# Properties

 Label 273.2.i.c Level $273$ Weight $2$ Character orbit 273.i Analytic conductor $2.180$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.64827.1 Defining polynomial: $$x^{6} - x^{5} + 3 x^{4} + 5 x^{2} - 2 x + 1$$ 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 basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 3 x^{4} + 5 x^{2} - 2 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + 3 \nu^{4} - 9 \nu^{3} + 5 \nu^{2} - 2 \nu + 6$$$$)/13$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{5} + 9 \nu^{4} - 14 \nu^{3} + 15 \nu^{2} - 6 \nu + 18$$$$)/13$$ $$\beta_{4}$$ $$=$$ $$($$$$-4 \nu^{5} - \nu^{4} - 10 \nu^{3} - 6 \nu^{2} - 34 \nu - 2$$$$)/13$$ $$\beta_{5}$$ $$=$$ $$($$$$-6 \nu^{5} + 5 \nu^{4} - 15 \nu^{3} - 9 \nu^{2} - 25 \nu + 10$$$$)/13$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3 \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{5} - 3 \beta_{4} - 4 \beta_{1} - 2$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 9 \beta_{2} - 9 \beta_{1}$$

## Character values

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

 $$n$$ $$92$$ $$106$$ $$157$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1 + \beta_{5}$$

## 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
 0.222521 − 0.385418i −0.623490 + 1.07992i 0.900969 − 1.56052i 0.222521 + 0.385418i −0.623490 − 1.07992i 0.900969 + 1.56052i
−0.400969 + 0.694498i −0.500000 0.866025i 0.678448 + 1.17511i −0.346011 + 0.599308i 0.801938 2.20291 1.46533i −2.69202 −0.500000 + 0.866025i −0.277479 0.480608i
79.2 0.277479 0.480608i −0.500000 0.866025i 0.846011 + 1.46533i 2.02446 3.50647i −0.554958 0.167563 + 2.64044i 2.04892 −0.500000 + 0.866025i −1.12349 1.94594i
79.3 1.12349 1.94594i −0.500000 0.866025i −1.52446 2.64044i −0.178448 + 0.309081i −2.24698 −2.37047 1.17511i −2.35690 −0.500000 + 0.866025i 0.400969 + 0.694498i
235.1 −0.400969 0.694498i −0.500000 + 0.866025i 0.678448 1.17511i −0.346011 0.599308i 0.801938 2.20291 + 1.46533i −2.69202 −0.500000 0.866025i −0.277479 + 0.480608i
235.2 0.277479 + 0.480608i −0.500000 + 0.866025i 0.846011 1.46533i 2.02446 + 3.50647i −0.554958 0.167563 2.64044i 2.04892 −0.500000 0.866025i −1.12349 + 1.94594i
235.3 1.12349 + 1.94594i −0.500000 + 0.866025i −1.52446 + 2.64044i −0.178448 0.309081i −2.24698 −2.37047 + 1.17511i −2.35690 −0.500000 0.866025i 0.400969 0.694498i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 235.3 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.c 6
3.b odd 2 1 819.2.j.d 6
7.c even 3 1 inner 273.2.i.c 6
7.c even 3 1 1911.2.a.m 3
7.d odd 6 1 1911.2.a.l 3
21.g even 6 1 5733.2.a.ba 3
21.h odd 6 1 819.2.j.d 6
21.h odd 6 1 5733.2.a.bb 3

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - 2 T_{2}^{5} + 5 T_{2}^{4} + 3 T_{2}^{2} - T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + 3 T^{2} + 5 T^{4} - 2 T^{5} + T^{6}$$
$3$ $$( 1 + T + T^{2} )^{3}$$
$5$ $$1 + 4 T + 13 T^{2} + 14 T^{3} + 13 T^{4} - 3 T^{5} + T^{6}$$
$7$ $$343 + 7 T^{3} + T^{6}$$
$11$ $$1849 + 1290 T + 943 T^{2} + 56 T^{3} + 31 T^{4} + T^{5} + T^{6}$$
$13$ $$( -1 + T )^{6}$$
$17$ $$169 - 286 T + 549 T^{2} + 84 T^{3} + 47 T^{4} - 5 T^{5} + T^{6}$$
$19$ $$1849 - 2021 T + 1650 T^{2} - 525 T^{3} + 122 T^{4} - 13 T^{5} + T^{6}$$
$23$ $$169 - 65 T + 103 T^{2} + 56 T^{3} + 31 T^{4} + 6 T^{5} + T^{6}$$
$29$ $$( 1 + 3 T - 4 T^{2} + T^{3} )^{2}$$
$31$ $$78961 - 36811 T + 11541 T^{2} - 2058 T^{3} + 269 T^{4} - 20 T^{5} + T^{6}$$
$37$ $$16129 - 13208 T + 8403 T^{2} - 1722 T^{3} + 257 T^{4} - 19 T^{5} + T^{6}$$
$41$ $$( 377 + 159 T + 22 T^{2} + T^{3} )^{2}$$
$43$ $$( -91 + 35 T + 14 T^{2} + T^{3} )^{2}$$
$47$ $$53824 + 34336 T + 22368 T^{2} + 168 T^{3} + 152 T^{4} + 2 T^{5} + T^{6}$$
$53$ $$15625 + 6250 T + 3125 T^{2} + 75 T^{4} + 5 T^{5} + T^{6}$$
$59$ $$41209 + 5684 T + 2205 T^{2} + 210 T^{3} + 77 T^{4} + 7 T^{5} + T^{6}$$
$61$ $$1849 + 1548 T + 1081 T^{2} + 266 T^{3} + 61 T^{4} - 5 T^{5} + T^{6}$$
$67$ $$1885129 - 203204 T + 34261 T^{2} - 1414 T^{3} + 229 T^{4} - 9 T^{5} + T^{6}$$
$71$ $$( -71 - 22 T + 9 T^{2} + T^{3} )^{2}$$
$73$ $$27889 - 14195 T + 9229 T^{2} + 686 T^{3} + 229 T^{4} - 12 T^{5} + T^{6}$$
$79$ $$729 - 729 T + 1053 T^{2} + 378 T^{3} + 117 T^{4} + 12 T^{5} + T^{6}$$
$83$ $$( 1 + 3 T - 4 T^{2} + T^{3} )^{2}$$
$89$ $$1164241 + 148902 T + 50335 T^{2} - 6160 T^{3} + 703 T^{4} - 29 T^{5} + T^{6}$$
$97$ $$( -7 - 84 T + 7 T^{2} + T^{3} )^{2}$$