Properties

Label 5586.2.a.bw
Level $5586$
Weight $2$
Character orbit 5586.a
Self dual yes
Analytic conductor $44.604$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.29268.1
Defining polynomial: \(x^{4} - x^{3} - 9 x^{2} + 5 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) 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} + ( -\beta_{1} + \beta_{3} ) q^{11} - q^{12} + 2 \beta_{2} q^{13} + \beta_{2} q^{15} + q^{16} + ( 3 - \beta_{2} + 2 \beta_{3} ) q^{17} - q^{18} + q^{19} -\beta_{2} q^{20} + ( \beta_{1} - \beta_{3} ) q^{22} + ( -2 + 3 \beta_{1} - \beta_{2} ) q^{23} + q^{24} + ( 1 + \beta_{1} + \beta_{3} ) q^{25} -2 \beta_{2} q^{26} - q^{27} -3 \beta_{1} q^{29} -\beta_{2} q^{30} + ( 3 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{31} - q^{32} + ( \beta_{1} - \beta_{3} ) q^{33} + ( -3 + \beta_{2} - 2 \beta_{3} ) q^{34} + q^{36} + ( -3 - 2 \beta_{1} + \beta_{2} ) q^{37} - q^{38} -2 \beta_{2} q^{39} + \beta_{2} q^{40} + ( 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{41} + ( 5 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{43} + ( -\beta_{1} + \beta_{3} ) q^{44} -\beta_{2} q^{45} + ( 2 - 3 \beta_{1} + \beta_{2} ) q^{46} + ( 2 - 2 \beta_{2} + \beta_{3} ) q^{47} - q^{48} + ( -1 - \beta_{1} - \beta_{3} ) q^{50} + ( -3 + \beta_{2} - 2 \beta_{3} ) q^{51} + 2 \beta_{2} q^{52} + ( -1 + 3 \beta_{3} ) q^{53} + q^{54} + ( 1 - 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{55} - q^{57} + 3 \beta_{1} q^{58} + ( 3 - 5 \beta_{1} + \beta_{2} ) q^{59} + \beta_{2} q^{60} + ( 6 + \beta_{3} ) q^{61} + ( -3 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{62} + q^{64} + ( -12 - 2 \beta_{1} - 2 \beta_{3} ) q^{65} + ( -\beta_{1} + \beta_{3} ) q^{66} + ( 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{67} + ( 3 - \beta_{2} + 2 \beta_{3} ) q^{68} + ( 2 - 3 \beta_{1} + \beta_{2} ) q^{69} + ( -1 + 4 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{71} - q^{72} + ( -3 + 5 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{73} + ( 3 + 2 \beta_{1} - \beta_{2} ) q^{74} + ( -1 - \beta_{1} - \beta_{3} ) q^{75} + q^{76} + 2 \beta_{2} q^{78} + ( -2 - \beta_{1} + 2 \beta_{3} ) q^{79} -\beta_{2} q^{80} + q^{81} + ( -3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{82} + ( 5 - 4 \beta_{1} + 2 \beta_{3} ) q^{83} + ( 2 - 3 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{85} + ( -5 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{86} + 3 \beta_{1} q^{87} + ( \beta_{1} - \beta_{3} ) q^{88} + ( 2 - \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{89} + \beta_{2} q^{90} + ( -2 + 3 \beta_{1} - \beta_{2} ) q^{92} + ( -3 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{93} + ( -2 + 2 \beta_{2} - \beta_{3} ) q^{94} -\beta_{2} q^{95} + q^{96} + ( -1 + 3 \beta_{1} - \beta_{2} ) q^{97} + ( -\beta_{1} + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{6} - 4 q^{8} + 4 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{6} - 4 q^{8} + 4 q^{9} - 2 q^{11} - 4 q^{12} + 4 q^{16} + 10 q^{17} - 4 q^{18} + 4 q^{19} + 2 