Properties

Label 5586.2.a.bt
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.

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

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
−1.65544
−0.210756
−1.00000 1.00000 1.00000 −2.34889 −1.00000 0 −1.00000 1.00000 2.34889
1.2 −1.00000 1.00000 1.00000 −1.39593 −1.00000 0 −1.00000 1.00000 1.39593
1.3 −1.00000 1.00000 1.00000 2.74483 −1.00000 0 −1.00000 1.00000 −2.74483
\(n\): 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.bt 3
7.b odd 2 1 5586.2.a.bs 3
7.d odd 6 2 798.2.j.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.j.j 6 7.d odd 6 2
5586.2.a.bs 3 7.b odd 2 1
5586.2.a.bt 3 1.a even 1 1 trivial

Hecke kernels

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

\( T_{5}^{3} + T_{5}^{2} - 7 T_{5} - 9 \)
\( T_{11}^{3} - 7 T_{11}^{2} - T_{11} + 43 \)
\( T_{13}^{3} + 2 T_{13}^{2} - 20 T_{13} + 8 \)
\( T_{17}^{3} - 10 T_{17}^{2} + 26 T_{17} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( -9 - 7 T + T^{2} + T^{3} \)
$7$ \( T^{3} \)
$11$ \( 43 - T - 7 T^{2} + T^{3} \)
$13$ \( 8 - 20 T + 2 T^{2} + T^{3} \)
$17$ \( -6 + 26 T - 10 T^{2} + T^{3} \)
$19$ \( ( -1 + T )^{3} \)
$23$ \( 82 + 8 T - 10 T^{2} + T^{3} \)
$29$ \( -27 - 25 T + 7 T^{2} + T^{3} \)
$31$ \( 479 - 37 T - 11 T^{2} + T^{3} \)
$37$ \( -66 - 74 T + T^{3} \)
$41$ \( 6 - 4 T - 2 T^{2} + T^{3} \)
$43$ \( 18 - 26 T + 6 T^{2} + T^{3} \)
$47$ \( -32 - 32 T + T^{3} \)
$53$ \( 29 - 21 T - T^{2} + T^{3} \)
$59$ \( -473 - 113 T + 3 T^{2} + T^{3} \)
$61$ \( -232 + 132 T - 22 T^{2} + T^{3} \)
$67$ \( 48 + 104 T + 20 T^{2} + T^{3} \)
$71$ \( 6 + 26 T + 10 T^{2} + T^{3} \)
$73$ \( -396 - 84 T + 10 T^{2} + T^{3} \)
$79$ \( 301 - 197 T + 3 T^{2} + T^{3} \)
$83$ \( 121 + 91 T + 19 T^{2} + T^{3} \)
$89$ \( 178 - 68 T - 14 T^{2} + T^{3} \)
$97$ \( -839 - 77 T + 13 T^{2} + T^{3} \)
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