Properties

Label 273.2.r.a
Level $273$
Weight $2$
Character orbit 273.r
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.r (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.

\(f(q)\) \(=\) \( q + ( -1 - \zeta_{6} ) q^{3} + 2 q^{4} + ( 1 + 2 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 - \zeta_{6} ) q^{3} + 2 q^{4} + ( 1 + 2 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} + ( -2 - 2 \zeta_{6} ) q^{12} + ( 1 - 4 \zeta_{6} ) q^{13} + 4 q^{16} + ( 10 - 5 \zeta_{6} ) q^{19} + ( 1 - 5 \zeta_{6} ) q^{21} + 5 \zeta_{6} q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + ( 2 + 4 \zeta_{6} ) q^{28} + ( -10 + 5 \zeta_{6} ) q^{31} + 6 \zeta_{6} q^{36} -11 q^{37} + ( -5 + 7 \zeta_{6} ) q^{39} + ( 8 - 8 \zeta_{6} ) q^{43} + ( -4 - 4 \zeta_{6} ) q^{48} + ( -3 + 8 \zeta_{6} ) q^{49} + ( 2 - 8 \zeta_{6} ) q^{52} -15 q^{57} + ( -18 + 9 \zeta_{6} ) q^{61} + ( -6 + 9 \zeta_{6} ) q^{63} + 8 q^{64} + ( -11 + 11 \zeta_{6} ) q^{67} + ( -16 + 8 \zeta_{6} ) q^{73} + ( 5 - 10 \zeta_{6} ) q^{75} + ( 20 - 10 \zeta_{6} ) q^{76} + ( 13 - 13 \zeta_{6} ) q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + ( 2 - 10 \zeta_{6} ) q^{84} + ( 9 - 10 \zeta_{6} ) q^{91} + 15 q^{93} + ( -3 - 3 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{3} + 4q^{4} + 4q^{7} + 3q^{9} + O(q^{10}) \) \( 2q - 3q^{3} + 4q^{4} + 4q^{7} + 3q^{9} - 6q^{12} - 2q^{13} + 8q^{16} + 15q^{19} - 3q^{21} + 5q^{25} + 8q^{28} - 15q^{31} + 6q^{36} - 22q^{37} - 3q^{39} + 8q^{43} - 12q^{48} + 2q^{49} - 4q^{52} - 30q^{57} - 27q^{61} - 3q^{63} + 16q^{64} - 11q^{67} - 24q^{73} + 30q^{76} + 13q^{79} - 9q^{81} - 6q^{84} + 8q^{91} + 30q^{93} - 9q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −1.50000 0.866025i 2.00000 0 0 2.00000 + 1.73205i 0 1.50000 + 2.59808i 0
269.1 0 −1.50000 + 0.866025i 2.00000 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.v odd 6 1 inner
273.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.r.a 2
3.b odd 2 1 CM 273.2.r.a 2
7.d odd 6 1 273.2.bf.a yes 2
13.c even 3 1 273.2.bf.a yes 2
21.g even 6 1 273.2.bf.a yes 2
39.i odd 6 1 273.2.bf.a yes 2
91.v odd 6 1 inner 273.2.r.a 2
273.r even 6 1 inner 273.2.r.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.r.a 2 1.a even 1 1 trivial
273.2.r.a 2 3.b odd 2 1 CM
273.2.r.a 2 91.v odd 6 1 inner
273.2.r.a 2 273.r even 6 1 inner
273.2.bf.a yes 2 7.d odd 6 1
273.2.bf.a yes 2 13.c even 3 1
273.2.bf.a yes 2 21.g even 6 1
273.2.bf.a yes 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

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 3 + 3 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 - 4 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 13 + 2 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 75 - 15 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 75 + 15 T + T^{2} \)
$37$ \( ( 11 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( 64 - 8 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 243 + 27 T + T^{2} \)
$67$ \( 121 + 11 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 192 + 24 T + T^{2} \)
$79$ \( 169 - 13 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 27 + 9 T + T^{2} \)
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