# Properties

 Label 273.2.t.a Level $273$ Weight $2$ Character orbit 273.t 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.t (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: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + 2 \zeta_{6} ) q^{2} + \zeta_{6} q^{3} - q^{4} + ( -4 + 2 \zeta_{6} ) q^{5} + ( -2 + \zeta_{6} ) q^{6} + ( 2 + \zeta_{6} ) q^{7} + ( -1 + 2 \zeta_{6} ) q^{8} + ( -1 + \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + ( -1 + 2 \zeta_{6} ) q^{2} + \zeta_{6} q^{3} - q^{4} + ( -4 + 2 \zeta_{6} ) q^{5} + ( -2 + \zeta_{6} ) q^{6} + ( 2 + \zeta_{6} ) q^{7} + ( -1 + 2 \zeta_{6} ) q^{8} + ( -1 + \zeta_{6} ) q^{9} -6 \zeta_{6} q^{10} + ( 4 - 2 \zeta_{6} ) q^{11} -\zeta_{6} q^{12} + ( -1 - 3 \zeta_{6} ) q^{13} + ( -4 + 5 \zeta_{6} ) q^{14} + ( -2 - 2 \zeta_{6} ) q^{15} -5 q^{16} -6 q^{17} + ( -1 - \zeta_{6} ) q^{18} + ( 1 + \zeta_{6} ) q^{19} + ( 4 - 2 \zeta_{6} ) q^{20} + ( -1 + 3 \zeta_{6} ) q^{21} + 6 \zeta_{6} q^{22} + 6 q^{23} + ( -2 + \zeta_{6} ) q^{24} + ( 7 - 7 \zeta_{6} ) q^{25} + ( 7 - 5 \zeta_{6} ) q^{26} - q^{27} + ( -2 - \zeta_{6} ) q^{28} + ( -6 + 6 \zeta_{6} ) q^{29} + ( 6 - 6 \zeta_{6} ) q^{30} + ( 6 + 6 \zeta_{6} ) q^{31} + ( 3 - 6 \zeta_{6} ) q^{32} + ( 2 + 2 \zeta_{6} ) q^{33} + ( 6 - 12 \zeta_{6} ) q^{34} + ( -10 + 2 \zeta_{6} ) q^{35} + ( 1 - \zeta_{6} ) q^{36} + ( 1 - 2 \zeta_{6} ) q^{37} + ( -3 + 3 \zeta_{6} ) q^{38} + ( 3 - 4 \zeta_{6} ) q^{39} -6 \zeta_{6} q^{40} + ( 2 + 2 \zeta_{6} ) q^{41} + ( -5 + \zeta_{6} ) q^{42} + \zeta_{6} q^{43} + ( -4 + 2 \zeta_{6} ) q^{44} + ( 2 - 4 \zeta_{6} ) q^{45} + ( -6 + 12 \zeta_{6} ) q^{46} + ( 12 - 6 \zeta_{6} ) q^{47} -5 \zeta_{6} q^{48} + ( 3 + 5 \zeta_{6} ) q^{49} + ( 7 + 7 \zeta_{6} ) q^{50} -6 \zeta_{6} q^{51} + ( 1 + 3 \zeta_{6} ) q^{52} + ( 6 - 6 \zeta_{6} ) q^{53} + ( 1 - 2 \zeta_{6} ) q^{54} + ( -12 + 12 \zeta_{6} ) q^{55} + ( -4 + 5 \zeta_{6} ) q^{56} + ( -1 + 2 \zeta_{6} ) q^{57} + ( -6 - 6 \zeta_{6} ) q^{58} + ( -2 + 4 \zeta_{6} ) q^{59} + ( 2 + 2 \zeta_{6} ) q^{60} + ( -1 + \zeta_{6} ) q^{61} + ( -18 + 18 \zeta_{6} ) q^{62} + ( -3 + 2 \zeta_{6} ) q^{63} - q^{64} + ( 10 + 4 \zeta_{6} ) q^{65} + ( -6 + 6 \zeta_{6} ) q^{66} + ( 4 - 2 \zeta_{6} ) q^{67} + 6 q^{68} + 6 \zeta_{6} q^{69} + ( 6 - 18 \zeta_{6} ) q^{70} + ( 4 - 2 \zeta_{6} ) q^{71} + ( -1 - \zeta_{6} ) q^{72} + ( -7 - 7 \zeta_{6} ) q^{73} + 3 q^{74} + 7 q^{75} + ( -1 - \zeta_{6} ) q^{76} + ( 10 - 2 \zeta_{6} ) q^{77} + ( 5 + 2 \zeta_{6} ) q^{78} + 8 \zeta_{6} q^{79} + ( 20 - 10 \zeta_{6} ) q^{80} -\zeta_{6} q^{81} + ( -6 + 6 \zeta_{6} ) q^{82} + ( 6 - 12 \zeta_{6} ) q^{83} + ( 1 - 3 \zeta_{6} ) q^{84} + ( 24 - 12 \zeta_{6} ) q^{85} + ( -2 + \zeta_{6} ) q^{86} -6 q^{87} + 6 \zeta_{6} q^{88} + ( -6 + 12 \zeta_{6} ) q^{89} + 6 q^{90} + ( 1 - 10 \zeta_{6} ) q^{91} -6 q^{92} + ( -6 + 12 \zeta_{6} ) q^{93} + 18 \zeta_{6} q^{94} -6 q^{95} + ( 6 - 3 \zeta_{6} ) q^{96} + ( -14 + 7 \zeta_{6} ) q^{97} + ( -13 + 11 \zeta_{6} ) q^{98} + ( -2 + 4 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} - 2q^{4} - 6q^{5} - 3q^{6} + 5q^{7} - q^{9} + O(q^{10})$$ $$2q + q^{3} - 2q^{4} - 6q^{5} - 3q^{6} + 5q^{7} - q^{9} - 6q^{10} + 6q^{11} - q^{12} - 5q^{13} - 3q^{14} - 6q^{15} - 10q^{16} - 12q^{17} - 3q^{18} + 3q^{19} + 6q^{20} + q^{21} + 6q^{22} + 12q^{23} - 3q^{24} + 7q^{25} + 9q^{26} - 2q^{27} - 5q^{28} - 6q^{29} + 6q^{30} + 18q^{31} + 6q^{33} - 18q^{35} + q^{36} - 3q^{38} + 2q^{39} - 6q^{40} + 6q^{41} - 9q^{42} + q^{43} - 6q^{44} + 18q^{47} - 5q^{48} + 11q^{49} + 21q^{50} - 6q^{51} + 5q^{52} + 6q^{53} - 12q^{55} - 3q^{56} - 18q^{58} + 6q^{60} - q^{61} - 18q^{62} - 4q^{63} - 2q^{64} + 24q^{65} - 6q^{66} + 6q^{67} + 12q^{68} + 6q^{69} - 6q^{70} + 6q^{71} - 3q^{72} - 21q^{73} + 6q^{74} + 14q^{75} - 3q^{76} + 18q^{77} + 12q^{78} + 8q^{79} + 30q^{80} - q^{81} - 6q^{82} - q^{84} + 36q^{85} - 3q^{86} - 12q^{87} + 6q^{88} + 12q^{90} - 8q^{91} - 12q^{92} + 18q^{94} - 12q^{95} + 9q^{96} - 21q^{97} - 15q^{98} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.73205i 0.500000 + 0.866025i −1.00000 −3.00000 + 1.73205i −1.50000 + 0.866025i 2.50000 + 0.866025i 1.73205i −0.500000 + 0.866025i −3.00000 5.19615i
205.1 1.73205i 0.500000 0.866025i −1.00000 −3.00000 1.73205i −1.50000 0.866025i 2.50000 0.866025i 1.73205i −0.500000 0.866025i −3.00000 + 5.19615i
 $$n$$: 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.k 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.t.a 2
3.b odd 2 1 819.2.bm.b 2
7.c even 3 1 273.2.bl.a yes 2
13.e even 6 1 273.2.bl.a yes 2
21.h odd 6 1 819.2.do.a 2
39.h odd 6 1 819.2.do.a 2
91.k even 6 1 inner 273.2.t.a 2
273.bp odd 6 1 819.2.bm.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.t.a 2 1.a even 1 1 trivial
273.2.t.a 2 91.k even 6 1 inner
273.2.bl.a yes 2 7.c even 3 1
273.2.bl.a yes 2 13.e even 6 1
819.2.bm.b 2 3.b odd 2 1
819.2.bm.b 2 273.bp odd 6 1
819.2.do.a 2 21.h odd 6 1
819.2.do.a 2 39.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 3$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 + T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$12 + 6 T + T^{2}$$
$7$ $$7 - 5 T + T^{2}$$
$11$ $$12 - 6 T + T^{2}$$
$13$ $$13 + 5 T + T^{2}$$
$17$ $$( 6 + T )^{2}$$
$19$ $$3 - 3 T + T^{2}$$
$23$ $$( -6 + T )^{2}$$
$29$ $$36 + 6 T + T^{2}$$
$31$ $$108 - 18 T + T^{2}$$
$37$ $$3 + T^{2}$$
$41$ $$12 - 6 T + T^{2}$$
$43$ $$1 - T + T^{2}$$
$47$ $$108 - 18 T + T^{2}$$
$53$ $$36 - 6 T + T^{2}$$
$59$ $$12 + T^{2}$$
$61$ $$1 + T + T^{2}$$
$67$ $$12 - 6 T + T^{2}$$
$71$ $$12 - 6 T + T^{2}$$
$73$ $$147 + 21 T + T^{2}$$
$79$ $$64 - 8 T + T^{2}$$
$83$ $$108 + T^{2}$$
$89$ $$108 + T^{2}$$
$97$ $$147 + 21 T + T^{2}$$