Properties

Label 273.2.bl.b
Level $273$
Weight $2$
Character orbit 273.bl
Analytic conductor $2.180$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.bl (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-7})\)
Defining polynomial: \(x^{4} - x^{3} - x^{2} - 2 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{3} ) q^{2} + q^{3} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4} + ( -1 - \beta_{2} ) q^{5} + ( 1 - \beta_{3} ) q^{6} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{7} + ( 1 - 2 \beta_{2} ) q^{8} + q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{3} ) q^{2} + q^{3} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4} + ( -1 - \beta_{2} ) q^{5} + ( 1 - \beta_{3} ) q^{6} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{7} + ( 1 - 2 \beta_{2} ) q^{8} + q^{9} + ( -2 + \beta_{1} + \beta_{3} ) q^{10} + ( -2 + 4 \beta_{2} ) q^{11} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{12} + ( -3 + 4 \beta_{2} ) q^{13} + ( -1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{14} + ( -1 - \beta_{2} ) q^{15} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{16} + ( -1 + \beta_{2} ) q^{17} + ( 1 - \beta_{3} ) q^{18} + ( -2 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{19} + ( -1 - \beta_{2} + 3 \beta_{3} ) q^{20} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{21} + ( 2 - 4 \beta_{1} + 2 \beta_{3} ) q^{22} + ( 2 - 4 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{23} + ( 1 - 2 \beta_{2} ) q^{24} -2 \beta_{2} q^{25} + ( 1 - 4 \beta_{1} + 3 \beta_{3} ) q^{26} + q^{27} + ( -4 + \beta_{1} - 3 \beta_{2} ) q^{28} + ( 7 - 7 \beta_{2} ) q^{29} + ( -2 + \beta_{1} + \beta_{3} ) q^{30} + ( -4 + \beta_{2} + 2 \beta_{3} ) q^{31} + ( 4 - \beta_{1} + 3 \beta_{2} ) q^{32} + ( -2 + 4 \beta_{2} ) q^{33} + ( -\beta_{1} + \beta_{3} ) q^{34} + ( 1 - 2 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{35} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{36} + ( -2 - \beta_{2} + 4 \beta_{3} ) q^{37} + ( -2 + 4 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{38} + ( -3 + 4 \beta_{2} ) q^{39} + ( -3 + 3 \beta_{2} ) q^{40} + ( -1 + 4 \beta_{1} + 3 \beta_{2} ) q^{41} + ( -1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{42} + ( -2 + 4 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{43} + ( 4 - 6 \beta_{1} - 2 \beta_{2} ) q^{44} + ( -1 - \beta_{2} ) q^{45} + ( 9 - 7 \beta_{1} + 2 \beta_{2} ) q^{46} + ( 1 + 2 \beta_{1} + 3 \beta_{2} ) q^{47} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{48} -7 q^{49} + ( -2 + 2 \beta_{1} ) q^{50} + ( -1 + \beta_{2} ) q^{51} + ( 4 - 7 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{52} + ( 4 - 8 \beta_{1} - 9 \beta_{2} + 4 \beta_{3} ) q^{53} + ( 1 - \beta_{3} ) q^{54} + ( 6 - 6 \beta_{2} ) q^{55} + ( -1 + 2 \beta_{1} + 2 \beta_{3} ) q^{56} + ( -2 + 