# Properties

 Label 273.2.l.a Level $273$ Weight $2$ Character orbit 273.l Analytic conductor $2.180$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,2,Mod(16,273)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(273, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 2]))

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

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

chi := DirichletCharacter("273.16");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.l (of order $$3$$, degree $$2$$, 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: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{3} - 2 q^{4} + (2 \zeta_{6} - 3) q^{7} + (\zeta_{6} - 1) q^{9} +O(q^{10})$$ q - z * q^3 - 2 * q^4 + (2*z - 3) * q^7 + (z - 1) * q^9 $$q - \zeta_{6} q^{3} - 2 q^{4} + (2 \zeta_{6} - 3) q^{7} + (\zeta_{6} - 1) q^{9} + 6 \zeta_{6} q^{11} + 2 \zeta_{6} q^{12} + (4 \zeta_{6} - 1) q^{13} + 4 q^{16} - 6 q^{17} + ( - \zeta_{6} + 1) q^{19} + (\zeta_{6} + 2) q^{21} - 6 q^{23} + ( - 5 \zeta_{6} + 5) q^{25} + q^{27} + ( - 4 \zeta_{6} + 6) q^{28} + (6 \zeta_{6} - 6) q^{29} + (5 \zeta_{6} - 5) q^{31} + ( - 6 \zeta_{6} + 6) q^{33} + ( - 2 \zeta_{6} + 2) q^{36} - q^{37} + ( - 3 \zeta_{6} + 4) q^{39} + 4 \zeta_{6} q^{43} - 12 \zeta_{6} q^{44} - 6 \zeta_{6} q^{47} - 4 \zeta_{6} q^{48} + ( - 8 \zeta_{6} + 5) q^{49} + 6 \zeta_{6} q^{51} + ( - 8 \zeta_{6} + 2) q^{52} + (6 \zeta_{6} - 6) q^{53} - q^{57} + ( - 13 \zeta_{6} + 13) q^{61} + ( - 3 \zeta_{6} + 1) q^{63} - 8 q^{64} + 13 \zeta_{6} q^{67} + 12 q^{68} + 6 \zeta_{6} q^{69} + 12 \zeta_{6} q^{71} + ( - 10 \zeta_{6} + 10) q^{73} - 5 q^{75} + (2 \zeta_{6} - 2) q^{76} + ( - 6 \zeta_{6} - 12) q^{77} + \zeta_{6} q^{79} - \zeta_{6} q^{81} - 6 q^{83} + ( - 2 \zeta_{6} - 4) q^{84} + 6 q^{87} + 6 q^{89} + ( - 6 \zeta_{6} - 5) q^{91} + 12 q^{92} + 5 q^{93} - 5 \zeta_{6} q^{97} - 6 q^{99} +O(q^{100})$$ q - z * q^3 - 2 * q^4 + (2*z - 3) * q^7 + (z - 1) * q^9 + 6*z * q^11 + 2*z * q^12 + (4*z - 1) * q^13 + 4 * q^16 - 6 * q^17 + (-z + 1) * q^19 + (z + 2) * q^21 - 6 * q^23 + (-5*z + 5) * q^25 + q^27 + (-4*z + 6) * q^28 + (6*z - 6) * q^29 + (5*z - 5) * q^31 + (-6*z + 6) * q^33 + (-2*z + 2) * q^36 - q^37 + (-3*z + 4) * q^39 + 4*z * q^43 - 12*z * q^44 - 6*z * q^47 - 4*z * q^48 + (-8*z + 5) * q^49 + 6*z * q^51 + (-8*z + 2) * q^52 + (6*z - 6) * q^53 - q^57 + (-13*z + 13) * q^61 + (-3*z + 1) * q^63 - 8 * q^64 + 13*z * q^67 + 12 * q^68 + 6*z * q^69 + 12*z * q^71 + (-10*z + 10) * q^73 - 5 * q^75 + (2*z - 2) * q^76 + (-6*z - 12) * q^77 + z * q^79 - z * q^81 - 6 * q^83 + (-2*z - 4) * q^84 + 6 * q^87 + 6 * q^89 + (-6*z - 5) * q^91 + 12 * q^92 + 5 * q^93 - 5*z * q^97 - 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - 4 q^{4} - 4 q^{7} - q^{9}+O(q^{10})$$ 2 * q - q^3 - 4 * q^4 - 4 * q^7 - q^9 $$2 q - q^{3} - 4 q^{4} - 4 q^{7} - q^{9} + 6 q^{11} + 2 q^{12} + 2 q^{13} + 8 q^{16} - 12 q^{17} + q^{19} + 5 q^{21} - 12 q^{23} + 5 q^{25} + 2 q^{27} + 8 q^{28} - 6 q^{29} - 5 q^{31} + 6 q^{33} + 2 q^{36} - 2 q^{37} + 5 q^{39} + 4 q^{43} - 12 q^{44} - 6 q^{47} - 4 q^{48} + 2 q^{49} + 6 q^{51} - 4 q^{52} - 6 q^{53} - 2 q^{57} + 13 q^{61} - q^{63} - 16 q^{64} + 13 q^{67} + 24 q^{68} + 6 q^{69} + 12 q^{71} + 10 q^{73} - 10 q^{75} - 2 q^{76} - 30 q^{77} + q^{79} - q^{81} - 12 q^{83} - 10 q^{84} + 12 q^{87} + 12 q^{89} - 16 q^{91} + 24 q^{92} + 10 q^{93} - 5 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q - q^3 - 4 * q^4 - 4 * q^7 - q^9 + 6 * q^11 + 2 * q^12 + 2 * q^13 + 8 * q^16 - 12 * q^17 + q^19 + 5 * q^21 - 12 * q^23 + 5 * q^25 + 2 * q^27 + 8 * q^28 - 6 * q^29 - 5 * q^31 + 6 * q^33 + 2 * q^36 - 2 * q^37 + 5 * q^39 + 4 * q^43 - 12 * q^44 - 6 * q^47 - 4 * q^48 + 2 * q^49 + 6 * q^51 - 4 * q^52 - 6 * q^53 - 2 * q^57 + 13 * q^61 - q^63 - 16 * q^64 + 13 * q^67 + 24 * q^68 + 6 * q^69 + 12 * q^71 + 10 * q^73 - 10 * q^75 - 2 * q^76 - 30 * q^77 + q^79 - q^81 - 12 * q^83 - 10 * q^84 + 12 * q^87 + 12 * q^89 - 16 * q^91 + 24 * q^92 + 10 * q^93 - 5 * q^97 - 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/273\mathbb{Z}\right)^\times$$.

