# Properties

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

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.50000 + 0.866025i 0.500000 0.866025i 0.500000 0.866025i −3.00000 + 1.73205i 1.73205i 0.500000 2.59808i 1.73205i −0.500000 0.866025i 3.00000 5.19615i
142.1 −1.50000 0.866025i 0.500000 + 0.866025i 0.500000 + 0.866025i −3.00000 1.73205i 1.73205i 0.500000 + 2.59808i 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.r 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.bj.a 2
3.b odd 2 1 819.2.dl.d 2
7.c even 3 1 273.2.bj.b yes 2
7.c even 3 1 1911.2.c.b 2
7.d odd 6 1 1911.2.c.e 2
13.b even 2 1 273.2.bj.b yes 2
21.h odd 6 1 819.2.dl.a 2
39.d odd 2 1 819.2.dl.a 2
91.r even 6 1 inner 273.2.bj.a 2
91.r even 6 1 1911.2.c.b 2
91.s odd 6 1 1911.2.c.e 2
273.w odd 6 1 819.2.dl.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bj.a 2 1.a even 1 1 trivial
273.2.bj.a 2 91.r even 6 1 inner
273.2.bj.b yes 2 7.c even 3 1
273.2.bj.b yes 2 13.b even 2 1
819.2.dl.a 2 21.h odd 6 1
819.2.dl.a 2 39.d odd 2 1
819.2.dl.d 2 3.b odd 2 1
819.2.dl.d 2 273.w odd 6 1
1911.2.c.b 2 7.c even 3 1
1911.2.c.b 2 91.r even 6 1
1911.2.c.e 2 7.d odd 6 1
1911.2.c.e 2 91.s odd 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 + 3 T + T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$12 + 6 T + T^{2}$$
$7$ $$7 - T + T^{2}$$
$11$ $$27 - 9 T + T^{2}$$
$13$ $$13 + 2 T + T^{2}$$
$17$ $$9 - 3 T + T^{2}$$
$19$ $$27 - 9 T + T^{2}$$
$23$ $$36 + 6 T + T^{2}$$
$29$ $$( -3 + T )^{2}$$
$31$ $$12 - 6 T + T^{2}$$
$37$ $$12 + 6 T + T^{2}$$
$41$ $$108 + T^{2}$$
$43$ $$( -8 + T )^{2}$$
$47$ $$3 + 3 T + T^{2}$$
$53$ $$9 + 3 T + T^{2}$$
$59$ $$75 + 15 T + T^{2}$$
$61$ $$1 + T + T^{2}$$
$67$ $$3 - 3 T + T^{2}$$
$71$ $$147 + T^{2}$$
$73$ $$108 + 18 T + T^{2}$$
$79$ $$16 + 4 T + T^{2}$$
$83$ $$12 + T^{2}$$
$89$ $$12 - 6 T + T^{2}$$
$97$ $$300 + T^{2}$$