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{57} + ( 7 \beta_{1} - 7 \beta_{3} ) q^{58} + ( -3 - 2 \beta_{1} - 5 \beta_{2} ) q^{59} + ( -1 - \beta_{2} + 3 \beta_{3} ) q^{60} + ( 2 + 4 \beta_{1} + 4 \beta_{3} ) q^{61} + ( -5 - 3 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} ) q^{62} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{63} + ( 9 - 2 \beta_{1} - 2 \beta_{3} ) q^{64} + ( 7 - 5 \beta_{2} ) q^{65} + ( 2 - 4 \beta_{1} + 2 \beta_{3} ) q^{66} + ( -2 + 8 \beta_{1} + 4 \beta_{2} - 8 \beta_{3} ) q^{67} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{68} + ( 2 - 4 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{69} + ( -2 - 3 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{70} + ( 4 - 5 \beta_{2} + 6 \beta_{3} ) q^{71} + ( 1 - 2 \beta_{2} ) q^{72} + ( 10 - 5 \beta_{2} ) q^{73} + ( -7 - 3 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} ) q^{74} -2 \beta_{2} q^{75} + ( -8 + 2 \beta_{1} - 6 \beta_{2} ) q^{76} + ( 2 - 4 \beta_{1} - 4 \beta_{3} ) q^{77} + ( 1 - 4 \beta_{1} + 3 \beta_{3} ) q^{78} + ( 7 - 2 \beta_{1} - 9 \beta_{2} + 4 \beta_{3} ) q^{79} + ( -2 + 3 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} ) q^{80} + q^{81} + ( -6 + \beta_{1} + \beta_{3} ) q^{82} + ( 2 - 4 \beta_{2} ) q^{83} + ( -4 + \beta_{1} - 3 \beta_{2} ) q^{84} + ( 2 - \beta_{2} ) q^{85} + ( -5 + 3 \beta_{1} - 2 \beta_{2} ) q^{86} + ( 7 - 7 \beta_{2} ) q^{87} + 6 q^{88} + ( -18 + 9 \beta_{2} ) q^{89} + ( -2 + \beta_{1} + \beta_{3} ) q^{90} + ( 3 - 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{91} + ( 15 - 5 \beta_{1} - 5 \beta_{3} ) q^{92} + ( -4 + \beta_{2} + 2 \beta_{3} ) q^{93} + ( -\beta_{1} - \beta_{3} ) q^{94} + ( 2 - 4 \beta_{1} - 6 \beta_{2} + 8 \beta_{3} ) q^{95} + ( 4 - \beta_{1} + 3 \beta_{2} ) q^{96} + ( 6 - \beta_{2} - 4 \beta_{3} ) q^{97} + ( -7 + 7 \beta_{3} ) q^{98} + ( -2 + 4 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 3q^{2} + 4q^{3} + q^{4} - 6q^{5} + 3q^{6} + 4q^{9} + O(q^{10}) \) \( 4q + 3q^{2} + 4q^{3} + q^{4} - 6q^{5} + 3q^{6} + 4q^{9} - 6q^{10} + q^{12} - 4q^{13} - 7q^{14} - 6q^{15} - q^{16} - 2q^{17} + 3q^{18} - 3q^{20} + 6q^{22} + 8q^{23} - 4q^{25} + 3q^{26} + 4q^{27} - 21q^{28} + 14q^{29} - 6q^{30} - 12q^{31} + 21q^{32} + q^{36} - 6q^{37} - 14q^{38} - 4q^{39} - 6q^{40} + 6q^{41} - 7q^{42} + 6q^{44} - 6q^{45} + 33q^{46} + 12q^{47} - q^{48} - 28q^{49} - 6q^{50} - 2q^{51} + 5q^{52} - 6q^{53} + 3q^{54} + 12q^{55} - 24q^{59} - 3q^{60} + 16q^{61} - 13q^{62} + 32q^{64} + 18q^{65} + 6q^{66} + q^{68} + 8q^{69} + 12q^{71} + 30q^{73} - 17q^{74} - 4q^{75} - 42q^{76} + 3q^{78} + 12q^{79} + 4q^{81} - 22q^{82} - 21q^{84} + 6q^{85} - 21q^{86} + 14q^{87} + 24q^{88} - 54q^{89} - 6q^{90} + 50q^{92} - 12q^{93} - 2q^{94} + 21q^{96} + 18q^{97} - 21q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - x^{2} - 2 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + \nu^{2} - \nu - 2 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + \nu + 2 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2}\)
