# Properties

 Label 1458.2.c.d Level $1458$ Weight $2$ Character orbit 1458.c Analytic conductor $11.642$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1458,2,Mod(487,1458)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1458, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4]))

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

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

chi := DirichletCharacter("1458.487");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1458 = 2 \cdot 3^{6}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1458.c (of order $$3$$, degree $$2$$, not minimal)

## 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: no Analytic conductor: $$11.6421886147$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{18})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 54) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_1 - 1) q^{4} + (\beta_{5} + \beta_1 - 1) q^{5} + (\beta_{2} + 2 \beta_1) q^{7} - q^{8}+O(q^{10})$$ q + b1 * q^2 + (b1 - 1) * q^4 + (b5 + b1 - 1) * q^5 + (b2 + 2*b1) * q^7 - q^8 $$q + \beta_1 q^{2} + (\beta_1 - 1) q^{4} + (\beta_{5} + \beta_1 - 1) q^{5} + (\beta_{2} + 2 \beta_1) q^{7} - q^{8} + (\beta_{4} + \beta_{3} - 1) q^{10} + (3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - \beta_1) q^{11} + (2 \beta_{5} - 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 3) q^{13} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{14} - \beta_1 q^{16} + ( - \beta_{4} + 3 \beta_{3} + 2) q^{17} + (3 \beta_{4} + 4 \beta_{3} - 3) q^{19} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_1) q^{20} + (3 \beta_{5} - 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{22} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{23} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_1) q^{25} + (2 \beta_{4} + 3) q^{26} + ( - \beta_{3} - 2) q^{28} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} - \beta_1) q^{29} + ( - 3 \beta_{5} - 4 \beta_1 + 4) q^{31} + ( - \beta_1 + 1) q^{32} + (\beta_{5} - \beta_{4} - \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{34} + (3 \beta_{4} + 2 \beta_{3} - 1) q^{35} + (\beta_{4} - 2 \beta_{3} - 5) q^{37} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + \beta_{2} - 3 \beta_1) q^{38} + ( - \beta_{5} - \beta_1 + 1) q^{40} + (3 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{41} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1) q^{43} + (3 \beta_{4} + \beta_{3} + 1) q^{44} + ( - 2 \beta_{3} + 1) q^{46} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 3 \beta_1) q^{47} + ( - \beta_{5} - 3 \beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{49} + ( - 2 \beta_{5} + \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{50} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 