Properties

 Label 273.2.k.b 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: 6.0.6040683.1 Defining polynomial: $$x^{6} + 5 x^{4} - 2 x^{3} + 25 x^{2} - 5 x + 1$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 5 x^{4} - 2 x^{3} + 25 x^{2} - 5 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 1$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{4} + 5 \nu^{2} - \nu + 15$$$$)/5$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{5} + 5 \nu^{3} - \nu^{2} + 25 \nu$$$$)/5$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{5} - 15 \nu^{3} + 8 \nu^{2} - 75 \nu + 15$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + 3 \beta_{4} - 3$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{2} + 1$$ $$\nu^{4}$$ $$=$$ $$-5 \beta_{5} - 15 \beta_{4} + 5 \beta_{3} + \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} + 8 \beta_{4} - 25 \beta_{2} - 25 \beta_{1} - 8$$

Character values

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

 $$n$$ $$92$$ $$106$$ $$157$$ $$\chi(n)$$ $$1$$ $$-\beta_{4}$$ $$1$$

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
 1.06421 + 1.84326i 0.100820 + 0.174625i −1.16503 − 2.01789i 1.06421 − 1.84326i 0.100820 − 0.174625i −1.16503 + 2.01789i
−1.06421 + 1.84326i 0.500000 0.866025i −1.26508 2.19119i −2.65859 1.06421 + 1.84326i −0.500000 0.866025i 1.12842 −0.500000 0.866025i 2.82929 4.90048i
22.2 −0.100820 + 0.174625i 0.500000 0.866025i 0.979671 + 1.69684i 3.75770 0.100820 + 0.174625i −0.500000 0.866025i −0.798360 −0.500000 0.866025i −0.378851 + 0.656189i
22.3 1.16503 2.01789i 0.500000 0.866025i −1.71459 2.96975i 0.900885 −1.16503 2.01789i −0.500000 0.866025i −3.33006 −0.500000 0.866025i 1.04956 1.81789i
211.1 −1.06421 1.84326i 0.500000 + 0.866025i −1.26508 + 2.19119i −2.65859 1.06421 1.84326i −0.500000 + 0.866025i 1.12842 −0.500000 + 0.866025i 2.82929 + 4.90048i
211.2 −0.100820 0.174625i 0.500000 + 0.866025i 0.979671 1.69684i 3.75770 0.100820 0.174625i −0.500000 + 0.866025i −0.798360 −0.500000 + 0.866025i −0.378851 0.656189i
211.3 1.16503 + 2.01789i 0.500000 + 0.866025i −1.71459 + 2.96975i 0.900885 −1.16503 + 2.01789i −0.500000 + 0.866025i −3.33006 −0.500000 + 0.866025i 1.04956 + 1.81789i
 $$n$$: 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.b 6
3.b odd 2 1 819.2.o.f 6
13.c even 3 1 inner 273.2.k.b 6
13.c even 3 1 3549.2.a.m 3
13.e even 6 1 3549.2.a.l 3
39.i odd 6 1 819.2.o.f 6

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 5 T_{2}^{4} + 2 T_{2}^{3} + 25 T_{2}^{2} + 5 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 5 T + 25 T^{2} + 2 T^{3} + 5 T^{4} + T^{6}$$
$3$ $$( 1 - T + T^{2} )^{3}$$
$5$ $$( 9 - 9 T - 2 T^{2} + T^{3} )^{2}$$
$7$ $$( 1 + T + T^{2} )^{3}$$
$11$ $$121 - 121 T + 209 T^{2} + 110 T^{3} + 53 T^{4} + 8 T^{5} + T^{6}$$
$13$ $$2197 - 1690 T + 806 T^{2} - 265 T^{3} + 62 T^{4} - 10 T^{5} + T^{6}$$
$17$ $$1849 + 1849 T + 1333 T^{2} + 430 T^{3} + 101 T^{4} + 12 T^{5} + T^{6}$$
$19$ $$729 + 459 T + 370 T^{2} + 3 T^{3} + 26 T^{4} + 3 T^{5} + T^{6}$$
$23$ $$225 + 120 T + 169 T^{2} - 86 T^{3} + 41 T^{4} - 7 T^{5} + T^{6}$$
$29$ $$169 + 286 T + 367 T^{2} + 172 T^{3} + 59 T^{4} + 9 T^{5} + T^{6}$$
$31$ $$( 169 - 36 T - 5 T^{2} + T^{3} )^{2}$$
$37$ $$16129 + 11303 T + 7921 T^{2} + 254 T^{3} + 89 T^{4} + T^{6}$$
$41$ $$46656 + 14688 T + 5920 T^{2} + 24 T^{3} + 104 T^{4} + 6 T^{5} + T^{6}$$
$43$ $$225 + 120 T + 169 T^{2} - 86 T^{3} + 41 T^{4} - 7 T^{5} + T^{6}$$
$47$ $$( 405 - 44 T - 9 T^{2} + T^{3} )^{2}$$
$53$ $$( 3 + 23 T + 13 T^{2} + T^{3} )^{2}$$
$59$ $$2673225 + 258330 T + 42949 T^{2} + 1532 T^{3} + 279 T^{4} + 11 T^{5} + T^{6}$$
$61$ $$75625 - 15125 T + 8250 T^{2} + 1595 T^{3} + 306 T^{4} + 19 T^{5} + T^{6}$$
$67$ $$1 + 102 T + 10383 T^{2} + 2140 T^{3} + 339 T^{4} + 21 T^{5} + T^{6}$$
$71$ $$47961 + 27813 T + 11311 T^{2} + 2356 T^{3} + 357 T^{4} + 22 T^{5} + T^{6}$$
$73$ $$( 71 + 57 T + 14 T^{2} + T^{3} )^{2}$$
$79$ $$( 163 - 62 T + T^{2} + T^{3} )^{2}$$
$83$ $$( -365 + 279 T - 32 T^{2} + T^{3} )^{2}$$
$89$ $$20449 - 11011 T + 6358 T^{2} - 55 T^{3} + 86 T^{4} - 3 T^{5} + T^{6}$$
$97$ $$139129 + 20142 T + 10749 T^{2} - 1880 T^{3} + 387 T^{4} - 21 T^{5} + T^{6}$$