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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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 \(S_{2}^{\mathrm{new}}(273, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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