3 \beta_1) q^{52} + ( - \beta_{4} + 2 \beta_{3} - 2) q^{53} + (4 \beta_{4} - \beta_{3} - 3) q^{55} + ( - \beta_{2} - 2 \beta_1) q^{56} + (2 \beta_{5} - 3 \beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{58} + (4 \beta_{5} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{59} + ( - 5 \beta_{5} + 5 \beta_{4} + 5 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{61} + ( - 3 \beta_{4} - 3 \beta_{3} + 4) q^{62} + q^{64} + ( - \beta_{5} + \beta_{4} + \beta_{3} - 4 \beta_{2} + \beta_1) q^{65} + ( - 3 \beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{67} + (\beta_{5} - 4 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 2) q^{68} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} - \beta_{2} - \beta_1) q^{70} + ( - \beta_{4} - 6 \beta_{3} - 4) q^{71} + (5 \beta_{4} - 2 \beta_{3} - 1) q^{73} + ( - \beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_{2} - 5 \beta_1) q^{74} + ( - 3 \beta_{5} - \beta_{3} + \beta_{2} - 3 \beta_1 + 3) q^{76} + (7 \beta_{5} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{77} + (3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + 8 \beta_{2}) q^{79} + ( - \beta_{4} - \beta_{3} + 1) q^{80} + (3 \beta_{4} + 5 \beta_{3} - 1) q^{82} + ( - 4 \beta_{5} + 4 \beta_{4} + 4 \beta_{3} - 2 \beta_{2}) q^{83} + (7 \beta_{5} - 5 \beta_{3} + 5 \beta_{2}) q^{85} + ( - 2 \beta_{5} - 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 4) q^{86} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + \beta_1) q^{88} + ( - \beta_{4} + 3 \beta_{3} + 5) q^{89} + (4 \beta_{4} + 5 \beta_{3} + 4) q^{91} + ( - 2 \beta_{2} + \beta_1) q^{92} + ( - \beta_{5} - \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{94} + ( - 5 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - 10 \beta_1 + 10) q^{95} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 6 \beta_1) q^{97} + ( - \beta_{4} - 4 \beta_{3} + 1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b1 - 1) * q^4 + (b5 + b1 - 1) * q^5 + (b2 + 2*b1) * q^7 - q^8 + (b4 + b3 - 1) * q^10 + (3*b5 - 3*b4 - 3*b3 + 2*b2 - b1) * q^11 + (2*b5 - 2*b3 + 2*b2 - 3*b1 + 3) * q^13 + (-b3 + b2 + 2*b1 - 2) * q^14 - b1 * q^16 + (-b4 + 3*b3 + 2) * q^17 + (3*b4 + 4*b3 - 3) * q^19 + (-b5 + b4 + b3 - b1) * q^20 + (3*b5 - 2*b3 + 2*b2 - b1 + 1) * q^22 + (-2*b3 + 2*b2 - b1 + 1) * q^23 + (-2*b5 + 2*b4 + 2*b3 - b2 + 2*b1) * q^25 + (2*b4 + 3) * q^26 + (-b3 - 2) * q^28 + (2*b5 - 2*b4 - 2*b3 + 3*b2 - b1) * q^29 + (-3*b5 - 4*b1 + 4) * q^31 + (-b1 + 1) * q^32 + (b5 - b4 - b3 + 4*b2 + 2*b1) * q^34 + (3*b4 + 2*b3 - 1) * q^35 + (b4 - 2*b3 - 5) * q^37 + (-3*b5 + 3*b4 + 3*b3 + b2 - 3*b1) * q^38 + (-b5 - b1 + 1) * q^40 + (3*b5 + 2*b3 - 2*b2 + b1 - 1) * q^41 + (-2*b5 + 2*b4 + 2*b3 + 2*b2 + 4*b1) * q^43 + (3*b4 + b3 + 1) * q^44 + (-2*b3 + 