Properties

Label 273.2.j.a
Level $273$
Weight $2$
Character orbit 273.j
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.j (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: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + q^{3} + ( 2 - 2 \zeta_{6} ) q^{4} + ( 3 - \zeta_{6} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( 2 - 2 \zeta_{6} ) q^{4} + ( 3 - \zeta_{6} ) q^{7} + q^{9} -6 q^{11} + ( 2 - 2 \zeta_{6} ) q^{12} + ( -1 + 4 \zeta_{6} ) q^{13} -4 \zeta_{6} q^{16} + ( 6 - 6 \zeta_{6} ) q^{17} - q^{19} + ( 3 - \zeta_{6} ) q^{21} + 6 \zeta_{6} q^{23} + 5 \zeta_{6} q^{25} + q^{27} + ( 4 - 6 \zeta_{6} ) q^{28} + ( -6 + 6 \zeta_{6} ) q^{29} -5 \zeta_{6} q^{31} -6 q^{33} + ( 2 - 2 \zeta_{6} ) q^{36} + \zeta_{6} q^{37} + ( -1 + 4 \zeta_{6} ) q^{39} + 4 \zeta_{6} q^{43} + ( -12 + 12 \zeta_{6} ) q^{44} + ( -6 + 6 \zeta_{6} ) q^{47} -4 \zeta_{6} q^{48} + ( 8 - 5 \zeta_{6} ) q^{49} + ( 6 - 6 \zeta_{6} ) q^{51} + ( 6 + 2 \zeta_{6} ) q^{52} -6 \zeta_{6} q^{53} - q^{57} -13 q^{61} + ( 3 - \zeta_{6} ) q^{63} -8 q^{64} -13 q^{67} -12 \zeta_{6} q^{68} + 6 \zeta_{6} q^{69} + 12 \zeta_{6} q^{71} + 10 \zeta_{6} q^{73} + 5 \zeta_{6} q^{75} + ( -2 + 2 \zeta_{6} ) q^{76} + ( -18 + 6 \zeta_{6} ) q^{77} + ( 1 - \zeta_{6} ) q^{79} + q^{81} -6 q^{83} + ( 4 - 6 \zeta_{6} ) q^{84} + ( -6 + 6 \zeta_{6} ) q^{87} -6 \zeta_{6} q^{89} + ( 1 + 9 \zeta_{6} ) q^{91} + 12 q^{92} -5 \zeta_{6} q^{93} -5 \zeta_{6} q^{97} -6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 2q^{4} + 5q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} + 2q^{4} + 5q^{7} + 2q^{9} - 12q^{11} + 2q^{12} + 2q^{13} - 4q^{16} + 6q^{17} - 2q^{19} + 5q^{21} + 6q^{23} + 5q^{25} + 2q^{27} + 2q^{28} - 6q^{29} - 5q^{31} - 12q^{33} + 2q^{36} + q^{37} + 2q^{39} + 4q^{43} - 12q^{44} - 6q^{47} - 4q^{48} + 11q^{49} + 6q^{51} + 14q^{52} - 6q^{53} - 2q^{57} - 26q^{61} + 5q^{63} - 16q^{64} - 26q^{67} - 12q^{68} + 6q^{69} + 12q^{71} + 10q^{73} + 5q^{75} - 2q^{76} - 30q^{77} + q^{79} + 2q^{81} - 12q^{83} + 2q^{84} - 6q^{87} - 6q^{89} + 11q^{91} + 24q^{92} - 5q^{93} - 5q^{97} - 12q^{99} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
100.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.00000 1.00000 1.73205i 0 0 2.50000 0.866025i 0 1.00000 0
172.1 0 1.00000 1.00000 + 1.73205i 0 0 2.50000 + 0.866025i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.g 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.j.a 2
3.b odd 2 1 819.2.n.b 2
7.c even 3 1 273.2.l.a yes 2
13.c even 3 1 273.2.l.a yes 2
21.h odd 6 1 819.2.s.b 2
39.i odd 6 1 819.2.s.b 2
91.g even 3 1 inner 273.2.j.a 2
273.bm odd 6 1 819.2.n.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.j.a 2 1.a even 1 1 trivial
273.2.j.a 2 91.g even 3 1 inner
273.2.l.a yes 2 7.c even 3 1
273.2.l.a yes 2 13.c even 3 1
819.2.n.b 2 3.b odd 2 1
819.2.n.b 2 273.bm odd 6 1
819.2.s.b 2 21.h odd 6 1
819.2.s.b 2 39.i odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 - 5 T + T^{2} \)
$11$ \( ( 6 + T )^{2} \)
$13$ \( 13 - 2 T + T^{2} \)
$17$ \( 36 - 6 T + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( 36 - 6 T + T^{2} \)
$29$ \( 36 + 6 T + T^{2} \)
$31$ \( 25 + 5 T + T^{2} \)
$37$ \( 1 - T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 16 - 4 T + T^{2} \)
$47$ \( 36 + 6 T + T^{2} \)
$53$ \( 36 + 6 T + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 13 + T )^{2} \)
$67$ \( ( 13 + T )^{2} \)
$71$ \( 144 - 12 T + T^{2} \)
$73$ \( 100 - 10 T + T^{2} \)
$79$ \( 1 - T + T^{2} \)
$83$ \( ( 6 + T )^{2} \)
$89$ \( 36 + 6 T + T^{2} \)
$97$ \( 25 + 5 T + T^{2} \)
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