Properties

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

\(f(q)\) \(=\) \( q + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{4} + ( -1 + 3 \zeta_{12}^{2} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{4} + ( -1 + 3 \zeta_{12}^{2} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + ( 2 + 2 \zeta_{12}^{2} ) q^{12} + ( -\zeta_{12} - 3 \zeta_{12}^{3} ) q^{13} + ( 4 - 4 \zeta_{12}^{2} ) q^{16} + ( -2 - 2 \zeta_{12} + 5 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{19} + ( -4 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{21} + ( 5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{25} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + ( 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{28} + ( -5 - \zeta_{12} + 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{31} + 6 \zeta_{12}^{3} q^{36} + ( -3 + 3 \zeta_{12} - 4 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{37} + ( 7 - 5 \zeta_{12}^{2} ) q^{39} + ( 7 - 14 \zeta_{12}^{2} ) q^{43} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{48} + ( -8 + 3 \zeta_{12}^{2} ) q^{49} + ( -2 - 6 \zeta_{12}^{2} ) q^{52} + ( 8 - 7 \zeta_{12} - 7 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{57} + ( 4 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{61} + ( -6 - 3 \zeta_{12}^{2} ) q^{63} -8 \zeta_{12}^{3} q^{64} + ( 7 - 9 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{67} + ( -9 - 8 \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{73} + ( 5 + 5 \zeta_{12}^{2} ) q^{75} + ( -4 + 6 \zeta_{12} - 6 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{76} + ( -7 \zeta_{12} + 14 \zeta_{12}^{3} ) q^{79} -9 \zeta_{12}^{2} q^{81} + ( -8 + 10 \zeta_{12}^{2} ) q^{84} + ( 10 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{91} + ( -11 - 11 \zeta_{12} + 4 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{93} + ( 11 - 8 \zeta_{12} - 8 \zeta_{12}^{2} + 11 \zeta_{12}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{7} - 6q^{9} + O(q^{10}) \) \( 4q + 2q^{7} - 6q^{9} + 12q^{12} + 8q^{16} + 2q^{19} - 8q^{31} - 20q^{37} + 18q^{39} - 26q^{49} - 20q^{52} + 18q^{57} - 30q^{63} + 32q^{67} - 34q^{73} + 30q^{75} - 28q^{76} - 18q^{81} - 12q^{84} - 36q^{93} + 28q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(-1\) \(\zeta_{12}^{3}\) \(-1 + \zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
44.1
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 −0.866025 1.50000i −1.73205 + 1.00000i 0 0 0.500000 + 2.59808i 0 −1.50000 + 2.59808i 0
86.1 0 0.866025 + 1.50000i 1.73205 1.00000i 0 0 0.500000 + 2.59808i 0 −1.50000 + 2.59808i 0
200.1 0 0.866025 1.50000i 1.73205 + 1.00000i 0 0 0.500000 2.59808i 0 −1.50000 2.59808i 0
242.1 0 −0.866025 + 1.50000i −1.73205 1.00000i 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.z odd 12 1 inner
273.cd 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.cd.b yes 4
3.b odd 2 1 CM 273.2.cd.b yes 4
7.c even 3 1 273.2.cd.a 4
13.d odd 4 1 273.2.cd.a 4
21.h odd 6 1 273.2.cd.a 4
39.f even 4 1 273.2.cd.a 4
91.z odd 12 1 inner 273.2.cd.b yes 4
273.cd even 12 1 inner 273.2.cd.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.cd.a 4 7.c even 3 1
273.2.cd.a 4 13.d odd 4 1
273.2.cd.a 4 21.h odd 6 1
273.2.cd.a 4 39.f even 4 1
273.2.cd.b yes 4 1.a even 1 1 trivial
273.2.cd.b yes 4 3.b odd 2 1 CM
273.2.cd.b yes 4 91.z odd 12 1 inner
273.2.cd.b yes 4 273.cd even 12 1 inner

Hecke kernels

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

\( T_{2} \)
\( T_{5} \)
\( T_{19}^{4} - 2 T_{19}^{3} + 65 T_{19}^{2} + 176 T_{19} + 121 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 + 3 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 7 - T + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( 169 + 23 T^{2} + T^{4} \)
$17$ \( T^{4} \)
$19$ \( 121 + 176 T + 65 T^{2} - 2 T^{3} + T^{4} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( 169 + 286 T + 137 T^{2} + 8 T^{3} + T^{4} \)
$37$ \( 5329 + 1606 T + 221 T^{2} + 20 T^{3} + T^{4} \)
$41$ \( T^{4} \)
$43$ \( ( 147 + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( 2304 + 48 T^{2} + T^{4} \)
$67$ \( 169 - 130 T + 281 T^{2} - 32 T^{3} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( 9409 + 1940 T + 389 T^{2} + 34 T^{3} + T^{4} \)
$79$ \( 21609 + 147 T^{2} + T^{4} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( 4 - 56 T + 392 T^{2} - 28 T^{3} + T^{4} \)
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