1) * q^46 + (-b5 + b4 + b3 + b2 + 3*b1) * q^47 + (-b5 - 3*b3 + 3*b2 - b1 + 1) * q^49 + (-2*b5 + b3 - b2 + 2*b1 - 2) * q^50 + (-2*b5 + 2*b4 + 2*b3 - 2*b2 + 3*b1) * q^52 + (-b4 + 2*b3 - 2) * q^53 + (4*b4 - b3 - 3) * q^55 + (-b2 - 2*b1) * q^56 + (2*b5 - 3*b3 + 3*b2 - b1 + 1) * q^58 + (4*b5 - b3 + b2 - b1 + 1) * q^59 + (-5*b5 + 5*b4 + 5*b3 - 4*b2 - 2*b1) * q^61 + (-3*b4 - 3*b3 + 4) * q^62 + q^64 + (-b5 + b4 + b3 - 4*b2 + b1) * q^65 + (-3*b3 + 3*b2 - b1 + 1) * q^67 + (b5 - 4*b3 + 4*b2 + 2*b1 - 2) * q^68 + (-3*b5 + 3*b4 + 3*b3 - b2 - b1) * q^70 + (-b4 - 6*b3 - 4) * q^71 + (5*b4 - 2*b3 - 1) * q^73 + (-b5 + b4 + b3 - 3*b2 - 5*b1) * q^74 + (-3*b5 - b3 + b2 - 3*b1 + 3) * q^76 + (7*b5 - b3 + b2 - b1 + 1) * q^77 + (3*b5 - 3*b4 - 3*b3 + 8*b2) * q^79 + (-b4 - b3 + 1) * q^80 + (3*b4 + 5*b3 - 1) * q^82 + (-4*b5 + 4*b4 + 4*b3 - 2*b2) * q^83 + (7*b5 - 5*b3 + 5*b2) * q^85 + (-2*b5 - 2*b3 + 2*b2 + 4*b1 - 4) * q^86 + (-3*b5 + 3*b4 + 3*b3 - 2*b2 + b1) * q^88 + (-b4 + 3*b3 + 5) * q^89 + (4*b4 + 5*b3 + 4) * q^91 + (-2*b2 + b1) * q^92 + (-b5 - b3 + b2 + 3*b1 - 3) * q^94 + (-5*b5 + 2*b3 - 2*b2 - 10*b1 + 10) * q^95 + (-2*b5 + 2*b4 + 2*b3 + b2 + 6*b1) * q^97 + (-b4 - 4*b3 + 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{2} - 3 q^{4} - 3 q^{5} + 6 q^{7} - 6 q^{8}+O(q^{10})$$ 6 * q + 3 * q^2 - 3 * q^4 - 3 * q^5 + 6 * q^7 - 6 * q^8 $$6 q + 3 q^{2} - 3 q^{4} - 3 q^{5} + 6 q^{7} - 6 q^{8} - 6 q^{10} - 3 q^{11} + 9 q^{13} - 6 q^{14} - 3 q^{16} + 12 q^{17} - 18 q^{19} - 3 q^{20} + 3 q^{22} + 3 q^{23} + 6 q^{25} + 18 q^{26} - 12 q^{28} - 3 q^{29} + 12 q^{31} + 3 q^{32} + 6 q^{34} - 6 q^{35} - 30 q^{37} - 9 q^{38} + 3 q^{40} - 3 q^{41} + 12 q^{43} + 6 q^{44} + 6 q^{46} + 9 q^{47} + 3 q^{49} - 6 q^{50} + 9 q^{52} - 12 q^{53} - 18 q^{55} - 6 q^{56} + 3 q^{58} + 3 q^{59} - 6 q^{61} + 24 q^{62} + 6 q^{64} + 3 q^{65} + 3 q^{67} - 6 q^{68} - 3 q^{70} - 24 q^{71} - 6 q^{73} - 15 q^{74} + 9 q^{76} + 3 q^{77} + 6 q^{80} - 6 q^{82} - 12 q^{86} + 3 q^{88} + 30 q^{89} + 24 q^{91} + 3 q^{92} - 9 q^{94} + 30 q^{95} + 18 q^{97} + 6 q^{98}+O(q^{100})$$ 6 * q + 3 * q^2 - 3 * q^4 - 3 * q^5 + 6 * q^7 - 6 * q^8 - 6 * q^10 - 3 * q^11 + 9 * q^13 - 6 * q^14 - 3 * q^16 + 12 * q^17 - 18 * q^19 - 3 * q^20 + 3 * q^22 + 3 * q^23 + 6 * q^25 + 18 * q^26 - 12 * q^28 - 3 * q^29 + 12 * q^31 + 3 * q^32 + 6 * q^34 - 6 * q^35 - 30 * q^37 - 9 * q^38 + 3 * q^40 - 3 * q^41 + 12 * q^43 + 6 * q^44 + 6 * q^46 + 9 * q^47 + 3 * q^49 - 6 * q^50 + 9 * q^52 - 12 * q^53 - 18 * q^55 - 6 * q^56 + 3 * q^58 + 3 * q^59 - 6 * q^61 + 24 * q^62 + 6 * q^64 + 3 * q^65 + 3 * q^67 - 6 * q^68 - 3 * q^70 - 24 * q^71 - 6 * q^73 - 15 * q^74 + 9 * q^76 + 3 * q^77 + 6 * q^80 - 6 * q^82 - 12 * q^86 + 3 * q^88 + 30 * q^89 + 24 * q^91 + 3 * q^92 - 9 * q^94 + 30 * q^95 + 18 * q^97 + 6 * q^98

