# Properties

 Label 273.1.s.a Level $273$ Weight $1$ Character orbit 273.s Analytic conductor $0.136$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,1,Mod(74,273)]

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([3, 4, 4]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("273.74");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 273.s (of order $$6$$, 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: $$0.136244748449$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ 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 Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.24843.1 Artin image: $C_3\times S_3$ Artin field: Galois closure of 6.0.223587.2

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{6}^{2} q^{3} + q^{4} + \zeta_{6}^{2} q^{7} - \zeta_{6} q^{9} +O(q^{10})$$ q + z^2 * q^3 + q^4 + z^2 * q^7 - z * q^9 $$q + \zeta_{6}^{2} q^{3} + q^{4} + \zeta_{6}^{2} q^{7} - \zeta_{6} q^{9} + \zeta_{6}^{2} q^{12} - \zeta_{6} q^{13} + q^{16} + \zeta_{6} q^{19} - \zeta_{6} q^{21} - \zeta_{6} q^{25} + q^{27} + \zeta_{6}^{2} q^{28} - \zeta_{6} q^{31} - \zeta_{6} q^{36} - q^{37} + q^{39} - \zeta_{6}^{2} q^{43} + \zeta_{6}^{2} q^{48} - \zeta_{6} q^{49} - \zeta_{6} q^{52} - q^{57} + \zeta_{6} q^{61} + q^{63} + q^{64} + \zeta_{6}^{2} q^{67} + \zeta_{6} q^{73} + q^{75} + \zeta_{6} q^{76} + \zeta_{6}^{2} q^{79} + \zeta_{6}^{2} q^{81} - \zeta_{6} q^{84} + q^{91} + 2 q^{93} - \zeta_{6}^{2} q^{97} +O(q^{100})$$ q + z^2 * q^3 + q^4 + z^2 * q^7 - z * q^9 + z^2 * q^12 - z * q^13 + q^16 + z * q^19 - z * q^21 - z * q^25 + q^27 + z^2 * q^28 - z * q^31 - z * q^36 - q^37 + q^39 - z^2 * q^43 + z^2 * q^48 - z * q^49 - z * q^52 - q^57 + z * q^61 + q^63 + q^64 + z^2 * q^67 + z * q^73 + q^75 + z * q^76 + z^2 * q^79 + z^2 * q^81 - z * q^84 + q^91 + 2 * q^93 - z^2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + 2 q^{4} - q^{7} - q^{9}+O(q^{10})$$ 2 * q - q^3 + 2 * q^4 - q^7 - q^9 $$2 q - q^{3} + 2 q^{4} - q^{7} - q^{9} - q^{12} - q^{13} + 2 q^{16} + q^{19} - q^{21} - q^{25} + 2 q^{27} - q^{28} - 2 q^{31} - q^{36} - 2 q^{37} + 2 q^{39} + q^{43} - q^{48} - q^{49} - q^{52} - 2 q^{57} + q^{61} + 2 q^{63} + 2 q^{64} - 2 q^{67} + q^{73} + 2 q^{75} + q^{76} - 2 q^{79} - q^{81} - q^{84} + 2 q^{91} + 4 q^{93} + q^{97}+O(q^{100})$$ 2 * q - q^3 + 2 * q^4 - q^7 - q^9 - q^12 - q^13 + 2 * q^16 + q^19 - q^21 - q^25 + 2 * q^27 - q^28 - 2 * q^31 - q^36 - 2 * q^37 + 2 * q^39 + q^43 - q^48 - q^49 - q^52 - 2 * q^57 + q^61 + 2 * q^63 + 2 * q^64 - 2 * q^67 + q^73 + 2 * q^75 + q^76 - 2 * q^79 - q^81 - q^84 + 2 * q^91 + 4 * q^93 + q^97

