Properties

Label 273.2.u.a
Level $273$
Weight $2$
Character orbit 273.u
Analytic conductor $2.180$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.u (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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 - \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{4} + ( -3 + 2 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 - \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{4} + ( -3 + 2 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} + ( 2 - 4 \zeta_{6} ) q^{12} + ( -4 + 3 \zeta_{6} ) q^{13} + ( -4 + 4 \zeta_{6} ) q^{16} + 8 \zeta_{6} q^{19} + ( 5 - \zeta_{6} ) q^{21} + 5 q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + ( -4 - 2 \zeta_{6} ) q^{28} -7 q^{31} + ( -6 + 6 \zeta_{6} ) q^{36} + ( 4 + 4 \zeta_{6} ) q^{37} + ( 7 - 2 \zeta_{6} ) q^{39} -13 \zeta_{6} q^{43} + ( 8 - 4 \zeta_{6} ) q^{48} + ( 5 - 8 \zeta_{6} ) q^{49} + ( -6 - 2 \zeta_{6} ) q^{52} + ( 8 - 16 \zeta_{6} ) q^{57} + ( 10 - 5 \zeta_{6} ) q^{61} + ( -6 - 3 \zeta_{6} ) q^{63} -8 q^{64} + ( -7 - 7 \zeta_{6} ) q^{67} + 17 q^{73} + ( -5 - 5 \zeta_{6} ) q^{75} + ( -16 + 16 \zeta_{6} ) q^{76} + 13 q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + ( 2 + 8 \zeta_{6} ) q^{84} + ( 6 - 11 \zeta_{6} ) q^{91} + ( 7 + 7 \zeta_{6} ) q^{93} + 5 \zeta_{6} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{3} + 2q^{4} - 4q^{7} + 3q^{9} + O(q^{10}) \) \( 2q - 3q^{3} + 2q^{4} - 4q^{7} + 3q^{9} - 5q^{13} - 4q^{16} + 8q^{19} + 9q^{21} + 10q^{25} - 10q^{28} - 14q^{31} - 6q^{36} + 12q^{37} + 12q^{39} - 13q^{43} + 12q^{48} + 2q^{49} - 14q^{52} + 15q^{61} - 15q^{63} - 16q^{64} - 21q^{67} + 34q^{73} - 15q^{75} - 16q^{76} + 26q^{79} - 9q^{81} + 12q^{84} + q^{91} + 21q^{93} + 5q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
62.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −1.50000 + 0.866025i 1.00000 1.73205i 0 0 −2.00000 1.73205i 0 1.50000 2.59808i 0
251.1 0 −1.50000 0.866025i 1.00000 + 1.73205i 0 0 −2.00000 + 1.73205i 0 1.50000 + 2.59808i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
91.t odd 6 1 inner
273.u 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.u.a 2
3.b odd 2 1 CM 273.2.u.a 2
7.b odd 2 1 273.2.u.b yes 2
13.e even 6 1 273.2.u.b yes 2
21.c even 2 1 273.2.u.b yes 2
39.h odd 6 1 273.2.u.b yes 2
91.t odd 6 1 inner 273.2.u.a 2
273.u even 6 1 inner 273.2.u.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.u.a 2 1.a even 1 1 trivial
273.2.u.a 2 3.b odd 2 1 CM
273.2.u.a 2 91.t odd 6 1 inner
273.2.u.a 2 273.u even 6 1 inner
273.2.u.b yes 2 7.b odd 2 1
273.2.u.b yes 2 13.e even 6 1
273.2.u.b yes 2 21.c even 2 1
273.2.u.b yes 2 39.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\):

\( T_{2} \)
\( T_{19}^{2} - 8 T_{19} + 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 3 + 3 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 + 4 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 13 + 5 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 64 - 8 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( 7 + T )^{2} \)
$37$ \( 48 - 12 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 169 + 13 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 75 - 15 T + T^{2} \)
$67$ \( 147 + 21 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -17 + T )^{2} \)
$79$ \( ( -13 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 25 - 5 T + T^{2} \)
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