q^{22} - 5 q^{23} + 4 q^{24} + 4 q^{25} - 4 q^{27} - 3 q^{29} + 9 q^{31} - 4 q^{32} + 2 q^{33} - 10 q^{34} + 4 q^{36} - 14 q^{37} - 4 q^{38} + 4 q^{41} + 21 q^{43} - 2 q^{44} + 5 q^{46} + 7 q^{47} - 4 q^{48} - 4 q^{50} - 10 q^{51} - 7 q^{53} + 4 q^{54} - 4 q^{57} + 3 q^{58} + 7 q^{59} + 23 q^{61} - 9 q^{62} + 4 q^{64} - 48 q^{65} - 2 q^{66} + 6 q^{67} + 10 q^{68} + 5 q^{69} + 2 q^{71} - 4 q^{72} - 5 q^{73} + 14 q^{74} - 4 q^{75} + 4 q^{76} - 11 q^{79} + 4 q^{81} - 4 q^{82} + 14 q^{83} + 6 q^{85} - 21 q^{86} + 3 q^{87} + 2 q^{88} + 10 q^{89} - 5 q^{92} - 9 q^{93} - 7 q^{94} + 4 q^{96} - q^{97} - 2 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 9 x^{2} + 5 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 5 \nu - 2 \)\()/2\)
\(\beta_{3}\)\(=\)\( \nu^{2} - 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 5\)
\(\nu^{3}\)\(=\)\(2 \beta_{2} + 5 \beta_{1} + 2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.85121
−1.25548
−2.43292
1.83719
−1.00000 −1.00000 1.00000 −3.46130 1.00000 0 −1.00000 1.00000 3.46130
1.2 −1.00000 −1.00000 1.00000 −1.14924 1.00000 0 −1.00000 1.00000 1.14924
1.3 −1.00000 −1.00000 1.00000 2.11806 1.00000 0 −1.00000 1.00000 −2.11806
1.4 −1.00000 −1.00000 1.00000 2.49248 1.00000 0 −1.00000 1.00000 −2.49248
\(n\): 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.bw 4
7.b odd 2 1 5586.2.a.bz 4
7.c even 3 2 798.2.j.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.j.l 8 7.c even 3 2
5586.2.a.bw 4 1.a even 1 1 trivial
5586.2.a.bz 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5586))\):

\( T_{5}^{4} - 12 T_{5}^{2} + 6 T_{5} + 21 \)
\( T_{11}^{4} + 2 T_{11}^{3} - 12 T_{11}^{2} - 22 T_{11} + 7 \)
\( T_{13}^{4} - 48 T_{13}^{2} - 48 T_{13} + 336 \)
\( T_{17}^{4} - 10 T_{17}^{3} - 6 T_{17}^{2} + 254 T_{17} - 452 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( 21 + 6 T - 12 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 7 - 22 T - 12 T^{2} + 2 T^{3} + T^{4} \)
$13$ \( 336 - 48 T - 48 T^{2} + T^{4} \)
$17$ \( -452 + 254 T - 6 T^{2} - 10 T^{3} + T^{4} \)
$19$ \( ( -1 + T )^{4} \)
$23$ \( 922 - 190 T - 60 T^{2} + 5 T^{3} + T^{4} \)
$29$ \( 1296 - 135 T - 81 T^{2} + 3 T^{3} + T^{4} \)
$31$ \( -1068 + 477 T - 33 T^{2} - 9 T^{3} + T^{4} \)
$37$ \( -8 - 22 T + 42 T^{2} + 14 T^{3} + T^{4} \)
$41$ \( -344 + 506 T - 96 T^{2} - 4 T^{3} + T^{4} \)
$43$ \( -654 + 24 T + 114 T^{2} - 21 T^{3} + T^{4} \)
$47$ \( 256 + 128 T - 24 T^{2} - 7 T^{3} + T^{4} \)
$53$ \( 976 - 419 T - 93 T^{2} + 7 T^{3} + T^{4} \)
$59$ \( 9202 + 653 T - 183 T^{2} - 7 T^{3} + T^{4} \)
$61$ \( 712 - 620 T + 186 T^{2} - 23 T^{3} + T^{4} \)
$67$ \( 7968 + 1008 T - 240 T^{2} - 6 T^{3} + T^{4} \)
$71$ \( 28 + 46 T - 126 T^{2} - 2 T^{3} + T^{4} \)
$73$ \( -1676 - 1240 T - 186 T^{2} + 5 T^{3} + T^{4} \)
$79$ \( 172 - 151 T + 3 T^{2} + 11 T^{3} + T^{4} \)
$83$ \( 43 + 286 T - 60 T^{2} - 14 T^{3} + T^{4} \)
$89$ \( -4844 + 1958 T - 156 T^{2} - 10 T^{3} + T^{4} \)
$97$ \( 1048 - 59 T - 69 T^{2} + T^{3} + T^{4} \)
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