# Properties

 Label 273.2.u.b Level $273$ Weight $2$ Character orbit 273.u 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.u (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: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{4} + ( -3 + \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 + \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{4} + ( -3 + \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} + ( -2 + 4 \zeta_{6} ) q^{12} + ( 4 - 3 \zeta_{6} ) q^{13} + ( -4 + 4 \zeta_{6} ) q^{16} -8 \zeta_{6} q^{19} + ( -4 - \zeta_{6} ) q^{21} + 5 q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( -2 - 4 \zeta_{6} ) q^{28} + 7 q^{31} + ( -6 + 6 \zeta_{6} ) q^{36} + ( 4 + 4 \zeta_{6} ) q^{37} + ( 7 - 2 \zeta_{6} ) q^{39} -13 \zeta_{6} q^{43} + ( -8 + 4 \zeta_{6} ) q^{48} + ( 8 - 5 \zeta_{6} ) q^{49} + ( 6 + 2 \zeta_{6} ) q^{52} + ( 8 - 16 \zeta_{6} ) q^{57} + ( -10 + 5 \zeta_{6} ) q^{61} + ( -3 - 6 \zeta_{6} ) q^{63} -8 q^{64} + ( -7 - 7 \zeta_{6} ) q^{67} -17 q^{73} + ( 5 + 5 \zeta_{6} ) q^{75} + ( 16 - 16 \zeta_{6} ) q^{76} + 13 q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + ( 2 - 10 \zeta_{6} ) q^{84} + ( -9 + 10 \zeta_{6} ) q^{91} + ( 7 + 7 \zeta_{6} ) q^{93} -5 \zeta_{6} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{3} + 2q^{4} - 5q^{7} + 3q^{9} + O(q^{10})$$ $$2q + 3q^{3} + 2q^{4} - 5q^{7} + 3q^{9} + 5q^{13} - 4q^{16} - 8q^{19} - 9q^{21} + 10q^{25} - 8q^{28} + 14q^{31} - 6q^{36} + 12q^{37} + 12q^{39} - 13q^{43} - 12q^{48} + 11q^{49} + 14q^{52} - 15q^{61} - 12q^{63} - 16q^{64} - 21q^{67} - 34q^{73} + 15q^{75} + 16q^{76} + 26q^{79} - 9q^{81} - 6q^{84} - 8q^{91} + 21q^{93} - 5q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
62.1
 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
251.1 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.t odd 6 1 inner
273.u 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.u.b yes 2
3.b odd 2 1 CM 273.2.u.b yes 2
7.b odd 2 1 273.2.u.a 2
13.e even 6 1 273.2.u.a 2
21.c even 2 1 273.2.u.a 2
39.h odd 6 1 273.2.u.a 2
91.t odd 6 1 inner 273.2.u.b yes 2
273.u even 6 1 inner 273.2.u.b yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.u.a 2 7.b odd 2 1
273.2.u.a 2 13.e even 6 1
273.2.u.a 2 21.c even 2 1
273.2.u.a 2 39.h odd 6 1
273.2.u.b yes 2 1.a even 1 1 trivial
273.2.u.b yes 2 3.b odd 2 1 CM
273.2.u.b yes 2 91.t odd 6 1 inner
273.2.u.b yes 2 273.u 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 $$S_{2}^{\mathrm{new}}(273, [\chi])$$:

 $$T_{2}$$ $$T_{19}^{2} + 8 T_{19} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$3 - 3 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$7 + 5 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$13 - 5 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$64 + 8 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$( -7 + T )^{2}$$
$37$ $$48 - 12 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$169 + 13 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$75 + 15 T + T^{2}$$
$67$ $$147 + 21 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 17 + T )^{2}$$
$79$ $$( -13 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$25 + 5 T + T^{2}$$