# Properties

 Label 273.2.i.a Level $273$ Weight $2$ Character orbit 273.i 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.i (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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{4} -4 \zeta_{6} q^{5} - q^{6} + ( -2 - \zeta_{6} ) q^{7} + 3 q^{8} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{4} -4 \zeta_{6} q^{5} - q^{6} + ( -2 - \zeta_{6} ) q^{7} + 3 q^{8} -\zeta_{6} q^{9} + ( 4 - 4 \zeta_{6} ) q^{10} + ( 5 - 5 \zeta_{6} ) q^{11} + \zeta_{6} q^{12} - q^{13} + ( 1 - 3 \zeta_{6} ) q^{14} + 4 q^{15} + \zeta_{6} q^{16} + ( -3 + 3 \zeta_{6} ) q^{17} + ( 1 - \zeta_{6} ) q^{18} + 5 \zeta_{6} q^{19} -4 q^{20} + ( 3 - 2 \zeta_{6} ) q^{21} + 5 q^{22} -6 \zeta_{6} q^{23} + ( -3 + 3 \zeta_{6} ) q^{24} + ( -11 + 11 \zeta_{6} ) q^{25} -\zeta_{6} q^{26} + q^{27} + ( -3 + 2 \zeta_{6} ) q^{28} + 7 q^{29} + 4 \zeta_{6} q^{30} + ( 5 - 5 \zeta_{6} ) q^{32} + 5 \zeta_{6} q^{33} -3 q^{34} + ( -4 + 12 \zeta_{6} ) q^{35} - q^{36} + ( -5 + 5 \zeta_{6} ) q^{38} + ( 1 - \zeta_{6} ) q^{39} -12 \zeta_{6} q^{40} + 8 q^{41} + ( 2 + \zeta_{6} ) q^{42} + 2 q^{43} -5 \zeta_{6} q^{44} + ( -4 + 4 \zeta_{6} ) q^{45} + ( 6 - 6 \zeta_{6} ) q^{46} -9 \zeta_{6} q^{47} - q^{48} + ( 3 + 5 \zeta_{6} ) q^{49} -11 q^{50} -3 \zeta_{6} q^{51} + ( -1 + \zeta_{6} ) q^{52} + ( -9 + 9 \zeta_{6} ) q^{53} + \zeta_{6} q^{54} -20 q^{55} + ( -6 - 3 \zeta_{6} ) q^{56} -5 q^{57} + 7 \zeta_{6} q^{58} + ( 9 - 9 \zeta_{6} ) q^{59} + ( 4 - 4 \zeta_{6} ) q^{60} -\zeta_{6} q^{61} + ( -1 + 3 \zeta_{6} ) q^{63} + 7 q^{64} + 4 \zeta_{6} q^{65} + ( -5 + 5 \zeta_{6} ) q^{66} + ( -7 + 7 \zeta_{6} ) q^{67} + 3 \zeta_{6} q^{68} + 6 q^{69} + ( -12 + 8 \zeta_{6} ) q^{70} -3 q^{71} -3 \zeta_{6} q^{72} + ( 6 - 6 \zeta_{6} ) q^{73} -11 \zeta_{6} q^{75} + 5 q^{76} + ( -15 + 10 \zeta_{6} ) q^{77} + q^{78} + 10 \zeta_{6} q^{79} + ( 4 - 4 \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} + 8 \zeta_{6} q^{82} + ( 1 - 3 \zeta_{6} ) q^{84} + 12 q^{85} + 2 \zeta_{6} q^{86} + ( -7 + 7 \zeta_{6} ) q^{87} + ( 15 - 15 \zeta_{6} ) q^{88} + 8 \zeta_{6} q^{89} -4 q^{90} + ( 2 + \zeta_{6} ) q^{91} -6 q^{92} + ( 9 - 9 \zeta_{6} ) q^{94} + ( 20 - 20 \zeta_{6} ) q^{95} + 5 \zeta_{6} q^{96} + 18 q^{97} + ( -5 + 8 \zeta_{6} ) q^{98} -5 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - q^{3} + q^{4} - 4q^{5} - 2q^{6} - 5q^{7} + 6q^{8} - q^{9} + O(q^{10})$$ $$2q + q^{2} - q^{3} + q^{4} - 4q^{5} - 