Properties

Label 273.2.bj.a
Level $273$
Weight $2$
Character orbit 273.bj
Analytic conductor $2.180$
Analytic rank $0$
Dimension $2$
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.bj (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(1\) \(-1\) \(-1 + \zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
0.500000 0.866025i
0.500000 + 0.866025i
−1.50000 + 0.866025i 0.500000 0.866025i 0.500000 0.866025i −3.00000 + 1.73205i 1.73205i 0.500000 2.59808i 1.73205i −0.500000 0.866025i 3.00000 5.19615i
142.1 −1.50000 0.866025i 0.500000 + 0.866025i 0.500000 + 0.866025i −3.00000 1.73205i 1.73205i 0.500000 + 2.59808i 1.73205i −0.500000 + 0.866025i 3.00000 + 5.19615i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.r even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.bj.a 2
3.b odd 2 1 819.2.dl.d 2
7.c even 3 1 273.2.bj.b yes 2
7.c even 3 1 1911.2.c.b 2
7.d odd 6 1 1911.2.c.e 2
13.b even 2 1 273.2.bj.b yes 2
21.h odd 6 1 819.2.dl.a 2
39.d odd 2 1 819.2.dl.a 2
91.r even 6 1 inner 273.2.bj.a 2
91.r even 6 1 1911.2.c.b 2
91.s odd 6 1 1911.2.c.e 2
273.w odd 6 1 819.2.dl.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bj.a 2 1.a even 1 1 trivial
273.2.bj.a 2 91.r even 6 1 inner
273.2.bj.b yes 2 7.c even 3 1
273.2.bj.b yes 2 13.b even 2 1
819.2.dl.a 2 21.h odd 6 1
819.2.dl.a 2 39.d odd 2 1
819.2.dl.d 2 3.b odd 2 1
819.2.dl.d 2 273.w odd 6 1
1911.2.c.b 2 7.c even 3 1
1911.2.c.b 2 91.r even 6 1
1911.2.c.e 2 7.d odd 6 1
1911.2.c.e 2 91.s odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3 + 3 T + T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( 12 + 6 T + T^{2} \)
$7$ \( 7 - T + T^{2} \)
$11$ \( 27 - 9 T + T^{2} \)
$13$ \( 13 + 2 T + T^{2} \)
$17$ \( 9 - 3 T + T^{2} \)
$19$ \( 27 - 9 T + T^{2} \)
$23$ \( 36 + 6 T + T^{2} \)
$29$ \( ( -3 + T )^{2} \)
$31$ \( 12 - 6 T + T^{2} \)
$37$ \( 12 + 6 T + T^{2} \)
$41$ \( 108 + T^{2} \)
$43$ \( ( -8 + T )^{2} \)
$47$ \( 3 + 3 T + T^{2} \)
$53$ \( 9 + 3 T + T^{2} \)
$59$ \( 75 + 15 T + T^{2} \)
$61$ \( 1 + T + T^{2} \)
$67$ \( 3 - 3 T + T^{2} \)
$71$ \( 147 + T^{2} \)
$73$ \( 108 + 18 T + T^{2} \)
$79$ \( 16 + 4 T + T^{2} \)
$83$ \( 12 + T^{2} \)
$89$ \( 12 - 6 T + T^{2} \)
$97$ \( 300 + T^{2} \)
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