# Properties

 Label 273.2.l.a Level $273$ Weight $2$ Character orbit 273.l Analytic conductor $2.180$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.l (of order $$3$$, 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: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{3} -2 q^{4} + ( -3 + 2 \zeta_{6} ) q^{7} + ( -1 + \zeta_{6} ) q^{9} +O(q^{10})$$ $$q -\zeta_{6} q^{3} -2 q^{4} + ( -3 + 2 \zeta_{6} ) q^{7} + ( -1 + \zeta_{6} ) q^{9} + 6 \zeta_{6} q^{11} + 2 \zeta_{6} q^{12} + ( -1 + 4 \zeta_{6} ) q^{13} + 4 q^{16} -6 q^{17} + ( 1 - \zeta_{6} ) q^{19} + ( 2 + \zeta_{6} ) q^{21} -6 q^{23} + ( 5 - 5 \zeta_{6} ) q^{25} + q^{27} + ( 6 - 4 \zeta_{6} ) q^{28} + ( -6 + 6 \zeta_{6} ) q^{29} + ( -5 + 5 \zeta_{6} ) q^{31} + ( 6 - 6 \zeta_{6} ) q^{33} + ( 2 - 2 \zeta_{6} ) q^{36} - q^{37} + ( 4 - 3 \zeta_{6} ) q^{39} + 4 \zeta_{6} q^{43} -12 \zeta_{6} q^{44} -6 \zeta_{6} q^{47} -4 \zeta_{6} q^{48} + ( 5 - 8 \zeta_{6} ) q^{49} + 6 \zeta_{6} q^{51} + ( 2 - 8 \zeta_{6} ) q^{52} + ( -6 + 6 \zeta_{6} ) q^{53} - q^{57} + ( 13 - 13 \zeta_{6} ) q^{61} + ( 1 - 3 \zeta_{6} ) q^{63} -8 q^{64} + 13 \zeta_{6} q^{67} + 12 q^{68} + 6 \zeta_{6} q^{69} + 12 \zeta_{6} q^{71} + ( 10 - 10 \zeta_{6} ) q^{73} -5 q^{75} + ( -2 + 2 \zeta_{6} ) q^{76} + ( -12 - 6 \zeta_{6} ) q^{77} + \zeta_{6} q^{79} -\zeta_{6} q^{81} -6 q^{83} + ( -4 - 2 \zeta_{6} ) q^{84} + 6 q^{87} + 6 q^{89} + ( -5 - 6 \zeta_{6} ) q^{91} + 12 q^{92} + 5 q^{93} -5 \zeta_{6} q^{97} -6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{3} - 4q^{4} - 4q^{7} - q^{9} + O(q^{10})$$ $$2q - q^{3} - 4q^{4} - 4q^{7} - q^{9} + 6q^{11} + 2q^{12} + 2q^{13} + 8q^{16} - 12q^{17} + q^{19} + 5q^{21} - 12q^{23} + 5q^{25} + 2q^{27} + 8q^{28} - 6q^{29} - 5q^{31} + 6q^{33} + 2q^{36} - 2q^{37} + 5q^{39} + 4q^{43} - 12q^{44} - 6q^{47} - 4q^{48} + 2q^{49} + 6q^{51} - 4q^{52} - 6q^{53} - 2q^{57} + 13q^{61} - q^{63} - 16q^{64} + 13q^{67} + 24q^{68} + 6q^{69} + 12q^{71} + 10q^{73} - 10q^{75} - 2q^{76} - 30q^{77} + q^{79} - q^{81} - 12q^{83} - 10q^{84} + 12q^{87} + 12q^{89} - 16q^{91} + 24q^{92} + 10q^{93} - 5q^{97} - 12q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −0.500000 + 0.866025i −2.00000 0 0 −2.00000 1.73205i 0 −0.500000 0.866025i 0
256.1 0 −0.500000 0.866025i −2.00000 0 0 −2.00000 + 1.73205i 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
91.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.l.a yes 2
3.b odd 2 1 819.2.s.b 2
7.c even 3 1 273.2.j.a 2
13.c even 3 1 273.2.j.a 2
21.h odd 6 1 819.2.n.b 2
39.i odd 6 1 819.2.n.b 2
91.h even 3 1 inner 273.2.l.a yes 2
273.s odd 6 1 819.2.s.b 2

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

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$7 + 4 T + T^{2}$$
$11$ $$36 - 6 T + T^{2}$$
$13$ $$13 - 2 T + T^{2}$$
$17$ $$( 6 + T )^{2}$$
$19$ $$1 - T + T^{2}$$
$23$ $$( 6 + T )^{2}$$
$29$ $$36 + 6 T + T^{2}$$
$31$ $$25 + 5 T + T^{2}$$
$37$ $$( 1 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$16 - 4 T + T^{2}$$
$47$ $$36 + 6 T + T^{2}$$
$53$ $$36 + 6 T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$169 - 13 T + T^{2}$$
$67$ $$169 - 13 T + T^{2}$$
$71$ $$144 - 12 T + T^{2}$$
$73$ $$100 - 10 T + T^{2}$$
$79$ $$1 - T + T^{2}$$
$83$ $$( 6 + T )^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$25 + 5 T + T^{2}$$