Properties

Label 273.2.l.a
Level $273$
Weight $2$
Character orbit 273.l
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.l (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: \(\Z[a_1, a_2, a_3]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −0.500000 + 0.866025i −2.00000 0 0 −2.00000 1.73205i 0 −0.500000 0.866025i 0
256.1 0 −0.500000 0.866025i −2.00000 0 0 −2.00000 + 1.73205i 0 −0.500000 + 0.866025i 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.h 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.l.a yes 2
3.b odd 2 1 819.2.s.b 2
7.c even 3 1 273.2.j.a 2
13.c even 3 1 273.2.j.a 2
21.h odd 6 1 819.2.n.b 2
39.i odd 6 1 819.2.n.b 2
91.h even 3 1 inner 273.2.l.a yes 2
273.s odd 6 1 819.2.s.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.j.a 2 7.c even 3 1
273.2.j.a 2 13.c even 3 1
273.2.l.a yes 2 1.a even 1 1 trivial
273.2.l.a yes 2 91.h even 3 1 inner
819.2.n.b 2 21.h odd 6 1
819.2.n.b 2 39.i odd 6 1
819.2.s.b 2 3.b odd 2 1
819.2.s.b 2 273.s 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

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 + 4 T + T^{2} \)
$11$ \( 36 - 6 T + T^{2} \)
$13$ \( 13 - 2 T + T^{2} \)
$17$ \( ( 6 + T )^{2} \)
$19$ \( 1 - T + T^{2} \)
$23$ \( ( 6 + T )^{2} \)
$29$ \( 36 + 6 T + T^{2} \)
$31$ \( 25 + 5 T + T^{2} \)
$37$ \( ( 1 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( 16 - 4 T + T^{2} \)
$47$ \( 36 + 6 T + T^{2} \)
$53$ \( 36 + 6 T + T^{2} \)
$59$ \( T^{2} \)
$61$ \( 169 - 13 T + T^{2} \)
$67$ \( 169 - 13 T + T^{2} \)
$71$ \( 144 - 12 T + T^{2} \)
$73$ \( 100 - 10 T + T^{2} \)
$79$ \( 1 - T + T^{2} \)
$83$ \( ( 6 + T )^{2} \)
$89$ \( ( -6 + T )^{2} \)
$97$ \( 25 + 5 T + T^{2} \)
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