1) * q^40 + (2*b2 + 3*b1 - 1) * q^41 + (-2*b2 - 2*b1 - 4) * q^43 + (2*b2 - 3*b1 - 1) * q^44 + (-2*b2 + 1) * q^46 + (b2 + b1 + 3) * q^47 + (3*b2 + b1 - 1) * q^49 + (b2 - 2*b1 - 2) * q^50 + (2*b2 - 2*b1 - 3) * q^52 + (-3*b2 + b1 + 2) * q^53 + (-5*b2 + 4*b1 - 3) * q^55 + (-b2 - 2) * q^56 + (3*b2 - 2*b1 - 1) * q^58 + (-b2 + 4*b1 + 1) * q^59 + (4*b2 - 5*b1 + 2) * q^61 + (3*b1 - 4) * q^62 + q^64 + (-4*b2 + b1 + 1) * q^65 + (3*b2 - 1) * q^67 + (-4*b2 + b1 - 2) * q^68 + (b2 - 3*b1 + 1) * q^70 + (5*b2 + b1 + 4) * q^71 + (-7*b2 + 5*b1 - 1) * q^73 + (-3*b2 + b1 - 5) * q^74 + (b2 + 3*b1 - 3) * q^76 + (-b2 + 7*b1 + 1) * q^77 + (-8*b2 + 3*b1) * q^79 + (b1 - 1) * q^80 + (2*b2 + 3*b1 - 1) * q^82 + (-2*b2 + 4*b1) * q^83 + (5*b2 - 7*b1) * q^85 + (-2*b2 - 2*b1 - 4) * q^86 + (2*b2 - 3*b1 - 1) * q^88 + (-4*b2 + b1 - 5) * q^89 + (b2 + 4*b1 + 4) * q^91 + (-2*b2 + 1) * q^92 + (b2 + b1 + 3) * q^94 + (2*b2 - 5*b1 + 10) * q^95 + (-b2 - 2*b1 - 6) * q^97 + (3*b2 + b1 - 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - 6 q^{7} + 3 q^{8}+O(q^{10})$$ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 - 6 * q^7 + 3 * q^8 $$3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - 6 q^{7} + 3 q^{8} - 3 q^{10} - 3 q^{11} - 9 q^{13} - 6 q^{14} + 3 q^{16} - 6 q^{17} - 9 q^{19} - 3 q^{20} - 3 q^{22} + 3 q^{23} - 6 q^{25} - 9 q^{26} - 6 q^{28} - 3 q^{29} - 12 q^{31} + 3 q^{32} - 6 q^{34} + 3 q^{35} - 15 q^{37} - 9 q^{38} - 3 q^{40} - 3 q^{41} - 12 q^{43} - 3 q^{44} + 3 q^{46} + 9 q^{47} - 3 q^{49} - 6 q^{50} - 9 q^{52} + 6 q^{53} - 9 q^{55} - 6 q^{56} - 3 q^{58} + 3 q^{59} + 6 q^{61} - 12 q^{62} + 3 q^{64} + 3 q^{65} - 3 q^{67} - 6 q^{68} + 3 q^{70} + 12 q^{71} - 3 q^{73} - 15 q^{74} - 9 q^{76} + 3 q^{77} - 3 q^{80} - 3 q^{82} - 12 q^{86} - 3 q^{88} - 15 q^{89} + 12 q^{91} + 3 q^{92} + 9 q^{94} + 30 q^{95} - 18 q^{97} - 3 q^{98}+O(q^{100})$$ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 - 6 * q^7 + 3 * q^8 - 3 * q^10 - 3 * q^11 - 9 * q^13 - 6 * q^14 + 3 * q^16 - 6 * q^17 - 9 * q^19 - 3 * q^20 - 3 * q^22 + 3 * q^23 - 6 * q^25 - 9 * q^26 - 6 * q^28 - 3 * q^29 - 12 * q^31 + 3 * q^32 - 6 * q^34 + 3 * q^35 - 15 * q^37 - 9 * q^38 - 3 * q^40 - 3 * q^41 - 12 * q^43 - 3 * q^44 + 3 * q^46 + 9 * q^47 - 3 * q^49 - 6 * q^50 - 9 * q^52 + 6 * q^53 - 9 * q^55 - 6 * q^56 - 3 * q^58 + 3 * q^59 + 6 * q^61 - 12 * q^62 + 3 * q^64 + 3 * q^65 - 3 * q^67 - 6 * q^68 + 3 * q^70 + 12 * q^71 - 3 * q^73 - 15 * q^74 - 9 * q^76 + 3 * q^77 - 3 * q^80 - 3 * q^82 - 12 * q^86 - 3 * q^88 - 15 * q^89 + 12 * q^91 + 3 * q^92 + 9 * q^94 + 30 * q^95 - 18 * q^97 - 3 * q^98

