Properties

Label 273.2.bv.a
Level $273$
Weight $2$
Character orbit 273.bv
Analytic conductor $2.180$
Analytic rank $0$
Dimension $4$
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.bv (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{3} -2 \zeta_{12}^{3} q^{4} + ( 1 + 2 \zeta_{12}^{2} ) q^{7} -3 \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{3} -2 \zeta_{12}^{3} q^{4} + ( 1 + 2 \zeta_{12}^{2} ) q^{7} -3 \zeta_{12}^{2} q^{9} + ( -2 - 2 \zeta_{12}^{2} ) q^{12} + ( 4 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{13} -4 q^{16} + ( -5 - 3 \zeta_{12} + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{19} + ( 5 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{21} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{25} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + ( 4 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{28} + ( 5 + 6 \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{31} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{36} + ( 4 + 3 \zeta_{12} + 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{37} + ( 5 - 7 \zeta_{12}^{2} ) q^{39} + ( 6 + 6 \zeta_{12}^{2} ) q^{43} + ( -4 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{48} + ( -3 + 8 \zeta_{12}^{2} ) q^{49} + ( 2 - 8 \zeta_{12}^{2} ) q^{52} + ( -8 - \zeta_{12} + \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{57} + ( -9 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{61} + ( 6 - 9 \zeta_{12}^{2} ) q^{63} + 8 \zeta_{12}^{3} q^{64} + ( 9 - 9 \zeta_{12} - 2 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{67} + ( -8 + 9 \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{73} + ( -5 + 10 \zeta_{12}^{2} ) q^{75} + ( -10 + 4 \zeta_{12} + 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{76} + ( -7 \zeta_{12} + 14 \zeta_{12}^{3} ) q^{79} + ( -9 + 9 \zeta_{12}^{2} ) q^{81} + ( 2 - 10 \zeta_{12}^{2} ) q^{84} + ( 10 \zeta_{12} - \zeta_{12}^{3} ) q^{91} + ( 11 + 7 \zeta_{12} - 7 \zeta_{12}^{2} - 11 \zeta_{12}^{3} ) q^{93} + ( 8 - 8 \zeta_{12} - 11 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{7} - 6q^{9} + O(q^{10}) \) \( 4q + 8q^{7} - 6q^{9} - 12q^{12} - 16q^{16} - 16q^{19} + 22q^{31} + 22q^{37} + 6q^{39} + 36q^{43} + 4q^{49} - 8q^{52} - 30q^{57} + 6q^{63} + 32q^{67} - 34q^{73} - 28q^{76} - 18q^{81} - 12q^{84} + 30q^{93} + 10q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(-1\) \(\zeta_{12}\) \(-1 + \zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0.866025 + 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
0 0.866025 1.50000i 2.00000i 0 0 2.00000 + 1.73205i 0 −1.50000 2.59808i 0
32.1 0 −0.866025 1.50000i 2.00000i 0 0 2.00000 1.73205i 0 −1.50000 + 2.59808i 0
128.1 0 −0.866025 + 1.50000i 2.00000i 0 0 2.00000 + 1.73205i 0 −1.50000 2.59808i 0
137.1 0 0.866025 + 1.50000i 2.00000i 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.x odd 12 1 inner
273.bv even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.bv.a 4
3.b odd 2 1 CM 273.2.bv.a 4
7.c even 3 1 273.2.bw.a yes 4
13.f odd 12 1 273.2.bw.a yes 4
21.h odd 6 1 273.2.bw.a yes 4
39.k even 12 1 273.2.bw.a yes 4
91.x odd 12 1 inner 273.2.bv.a 4
273.bv even 12 1 inner 273.2.bv.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bv.a 4 1.a even 1 1 trivial
273.2.bv.a 4 3.b odd 2 1 CM
273.2.bv.a 4 91.x odd 12 1 inner
273.2.bv.a 4 273.bv even 12 1 inner
273.2.bw.a yes 4 7.c even 3 1
273.2.bw.a yes 4 13.f odd 12 1
273.2.bw.a yes 4 21.h odd 6 1
273.2.bw.a yes 4 39.k even 12 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

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 + 3 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 7 - 4 T + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( 169 - 22 T^{2} + T^{4} \)
$17$ \( T^{4} \)
$19$ \( 1369 + 518 T + 113 T^{2} + 16 T^{3} + T^{4} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( 169 - 104 T + 137 T^{2} - 22 T^{3} + T^{4} \)
$37$ \( 2209 - 1034 T + 242 T^{2} - 22 T^{3} + T^{4} \)
$41$ \( T^{4} \)
$43$ \( ( 108 - 18 T + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( 59049 + 243 T^{2} + T^{4} \)
$67$ \( 169 - 130 T + 281 T^{2} - 32 T^{3} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( 2116 + 644 T + 338 T^{2} + 34 T^{3} + T^{4} \)
$79$ \( 21609 + 147 T^{2} + T^{4} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( 27889 - 4676 T + 221 T^{2} - 10 T^{3} + T^{4} \)
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