Properties

Label 5586.2.a.bu
Level $5586$
Weight $2$
Character orbit 5586.a
Self dual yes
Analytic conductor $44.604$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5586.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.6044345691\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.404.1
Defining polynomial: \(x^{3} - x^{2} - 5 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 798)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 5 x - 1\):

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.86620
−0.210756
−1.65544
1.00000 −1.00000 1.00000 −0.866198 −1.00000 0 1.00000 1.00000 −0.866198
1.2 1.00000 −1.00000 1.00000 2.21076 −1.00000 0 1.00000 1.00000 2.21076
1.3 1.00000 −1.00000 1.00000 3.65544 −1.00000 0 1.00000 1.00000 3.65544
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)
\(19\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5586.2.a.bu 3
7.b odd 2 1 5586.2.a.bv 3
7.d odd 6 2 798.2.j.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.j.i 6 7.d odd 6 2
5586.2.a.bu 3 1.a even 1 1 trivial
5586.2.a.bv 3 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5586))\):

\( T_{5}^{3} - 5 T_{5}^{2} + 3 T_{5} + 7 \)
\( T_{11}^{3} - T_{11}^{2} - 17 T_{11} + 21 \)
\( T_{13}^{3} - 6 T_{13}^{2} - 20 T_{13} + 88 \)
\( T_{17}^{3} - 14 T_{17}^{2} + 60 T_{17} - 74 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( 7 + 3 T - 5 T^{2} + T^{3} \)
$7$ \( T^{3} \)
$11$ \( 21 - 17 T - T^{2} + T^{3} \)
$13$ \( 88 - 20 T - 6 T^{2} + T^{3} \)
$17$ \( -74 + 60 T - 14 T^{2} + T^{3} \)
$19$ \( ( -1 + T )^{3} \)
$23$ \( -302 - 54 T + 6 T^{2} + T^{3} \)
$29$ \( 43 - 41 T + T^{2} + T^{3} \)
$31$ \( 7 + 19 T - 11 T^{2} + T^{3} \)
$37$ \( -194 - 44 T + 4 T^{2} + T^{3} \)
$41$ \( -42 + 46 T - 14 T^{2} + T^{3} \)
$43$ \( 606 - 92 T - 6 T^{2} + T^{3} \)
$47$ \( 144 - 96 T + 4 T^{2} + T^{3} \)
$53$ \( -101 - 5 T + 9 T^{2} + T^{3} \)
$59$ \( 63 - 3 T - 7 T^{2} + T^{3} \)
$61$ \( -56 + 12 T + 10 T^{2} + T^{3} \)
$67$ \( -32 + 16 T + 12 T^{2} + T^{3} \)
$71$ \( 206 - 92 T + 6 T^{2} + T^{3} \)
$73$ \( 196 - 56 T - 2 T^{2} + T^{3} \)
$79$ \( -1067 - 81 T + 15 T^{2} + T^{3} \)
$83$ \( 3279 - 157 T - 19 T^{2} + T^{3} \)
$89$ \( -42 + 46 T - 14 T^{2} + T^{3} \)
$97$ \( -873 + 301 T - 31 T^{2} + T^{3} \)
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