# Properties

 Label 273.2.k.c Level $273$ Weight $2$ Character orbit 273.k 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.k (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.

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

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 $$\left(\mathbb{Z}/273\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1
 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 −3.04892 −0.400969 0.694498i −0.500000 0.866025i −2.69202 −0.500000 0.866025i 1.22252 2.11747i
22.2 0.277479 0.480608i −0.500000 + 0.866025i 0.846011 + 1.46533i 1.35690 0.277479 + 0.480608i −0.500000 0.866025i 2.04892 −0.500000 0.866025i 0.376510 0.652135i
22.3 1.12349 1.94594i −0.500000 + 0.866025i −1.52446 2.64044i 1.69202 1.12349 + 1.94594i −0.500000 0.866025i −2.35690 −0.500000 0.866025i 1.90097 3.29257i
211.1 −0.400969 0.694498i −0.500000 0.866025i 0.678448 1.17511i −3.04892 −0.400969 + 0.694498i −0.500000 + 0.866025i −2.69202 −0.500000 + 0.866025i 1.22252 + 2.11747i
211.2 0.277479 + 0.480608i −0.500000 0.866025i 0.846011 1.46533i 1.35690 0.277479 0.480608i −0.500000 + 0.866025i 2.04892 −0.500000 + 0.866025i 0.376510 + 0.652135i
211.3 1.12349 + 1.94594i −0.500000 0.866025i −1.52446 + 2.64044i 1.69202 1.12349 1.94594i −0.500000 + 0.866025i −2.35690 −0.500000 + 0.866025i 1.90097 + 3.29257i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 211.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.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.k.c 6
3.b odd 2 1 819.2.o.e 6
13.c even 3 1 inner 273.2.k.c 6
13.c even 3 1 3549.2.a.i 3
13.e even 6 1 3549.2.a.u 3
39.i odd 6 1 819.2.o.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.k.c 6 1.a even 1 1 trivial
273.2.k.c 6 13.c even 3 1 inner
819.2.o.e 6 3.b odd 2 1
819.2.o.e 6 39.i odd 6 1
3549.2.a.i 3 13.c even 3 1
3549.2.a.u 3 13.e even 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

$p$ $F_p(T)$
$2$ $$1 - T + 3 T^{2} + 5 T^{4} - 2 T^{5} + T^{6}$$
$3$ $$( 1 + T + T^{2} )^{3}$$
$5$ $$( 7 - 7 T + T^{3} )^{2}$$
$7$ $$( 1 + T + T^{2} )^{3}$$
$11$ $$169 + 195 T + 199 T^{2} + 56 T^{3} + 19 T^{4} - 2 T^{5} + T^{6}$$
$13$ $$2197 + 1014 T + 390 T^{2} + 115 T^{3} + 30 T^{4} + 6 T^{5} + T^{6}$$
$17$ $$1 - T + 3 T^{2} + 5 T^{4} - 2 T^{5} + T^{6}$$
$19$ $$1849 - 2021 T + 1650 T^{2} - 525 T^{3} + 122 T^{4} - 13 T^{5} + T^{6}$$
$23$ $$19321 + 6394 T + 2533 T^{2} + 140 T^{3} + 55 T^{4} + 3 T^{5} + T^{6}$$
$29$ $$28561 + 12168 T + 5353 T^{2} + 266 T^{3} + 73 T^{4} + T^{5} + T^{6}$$
$31$ $$( -211 - 4 T + 11 T^{2} + T^{3} )^{2}$$
$37$ $$284089 - 58097 T + 14013 T^{2} - 630 T^{3} + 125 T^{4} - 4 T^{5} + T^{6}$$
$41$ $$64 + 288 T + 1312 T^{2} - 56 T^{3} + 40 T^{4} + 2 T^{5} + T^{6}$$
$43$ $$44521 - 844 T + 2337 T^{2} - 378 T^{3} + 125 T^{4} - 11 T^{5} + T^{6}$$
$47$ $$( 127 + 104 T + 19 T^{2} + T^{3} )^{2}$$
$53$ $$( 41 - 29 T - 5 T^{2} + T^{3} )^{2}$$
$59$ $$44521 - 844 T + 2337 T^{2} - 378 T^{3} + 125 T^{4} - 11 T^{5} + T^{6}$$
$61$ $$461041 + 71295 T + 15778 T^{2} + 623 T^{3} + 154 T^{4} + 7 T^{5} + T^{6}$$
$67$ $$841 + 754 T + 1111 T^{2} - 448 T^{3} + 199 T^{4} - 15 T^{5} + T^{6}$$
$71$ $$284089 + 9061 T + 9883 T^{2} - 1372 T^{3} + 307 T^{4} - 18 T^{5} + T^{6}$$
$73$ $$( -377 - 79 T + 6 T^{2} + T^{3} )^{2}$$
$79$ $$( 27 - 18 T - 3 T^{2} + T^{3} )^{2}$$
$83$ $$( 13 - 99 T + 2 T^{2} + T^{3} )^{2}$$
$89$ $$16129 - 11049 T + 5410 T^{2} - 1225 T^{3} + 202 T^{4} - 17 T^{5} + T^{6}$$
$97$ $$169 - 390 T + 1069 T^{2} + 364 T^{3} + 199 T^{4} - 13 T^{5} + T^{6}$$