# Properties

 Label 273.2.bn.a Level $273$ Weight $2$ Character orbit 273.bn Analytic conductor $2.180$ Analytic rank $1$ 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.bn (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.17991597518$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ 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

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.

 $$f(q)$$ $$=$$ $$q + ( -2 + \zeta_{6} ) q^{3} + ( -2 + 2 \zeta_{6} ) q^{4} + ( -2 - \zeta_{6} ) q^{7} + ( 3 - 3 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + ( -2 + \zeta_{6} ) q^{3} + ( -2 + 2 \zeta_{6} ) q^{4} + ( -2 - \zeta_{6} ) q^{7} + ( 3 - 3 \zeta_{6} ) q^{9} + ( 2 - 4 \zeta_{6} ) q^{12} + ( -1 - 3 \zeta_{6} ) q^{13} -4 \zeta_{6} q^{16} + ( -2 - 2 \zeta_{6} ) q^{19} + ( 5 - \zeta_{6} ) q^{21} -5 q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( 6 - 4 \zeta_{6} ) q^{28} + ( -5 + 10 \zeta_{6} ) q^{31} + 6 \zeta_{6} q^{36} -10 \zeta_{6} q^{37} + ( 5 + 2 \zeta_{6} ) q^{39} + ( -13 + 13 \zeta_{6} ) q^{43} + ( 4 + 4 \zeta_{6} ) q^{48} + ( 3 + 5 \zeta_{6} ) q^{49} + ( 8 - 2 \zeta_{6} ) q^{52} + 6 q^{57} + ( 5 + 5 \zeta_{6} ) q^{61} + ( -9 + 6 \zeta_{6} ) q^{63} + 8 q^{64} -11 \zeta_{6} q^{67} + ( -1 + 2 \zeta_{6} ) q^{73} + ( 10 - 5 \zeta_{6} ) q^{75} + ( 8 - 4 \zeta_{6} ) q^{76} -13 q^{79} -9 \zeta_{6} q^{81} + ( -8 + 10 \zeta_{6} ) q^{84} + ( -1 + 10 \zeta_{6} ) q^{91} -15 \zeta_{6} q^{93} + ( -11 - 11 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{3} - 2q^{4} - 5q^{7} + 3q^{9} + O(q^{10})$$ $$2q - 3q^{3} - 2q^{4} - 5q^{7} + 3q^{9} - 5q^{13} - 4q^{16} - 6q^{19} + 9q^{21} - 10q^{25} + 8q^{28} + 6q^{36} - 10q^{37} + 12q^{39} - 13q^{43} + 12q^{48} + 11q^{49} + 14q^{52} + 12q^{57} + 15q^{61} - 12q^{63} + 16q^{64} - 11q^{67} + 15q^{75} + 12q^{76} - 26q^{79} - 9q^{81} - 6q^{84} + 8q^{91} - 15q^{93} - 33q^{97} + O(q^{100})$$

## Character values

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

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

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
146.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −1.50000 + 0.866025i −1.00000 + 1.73205i 0 0 −2.50000 0.866025i 0 1.50000 2.59808i 0
230.1 0 −1.50000 0.866025i −1.00000 1.73205i 0 0 −2.50000 + 0.866025i 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.n odd 6 1 inner
273.bn 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.bn.a 2
3.b odd 2 1 CM 273.2.bn.a 2
7.b odd 2 1 273.2.bn.b yes 2
13.c even 3 1 273.2.bn.b yes 2
21.c even 2 1 273.2.bn.b yes 2
39.i odd 6 1 273.2.bn.b yes 2
91.n odd 6 1 inner 273.2.bn.a 2
273.bn even 6 1 inner 273.2.bn.a 2

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

## Hecke kernels

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

 $$T_{2}$$ $$T_{19}^{2} + 6 T_{19} + 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$3 + 3 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$7 + 5 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$13 + 5 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$12 + 6 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$75 + T^{2}$$
$37$ $$100 + 10 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$169 + 13 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$75 - 15 T + T^{2}$$
$67$ $$121 + 11 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$3 + T^{2}$$
$79$ $$( 13 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$363 + 33 T + T^{2}$$