\(\nu^{3}\)\(=\)\(-\beta_{3} + \beta_{2} + \beta_{1} + 2\)

Character values

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

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(-1 + \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
88.1
1.39564 0.228425i
−0.895644 + 1.09445i
1.39564 + 0.228425i
−0.895644 1.09445i
−0.395644 0.228425i 1.00000 −0.895644 1.55130i −1.50000 + 0.866025i −0.395644 0.228425i 2.64575i 1.73205i 1.00000 0.791288
88.2 1.89564 + 1.09445i 1.00000 1.39564 + 2.41733i −1.50000 + 0.866025i 1.89564 + 1.09445i 2.64575i 1.73205i 1.00000 −3.79129
121.1 −0.395644 + 0.228425i 1.00000 −0.895644 + 1.55130i −1.50000 0.866025i −0.395644 + 0.228425i 2.64575i 1.73205i 1.00000 0.791288
121.2 1.89564 1.09445i 1.00000 1.39564 2.41733i −1.50000 0.866025i 1.89564 1.09445i 2.64575i 1.73205i 1.00000 −3.79129
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.u 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.bl.b yes 4
3.b odd 2 1 819.2.do.d 4
7.c even 3 1 273.2.t.b 4
13.e even 6 1 273.2.t.b 4
21.h odd 6 1 819.2.bm.d 4
39.h odd 6 1 819.2.bm.d 4
91.u even 6 1 inner 273.2.bl.b yes 4
273.x odd 6 1 819.2.do.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.t.b 4 7.c even 3 1
273.2.t.b 4 13.e even 6 1
273.2.bl.b yes 4 1.a even 1 1 trivial
273.2.bl.b yes 4 91.u even 6 1 inner
819.2.bm.d 4 21.h odd 6 1
819.2.bm.d 4 39.h odd 6 1
819.2.do.d 4 3.b odd 2 1
819.2.do.d 4 273.x odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 3 T_{2}^{3} + 2 T_{2}^{2} + 3 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 2 T^{2} - 3 T^{3} + T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( ( 3 + 3 T + T^{2} )^{2} \)
$7$ \( ( 7 + T^{2} )^{2} \)
$11$ \( ( 12 + T^{2} )^{2} \)
$13$ \( ( 13 + 2 T + T^{2} )^{2} \)
$17$ \( ( 1 + T + T^{2} )^{2} \)
$19$ \( ( 28 + T^{2} )^{2} \)
$23$ \( 25 + 40 T + 69 T^{2} - 8 T^{3} + T^{4} \)
$29$ \( ( 49 - 7 T + T^{2} )^{2} \)
$31$ \( 25 + 60 T + 53 T^{2} + 12 T^{3} + T^{4} \)
$37$ \( 625 - 150 T - 13 T^{2} + 6 T^{3} + T^{4} \)
$41$ \( 625 + 150 T - 13 T^{2} - 6 T^{3} + T^{4} \)
$43$ \( 441 + 21 T^{2} + T^{4} \)
$47$ \( 25 - 60 T + 53 T^{2} - 12 T^{3} + T^{4} \)
$53$ \( 5625 - 450 T + 111 T^{2} + 6 T^{3} + T^{4} \)
$59$ \( 1681 + 984 T + 233 T^{2} + 24 T^{3} + T^{4} \)
$61$ \( ( -68 - 8 T + T^{2} )^{2} \)
$67$ \( 10000 + 248 T^{2} + T^{4} \)
$71$ \( 2601 + 612 T - 3 T^{2} - 12 T^{3} + T^{4} \)
$73$ \( ( 75 - 15 T + T^{2} )^{2} \)
$79$ \( 225 - 180 T + 129 T^{2} - 12 T^{3} + T^{4} \)
$83$ \( ( 12 + T^{2} )^{2} \)
$89$ \( ( 243 + 27 T + T^{2} )^{2} \)
$97$ \( 1 + 18 T + 107 T^{2} - 18 T^{3} + T^{4} \)
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