Properties

Label 273.2.ba.b
Level $273$
Weight $2$
Character orbit 273.ba
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.ba (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.

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
38.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.50000 + 0.866025i 1.00000 1.73205i 0 0 0.500000 2.59808i 0 1.50000 + 2.59808i 0
194.1 0 1.50000 0.866025i 1.00000 + 1.73205i 0 0 0.500000 + 2.59808i 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.s odd 6 1 inner
273.ba 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.ba.b yes 2
3.b odd 2 1 CM 273.2.ba.b yes 2
7.d odd 6 1 273.2.ba.a 2
13.b even 2 1 273.2.ba.a 2
21.g even 6 1 273.2.ba.a 2
39.d odd 2 1 273.2.ba.a 2
91.s odd 6 1 inner 273.2.ba.b yes 2
273.ba even 6 1 inner 273.2.ba.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.ba.a 2 7.d odd 6 1
273.2.ba.a 2 13.b even 2 1
273.2.ba.a 2 21.g even 6 1
273.2.ba.a 2 39.d odd 2 1
273.2.ba.b yes 2 1.a even 1 1 trivial
273.2.ba.b yes 2 3.b odd 2 1 CM
273.2.ba.b yes 2 91.s odd 6 1 inner
273.2.ba.b yes 2 273.ba even 6 1 inner

Hecke kernels

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

\( T_{2} \)
\( T_{19}^{2} - 7 T_{19} + 49 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 3 - 3 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 - T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 13 + 7 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( 49 - 7 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 49 + 7 T + T^{2} \)
$37$ \( 147 - 21 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 5 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 48 + 12 T + T^{2} \)
$67$ \( 147 + 21 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 49 + 7 T + T^{2} \)
$79$ \( 169 + 13 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( -14 + T )^{2} \)
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