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 $2$

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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:

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

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