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{18}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$\zeta_{18}^{5} + \zeta_{18}$$ v^5 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18}$$ -v^4 + v^2 + v $$\beta_{4}$$ $$=$$ $$-\zeta_{18}^{5} + \zeta_{18}^{4}$$ -v^5 + v^4 $$\beta_{5}$$ $$=$$ $$-\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}$$ -v^5 - v^4 + v
 $$\zeta_{18}$$ $$=$$ $$( \beta_{5} + \beta_{4} + 2\beta_{2} ) / 3$$ (b5 + b4 + 2*b2) / 3 $$\zeta_{18}^{2}$$ $$=$$ $$( -2\beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} ) / 3$$ (-2*b5 + b4 + 3*b3 - b2) / 3 $$\zeta_{18}^{3}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{18}^{4}$$ $$=$$ $$( -\beta_{5} + 2\beta_{4} + \beta_{2} ) / 3$$ (-b5 + 2*b4 + b2) / 3 $$\zeta_{18}^{5}$$ $$=$$ $$( -\beta_{5} - \beta_{4} + \beta_{2} ) / 3$$ (-b5 - b4 + b2) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1458\mathbb{Z}\right)^\times$$.

 $$n$$ $$731$$ $$\chi(n)$$ $$-1 + \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
487.1
 −0.766044 − 0.642788i −0.173648 + 0.984808i 0.939693 − 0.342020i −0.766044 + 0.642788i −0.173648 − 0.984808i 0.939693 + 0.342020i
0.500000 0.866025i 0 −0.500000 0.866025i −1.26604 2.19285i 0 1.17365 2.03282i −1.00000 0 −2.53209
487.2 0.500000 0.866025i 0 −0.500000 0.866025i −0.673648 1.16679i 0 0.0603074 0.104455i −1.00000 0 −1.34730
487.3 0.500000 0.866025i 0 −0.500000 0.866025i 0.439693 + 0.761570i 0 1.76604 3.05888i −1.00000 0 0.879385
973.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.26604 + 2.19285i 0 1.17365 + 2.03282i −1.00000 0 −2.53209
973.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.673648 + 1.16679i 0 0.0603074 + 0.104455i −1.00000 0 −1.34730
973.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.439693 0.761570i 0 1.76604 + 3.05888i −1.00000 0 0.879385
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 487.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} + 3T_{5}^{5} + 9T_{5}^{4} + 6T_{5}^{3} + 9T_{5}^{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(1458, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{3}$$
$3$ $$T^{6}$$
$5$ $$T^{6} + 3 T^{5} + 9 T^{4} + 6 T^{3} + \cdots + 9$$
$7$ $$T^{6} - 6 T^{5} + 27 T^{4} - 52 T^{3} + \cdots + 1$$
$11$ $$T^{6} + 3 T^{5} + 27 T^{4} + \cdots + 3249$$
$13$ $$T^{6} - 9 T^{5} + 66 T^{4} - 169 T^{3} + \cdots + 289$$
$17$ $$(T^{3} - 6 T^{2} - 27 T + 159)^{2}$$
$19$ $$(T^{3} + 9 T^{2} - 12 T - 179)^{2}$$
$23$ $$T^{6} - 3 T^{5} + 18 T^{4} + 21 T^{3} + \cdots + 9$$
$29$ $$T^{6} + 3 T^{5} + 27 T^{4} - 48 T^{3} + \cdots + 9$$
$31$ $$T^{6} - 12 T^{5} + 123 T^{4} + \cdots + 5041$$
$37$ $$(T^{3} + 15 T^{2} + 54 T - 17)^{2}$$
$41$ $$T^{6} + 3 T^{5} + 63 T^{4} + \cdots + 47961$$
$43$ $$T^{6} - 12 T^{5} + 132 T^{4} + \cdots + 64$$
$47$ $$T^{6} - 9 T^{5} + 63 T^{4} - 144 T^{3} + \cdots + 81$$
$53$ $$(T^{3} + 6 T^{2} - 9 T + 3)^{2}$$
$59$ $$T^{6} - 3 T^{5} + 45 T^{4} + \cdots + 3249$$
$61$ $$T^{6} + 6 T^{5} + 87 T^{4} + \cdots + 2809$$
$67$ $$T^{6} - 3 T^{5} + 33 T^{4} + 74 T^{3} + \cdots + 1$$
$71$ $$(T^{3} + 12 T^{2} - 45 T - 327)^{2}$$
$73$ $$(T^{3} + 3 T^{2} - 114 T - 269)^{2}$$
$79$ $$T^{6} + 147 T^{4} - 1366 T^{3} + \cdots + 466489$$
$83$ $$T^{6} + 36 T^{4} - 144 T^{3} + \cdots + 5184$$
$89$ $$(T^{3} - 15 T^{2} + 36 T + 159)^{2}$$
$97$ $$T^{6} - 18 T^{5} + 237 T^{4} + \cdots + 16129$$