Properties

Label 273.2.i.a
Level $273$
Weight $2$
Character orbit 273.i
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.i (of order \(3\), 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: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 0.866025i −0.500000 0.866025i 0.500000 + 0.866025i −2.00000 + 3.46410i −1.00000 −2.50000 + 0.866025i 3.00000 −0.500000 + 0.866025i 2.00000 + 3.46410i
235.1 0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 0.866025i −2.00000 3.46410i −1.00000 −2.50000 0.866025i 3.00000 −0.500000 0.866025i 2.00000 3.46410i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.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.i.a 2
3.b odd 2 1 819.2.j.a 2
7.c even 3 1 inner 273.2.i.a 2
7.c even 3 1 1911.2.a.c 1
7.d odd 6 1 1911.2.a.b 1
21.g even 6 1 5733.2.a.k 1
21.h odd 6 1 819.2.j.a 2
21.h odd 6 1 5733.2.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.i.a 2 1.a even 1 1 trivial
273.2.i.a 2 7.c even 3 1 inner
819.2.j.a 2 3.b odd 2 1
819.2.j.a 2 21.h odd 6 1
1911.2.a.b 1 7.d odd 6 1
1911.2.a.c 1 7.c even 3 1
5733.2.a.i 1 21.h odd 6 1
5733.2.a.k 1 21.g even 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( 16 + 4 T + T^{2} \)
$7$ \( 7 + 5 T + T^{2} \)
$11$ \( 25 - 5 T + T^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( 9 + 3 T + T^{2} \)
$19$ \( 25 - 5 T + T^{2} \)
$23$ \( 36 + 6 T + T^{2} \)
$29$ \( ( -7 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( ( -8 + T )^{2} \)
$43$ \( ( -2 + T )^{2} \)
$47$ \( 81 + 9 T + T^{2} \)
$53$ \( 81 + 9 T + T^{2} \)
$59$ \( 81 - 9 T + T^{2} \)
$61$ \( 1 + T + T^{2} \)
$67$ \( 49 + 7 T + T^{2} \)
$71$ \( ( 3 + T )^{2} \)
$73$ \( 36 - 6 T + T^{2} \)
$79$ \( 100 - 10 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( 64 - 8 T + T^{2} \)
$97$ \( ( -18 + T )^{2} \)
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