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 $2$

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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: 6.0.64827.1
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:

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

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