## Character values

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

 $$n$$ $$92$$ $$106$$ $$157$$ $$\chi(n)$$ $$-1$$ $$\zeta_{6}^{2}$$ $$\zeta_{6}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
74.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −0.500000 0.866025i 1.00000 0 0 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0
107.1 0 −0.500000 + 0.866025i 1.00000 0 0 −0.500000 + 0.866025i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
91.h even 3 1 inner
273.s odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.1.s.a 2
3.b odd 2 1 CM 273.1.s.a 2
7.b odd 2 1 1911.1.s.a 2
7.c even 3 1 273.1.bm.a yes 2
7.c even 3 1 1911.1.be.a 2
7.d odd 6 1 1911.1.be.b 2
7.d odd 6 1 1911.1.bm.a 2
13.b even 2 1 3549.1.s.a 2
13.c even 3 1 273.1.bm.a yes 2
13.c even 3 1 3549.1.bk.b 2
13.d odd 4 2 3549.1.bp.c 4
13.e even 6 1 3549.1.bk.a 2
13.e even 6 1 3549.1.bm.a 2
13.f odd 12 2 3549.1.w.d 4
13.f odd 12 2 3549.1.x.c 4
21.c even 2 1 1911.1.s.a 2
21.g even 6 1 1911.1.be.b 2
21.g even 6 1 1911.1.bm.a 2
21.h odd 6 1 273.1.bm.a yes 2
21.h odd 6 1 1911.1.be.a 2
39.d odd 2 1 3549.1.s.a 2
39.f even 4 2 3549.1.bp.c 4
39.h odd 6 1 3549.1.bk.a 2
39.h odd 6 1 3549.1.bm.a 2
39.i odd 6 1 273.1.bm.a yes 2
39.i odd 6 1 3549.1.bk.b 2
39.k even 12 2 3549.1.w.d 4
39.k even 12 2 3549.1.x.c 4
91.g even 3 1 1911.1.be.a 2
91.g even 3 1 3549.1.bk.b 2
91.h even 3 1 inner 273.1.s.a 2
91.k even 6 1 3549.1.s.a 2
91.m odd 6 1 1911.1.be.b 2
91.n odd 6 1 1911.1.bm.a 2
91.r even 6 1 3549.1.bm.a 2
91.u even 6 1 3549.1.bk.a 2
91.v odd 6 1 1911.1.s.a 2
91.x odd 12 2 3549.1.bp.c 4
91.z odd 12 2 3549.1.x.c 4
91.bd odd 12 2 3549.1.w.d 4
273.r even 6 1 1911.1.s.a 2
273.s odd 6 1 inner 273.1.s.a 2
273.w odd 6 1 3549.1.bm.a 2
273.x odd 6 1 3549.1.bk.a 2
273.bf even 6 1 1911.1.be.b 2
273.bm odd 6 1 1911.1.be.a 2
273.bm odd 6 1 3549.1.bk.b 2
273.bn even 6 1 1911.1.bm.a 2
273.bp odd 6 1 3549.1.s.a 2
273.bv even 12 2 3549.1.bp.c 4
273.bw even 12 2 3549.1.w.d 4
273.cd even 12 2 3549.1.x.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.1.s.a 2 1.a even 1 1 trivial
273.1.s.a 2 3.b odd 2 1 CM
273.1.s.a 2 91.h even 3 1 inner
273.1.s.a 2 273.s odd 6 1 inner
273.1.bm.a yes 2 7.c even 3 1
273.1.bm.a yes 2 13.c even 3 1
273.1.bm.a yes 2 21.h odd 6 1
273.1.bm.a yes 2 39.i odd 6 1
1911.1.s.a 2 7.b odd 2 1
1911.1.s.a 2 21.c even 2 1
1911.1.s.a 2 91.v odd 6 1
1911.1.s.a 2 273.r even 6 1
1911.1.be.a 2 7.c even 3 1
1911.1.be.a 2 21.h odd 6 1
1911.1.be.a 2 91.g even 3 1
1911.1.be.a 2 273.bm odd 6 1
1911.1.be.b 2 7.d odd 6 1
1911.1.be.b 2 21.g even 6 1
1911.1.be.b 2 91.m odd 6 1
1911.1.be.b 2 273.bf even 6 1
1911.1.bm.a 2 7.d odd 6 1
1911.1.bm.a 2 21.g even 6 1
1911.1.bm.a 2 91.n odd 6 1
1911.1.bm.a 2 273.bn even 6 1
3549.1.s.a 2 13.b even 2 1
3549.1.s.a 2 39.d odd 2 1
3549.1.s.a 2 91.k even 6 1
3549.1.s.a 2 273.bp odd 6 1
3549.1.w.d 4 13.f odd 12 2
3549.1.w.d 4 39.k even 12 2
3549.1.w.d 4 91.bd odd 12 2
3549.1.w.d 4 273.bw even 12 2
3549.1.x.c 4 13.f odd 12 2
3549.1.x.c 4 39.k even 12 2
3549.1.x.c 4 91.z odd 12 2
3549.1.x.c 4 273.cd even 12 2
3549.1.bk.a 2 13.e even 6 1
3549.1.bk.a 2 39.h odd 6 1
3549.1.bk.a 2 91.u even 6 1
3549.1.bk.a 2 273.x odd 6 1
3549.1.bk.b 2 13.c even 3 1
3549.1.bk.b 2 39.i odd 6 1
3549.1.bk.b 2 91.g even 3 1
3549.1.bk.b 2 273.bm odd 6 1
3549.1.bm.a 2 13.e even 6 1
3549.1.bm.a 2 39.h odd 6 1
3549.1.bm.a 2 91.r even 6 1
3549.1.bm.a 2 273.w odd 6 1
3549.1.bp.c 4 13.d odd 4 2
3549.1.bp.c 4 39.f even 4 2
3549.1.bp.c 4 91.x odd 12 2
3549.1.bp.c 4 273.bv even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{1}^{\mathrm{new}}(273, [\chi])$$.

## Hecke characteristic polynomials

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