Properties

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

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.y (of order $$6$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

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.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \zeta_{6} ) q^{3} + ( 2 - 2 \zeta_{6} ) q^{4} + ( 3 - \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q + ( 1 - 2 \zeta_{6} ) q^{3} + ( 2 - 2 \zeta_{6} ) q^{4} + ( 3 - \zeta_{6} ) q^{7} -3 q^{9} + ( -2 - 2 \zeta_{6} ) q^{12} + ( -3 + 4 \zeta_{6} ) q^{13} -4 \zeta_{6} q^{16} + q^{19} + ( 1 - 5 \zeta_{6} ) q^{21} -5 \zeta_{6} q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( 4 - 6 \zeta_{6} ) q^{28} + 7 \zeta_{6} q^{31} + ( -6 + 6 \zeta_{6} ) q^{36} + ( 6 - 3 \zeta_{6} ) q^{37} + ( 5 + 2 \zeta_{6} ) q^{39} + 8 \zeta_{6} q^{43} + ( -8 + 4 \zeta_{6} ) q^{48} + ( 8 - 5 \zeta_{6} ) q^{49} + ( 2 + 6 \zeta_{6} ) q^{52} + ( 1 - 2 \zeta_{6} ) q^{57} + ( -9 + 18 \zeta_{6} ) q^{61} + ( -9 + 3 \zeta_{6} ) q^{63} -8 q^{64} + ( 7 - 14 \zeta_{6} ) q^{67} + 10 \zeta_{6} q^{73} + ( -10 + 5 \zeta_{6} ) q^{75} + ( 2 - 2 \zeta_{6} ) q^{76} + ( -13 + 13 \zeta_{6} ) q^{79} + 9 q^{81} + ( -8 - 2 \zeta_{6} ) q^{84} + ( -5 + 11 \zeta_{6} ) q^{91} + ( 14 - 7 \zeta_{6} ) q^{93} -19 \zeta_{6} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{4} + 5q^{7} - 6q^{9} + O(q^{10})$$ $$2q + 2q^{4} + 5q^{7} - 6q^{9} - 6q^{12} - 2q^{13} - 4q^{16} + 2q^{19} - 3q^{21} - 5q^{25} + 2q^{28} + 7q^{31} - 6q^{36} + 9q^{37} + 12q^{39} + 8q^{43} - 12q^{48} + 11q^{49} + 10q^{52} - 15q^{63} - 16q^{64} + 10q^{73} - 15q^{75} + 2q^{76} - 13q^{79} + 18q^{81} - 18q^{84} + q^{91} + 21q^{93} - 19q^{97} + O(q^{100})$$

Character values

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

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

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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.73205i 1.00000 + 1.73205i 0 0 2.50000 + 0.866025i 0 −3.00000 0
173.1 0 1.73205i 1.00000 1.73205i 0 0 2.50000 0.866025i 0 −3.00000 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.p odd 6 1 inner
273.y 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.y.a 2
3.b odd 2 1 CM 273.2.y.a 2
7.d odd 6 1 273.2.br.a yes 2
13.e even 6 1 273.2.br.a yes 2
21.g even 6 1 273.2.br.a yes 2
39.h odd 6 1 273.2.br.a yes 2
91.p odd 6 1 inner 273.2.y.a 2
273.y even 6 1 inner 273.2.y.a 2

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

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$3 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$7 - 5 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$13 + 2 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$49 - 7 T + T^{2}$$
$37$ $$27 - 9 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$64 - 8 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$243 + T^{2}$$
$67$ $$147 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$100 - 10 T + T^{2}$$
$79$ $$169 + 13 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$361 + 19 T + T^{2}$$