 $$n$$ $$92$$ $$106$$ $$157$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −0.500000 + 0.866025i −2.00000 0 0 −2.00000 1.73205i 0 −0.500000 0.866025i 0
256.1 0 −0.500000 0.866025i −2.00000 0 0 −2.00000 + 1.73205i 0 −0.500000 + 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.l.a yes 2
3.b odd 2 1 819.2.s.b 2
7.c even 3 1 273.2.j.a 2
13.c even 3 1 273.2.j.a 2
21.h odd 6 1 819.2.n.b 2
39.i odd 6 1 819.2.n.b 2
91.h even 3 1 inner 273.2.l.a yes 2
273.s odd 6 1 819.2.s.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.j.a 2 7.c even 3 1
273.2.j.a 2 13.c even 3 1
273.2.l.a yes 2 1.a even 1 1 trivial
273.2.l.a yes 2 91.h even 3 1 inner
819.2.n.b 2 21.h odd 6 1
819.2.n.b 2 39.i odd 6 1
819.2.s.b 2 3.b odd 2 1
819.2.s.b 2 273.s odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4T + 7$$
$11$ $$T^{2} - 6T + 36$$
$13$ $$T^{2} - 2T + 13$$
$17$ $$(T + 6)^{2}$$
$19$ $$T^{2} - T + 1$$
$23$ $$(T + 6)^{2}$$
$29$ $$T^{2} + 6T + 36$$
$31$ $$T^{2} + 5T + 25$$
$37$ $$(T + 1)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} - 4T + 16$$
$47$ $$T^{2} + 6T + 36$$
$53$ $$T^{2} + 6T + 36$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 13T + 169$$
$67$ $$T^{2} - 13T + 169$$
$71$ $$T^{2} - 12T + 144$$
$73$ $$T^{2} - 10T + 100$$
$79$ $$T^{2} - T + 1$$
$83$ $$(T + 6)^{2}$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 5T + 25$$