# Properties

 Label 273.2.ba.b Level $273$ Weight $2$ Character orbit 273.ba Analytic conductor $2.180$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,2,Mod(38,273)]

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, 1, 3]))

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

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

chi := DirichletCharacter("273.38");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.ba (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: $$2.17991597518$$ 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: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} + 1) q^{3} + ( - 2 \zeta_{6} + 2) q^{4} + ( - 3 \zeta_{6} + 2) q^{7} + 3 \zeta_{6} q^{9}+O(q^{10})$$ q + (z + 1) * q^3 + (-2*z + 2) * q^4 + (-3*z + 2) * q^7 + 3*z * q^9 $$q + (\zeta_{6} + 1) q^{3} + ( - 2 \zeta_{6} + 2) q^{4} + ( - 3 \zeta_{6} + 2) q^{7} + 3 \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 4) q^{12} + ( - \zeta_{6} - 3) q^{13} - 4 \zeta_{6} q^{16} + 7 \zeta_{6} q^{19} + ( - 4 \zeta_{6} + 5) q^{21} + (5 \zeta_{6} - 5) q^{25} + (6 \zeta_{6} - 3) q^{27} + ( - 4 \zeta_{6} - 2) q^{28} + (7 \zeta_{6} - 7) q^{31} + 6 q^{36} + ( - 7 \zeta_{6} + 14) q^{37} + ( - 5 \zeta_{6} - 2) q^{39} - 5 q^{43} + ( - 8 \zeta_{6} + 4) q^{48} + ( - 3 \zeta_{6} - 5) q^{49} + (6 \zeta_{6} - 8) q^{52} + (14 \zeta_{6} - 7) q^{57} + (4 \zeta_{6} - 8) q^{61} + ( - 3 \zeta_{6} + 9) q^{63} - 8 q^{64} + ( - 7 \zeta_{6} - 7) q^{67} + (7 \zeta_{6} - 7) q^{73} + (5 \zeta_{6} - 10) q^{75} + 14 q^{76} - 13 \zeta_{6} q^{79} + (9 \zeta_{6} - 9) q^{81} + ( - 10 \zeta_{6} + 2) q^{84} + (10 \zeta_{6} - 9) q^{91} + (7 \zeta_{6} - 14) q^{93} + 14 q^{97} +O(q^{100})$$ q + (z + 1) * q^3 + (-2*z + 2) * q^4 + (-3*z + 2) * q^7 + 3*z * q^9 + (-2*z + 4) * q^12 + (-z - 3) * q^13 - 4*z * q^16 + 7*z * q^19 + (-4*z + 5) * q^21 + (5*z - 5) * q^25 + (6*z - 3) * q^27 + (-4*z - 2) * q^28 + (7*z - 7) * q^31 + 6 * q^36 + (-7*z + 14) * q^37 + (-5*z - 2) * q^39 - 5 * q^43 + (-8*z + 4) * q^48 + (-3*z - 5) * q^49 + (6*z - 8) * q^52 + (14*z - 7) * q^57 + (4*z - 8) * q^61 + (-3*z + 9) * q^63 - 8 * q^64 + (-7*z - 7) * q^67 + (7*z - 7) * q^73 + (5*z - 10) * q^75 + 14 * q^76 - 13*z * q^79 + (9*z - 9) * q^81 + (-10*z + 2) * q^84 + (10*z - 9) * q^91 + (7*z - 14) * q^93 + 14 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{3} + 2 q^{4} + q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q + 3 * q^3 + 2 * q^4 + q^7 + 3 * q^9 $$2 q + 3 q^{3} + 2 q^{4} + q^{7} + 3 q^{9} + 6 q^{12} - 7 q^{13} - 4 q^{16} + 7 q^{19} + 6 q^{21} - 5 q^{25} - 8 q^{28} - 7 q^{31} + 12 q^{36} + 21 q^{37} - 9 q^{39} - 10 q^{43} - 13 q^{49} - 10 q^{52} - 12 q^{61} + 15 q^{63} - 16 q^{64} - 21 q^{67} - 7 q^{73} - 15 q^{75} + 28 q^{76} - 13 q^{79} - 9 q^{81} - 6 q^{84} - 8 q^{91} - 21 q^{93} + 28 q^{97}+O(q^{100})$$ 2 * q + 3 * q^3 + 2 * q^4 + q^7 + 3 * q^9 + 6 * q^12 - 7 * q^13 - 4 * q^16 + 7 * q^19 + 6 * q^21 - 5 * q^25 - 8 * q^28 - 7 * q^31 + 12 * q^36 + 21 * q^37 - 9 * q^39 - 10 * q^43 - 13 * q^49 - 10 * q^52 - 12 * q^61 + 15 * q^63 - 16 * q^64 - 21 * q^67 - 7 * q^73 - 15 * q^75 + 28 * q^76 - 13 * q^79 - 9 * q^81 - 6 * q^84 - 8 * q^91 - 21 * q^93 + 28 * q^97

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
38.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.50000 + 0.866025i 1.00000 1.73205i 0 0 0.500000 2.59808i 0 1.50000 + 2.59808i 0
194.1 0 1.50000 0.866025i 1.00000 + 1.73205i 0 0 0.500000 + 2.59808i 0 1.50000 2.59808i 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.s odd 6 1 inner
273.ba even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.ba.b yes 2
3.b odd 2 1 CM 273.2.ba.b yes 2
7.d odd 6 1 273.2.ba.a 2
13.b even 2 1 273.2.ba.a 2
21.g even 6 1 273.2.ba.a 2
39.d odd 2 1 273.2.ba.a 2
91.s odd 6 1 inner 273.2.ba.b yes 2
273.ba even 6 1 inner 273.2.ba.b yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.ba.a 2 7.d odd 6 1
273.2.ba.a 2 13.b even 2 1
273.2.ba.a 2 21.g even 6 1
273.2.ba.a 2 39.d odd 2 1
273.2.ba.b yes 2 1.a even 1 1 trivial
273.2.ba.b yes 2 3.b odd 2 1 CM
273.2.ba.b yes 2 91.s odd 6 1 inner
273.2.ba.b yes 2 273.ba even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$:

 $$T_{2}$$ T2 $$T_{19}^{2} - 7T_{19} + 49$$ T19^2 - 7*T19 + 49

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 3T + 3$$
$5$ $$T^{2}$$
$7$ $$T^{2} - T + 7$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 7T + 13$$
$17$ $$T^{2}$$
$19$ $$T^{2} - 7T + 49$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 7T + 49$$
$37$ $$T^{2} - 21T + 147$$
$41$ $$T^{2}$$
$43$ $$(T + 5)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 12T + 48$$
$67$ $$T^{2} + 21T + 147$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 7T + 49$$
$79$ $$T^{2} + 13T + 169$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$(T - 14)^{2}$$