2q^{6} - 5q^{7} + 6q^{8} - q^{9} + 4q^{10} + 5q^{11} + q^{12} - 2q^{13} - q^{14} + 8q^{15} + q^{16} - 3q^{17} + q^{18} + 5q^{19} - 8q^{20} + 4q^{21} + 10q^{22} - 6q^{23} - 3q^{24} - 11q^{25} - q^{26} + 2q^{27} - 4q^{28} + 14q^{29} + 4q^{30} + 5q^{32} + 5q^{33} - 6q^{34} + 4q^{35} - 2q^{36} - 5q^{38} + q^{39} - 12q^{40} + 16q^{41} + 5q^{42} + 4q^{43} - 5q^{44} - 4q^{45} + 6q^{46} - 9q^{47} - 2q^{48} + 11q^{49} - 22q^{50} - 3q^{51} - q^{52} - 9q^{53} + q^{54} - 40q^{55} - 15q^{56} - 10q^{57} + 7q^{58} + 9q^{59} + 4q^{60} - q^{61} + q^{63} + 14q^{64} + 4q^{65} - 5q^{66} - 7q^{67} + 3q^{68} + 12q^{69} - 16q^{70} - 6q^{71} - 3q^{72} + 6q^{73} - 11q^{75} + 10q^{76} - 20q^{77} + 2q^{78} + 10q^{79} + 4q^{80} - q^{81} + 8q^{82} - q^{84} + 24q^{85} + 2q^{86} - 7q^{87} + 15q^{88} + 8q^{89} - 8q^{90} + 5q^{91} - 12q^{92} + 9q^{94} + 20q^{95} + 5q^{96} + 36q^{97} - 2q^{98} - 10q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 0.866025i −0.500000 0.866025i 0.500000 + 0.866025i −2.00000 + 3.46410i −1.00000 −2.50000 + 0.866025i 3.00000 −0.500000 + 0.866025i 2.00000 + 3.46410i
235.1 0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 0.866025i −2.00000 3.46410i −1.00000 −2.50000 0.866025i 3.00000 −0.500000 0.866025i 2.00000 3.46410i
 $$n$$: 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
7.c 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.i.a 2
3.b odd 2 1 819.2.j.a 2
7.c even 3 1 inner 273.2.i.a 2
7.c even 3 1 1911.2.a.c 1
7.d odd 6 1 1911.2.a.b 1
21.g even 6 1 5733.2.a.k 1
21.h odd 6 1 819.2.j.a 2
21.h odd 6 1 5733.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.i.a 2 1.a even 1 1 trivial
273.2.i.a 2 7.c even 3 1 inner
819.2.j.a 2 3.b odd 2 1
819.2.j.a 2 21.h odd 6 1
1911.2.a.b 1 7.d odd 6 1
1911.2.a.c 1 7.c even 3 1
5733.2.a.i 1 21.h odd 6 1
5733.2.a.k 1 21.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$16 + 4 T + T^{2}$$
$7$ $$7 + 5 T + T^{2}$$
$11$ $$25 - 5 T + T^{2}$$
$13$ $$( 1 + T )^{2}$$
$17$ $$9 + 3 T + T^{2}$$
$19$ $$25 - 5 T + T^{2}$$
$23$ $$36 + 6 T + T^{2}$$
$29$ $$( -7 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$( -8 + T )^{2}$$
$43$ $$( -2 + T )^{2}$$
$47$ $$81 + 9 T + T^{2}$$
$53$ $$81 + 9 T + T^{2}$$
$59$ $$81 - 9 T + T^{2}$$
$61$ $$1 + T + T^{2}$$
$67$ $$49 + 7 T + T^{2}$$
$71$ $$( 3 + T )^{2}$$
$73$ $$36 - 6 T + T^{2}$$
$79$ $$100 - 10 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$64 - 8 T + T^{2}$$
$97$ $$( -18 + T )^{2}$$