Basis of coefficient ring in terms of $$\nu = \zeta_{18} + \zeta_{18}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2

## 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
 −1.53209 −0.347296 1.87939
1.00000 0 1.00000 −2.53209 0 −2.34730 1.00000 0 −2.53209
1.2 1.00000 0 1.00000 −1.34730 0 −0.120615 1.00000 0 −1.34730
1.3 1.00000 0 1.00000 0.879385 0 −3.53209 1.00000 0 0.879385
 $$n$$: 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 1458.2.a.d 3
3.b odd 2 1 1458.2.a.a 3
9.c even 3 2 1458.2.c.a 6
9.d odd 6 2 1458.2.c.d 6
27.e even 9 2 162.2.e.a 6
27.e even 9 2 486.2.e.a 6
27.e even 9 2 486.2.e.c 6
27.f odd 18 2 54.2.e.a 6
27.f odd 18 2 486.2.e.b 6
27.f odd 18 2 486.2.e.d 6
108.l even 18 2 432.2.u.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.2.e.a 6 27.f odd 18 2
162.2.e.a 6 27.e even 9 2
432.2.u.a 6 108.l even 18 2
486.2.e.a 6 27.e even 9 2
486.2.e.b 6 27.f odd 18 2
486.2.e.c 6 27.e even 9 2
486.2.e.d 6 27.f odd 18 2
1458.2.a.a 3 3.b odd 2 1
1458.2.a.d 3 1.a even 1 1 trivial
1458.2.c.a 6 9.c even 3 2
1458.2.c.d 6 9.d odd 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{3} + 3T_{5}^{2} - 3$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1458))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{3}$$
$3$ $$T^{3}$$
$5$ $$T^{3} + 3T^{2} - 3$$
$7$ $$T^{3} + 6 T^{2} + 9 T + 1$$
$11$ $$T^{3} + 3 T^{2} - 18 T - 57$$
$13$ $$T^{3} + 9 T^{2} + 15 T - 17$$
$17$ $$T^{3} + 6 T^{2} - 27 T - 159$$
$19$ $$T^{3} + 9 T^{2} - 12 T - 179$$
$23$ $$T^{3} - 3 T^{2} - 9 T + 3$$
$29$ $$T^{3} + 3 T^{2} - 18 T - 3$$
$31$ $$T^{3} + 12 T^{2} + 21 T - 71$$
$37$ $$T^{3} + 15 T^{2} + 54 T - 17$$
$41$ $$T^{3} + 3 T^{2} - 54 T - 219$$
$43$ $$T^{3} + 12 T^{2} + 12 T - 8$$
$47$ $$T^{3} - 9 T^{2} + 18 T - 9$$
$53$ $$T^{3} - 6 T^{2} - 9 T - 3$$
$59$ $$T^{3} - 3 T^{2} - 36 T + 57$$
$61$ $$T^{3} - 6 T^{2} - 51 T - 53$$
$67$ $$T^{3} + 3 T^{2} - 24 T + 1$$
$71$ $$T^{3} - 12 T^{2} - 45 T + 327$$
$73$ $$T^{3} + 3 T^{2} - 114 T - 269$$
$79$ $$T^{3} - 147T - 683$$
$83$ $$T^{3} - 36T + 72$$
$89$ $$T^{3} + 15 T^{2} + 36 T - 159$$
$97$ $$T^{3} + 18 T^{2} + 87 T + 127$$