Properties

Label 5586.2.a.ca
Level $5586$
Weight $2$
Character orbit 5586.a
Self dual yes
Analytic conductor $44.604$
Analytic rank $1$
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.202932.1
Defining polynomial: \(x^{4} - 2 x^{3} - 12 x^{2} + 12 x + 33\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 12 x^{2} + 12 x + 33\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 7 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - 3 \nu^{2} - 5 \nu + 13 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 7\)
\(\nu^{3}\)\(=\)\(2 \beta_{3} + 3 \beta_{2} + 8 \beta_{1} + 8\)

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
3.71730
2.32952
−1.49111
−2.55571
1.00000 −1.00000 1.00000 −3.71730 −1.00000 0 1.00000 1.00000 −3.71730
1.2 1.00000 −1.00000 1.00000 −2.32952 −1.00000 0 1.00000 1.00000 −2.32952
1.3 1.00000 −1.00000 1.00000 1.49111 −1.00000 0 1.00000 1.00000 1.49111
1.4 1.00000 −1.00000 1.00000 2.55571 −1.00000 0 1.00000 1.00000 2.55571
\(n\): 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.ca 4
7.b odd 2 1 5586.2.a.cb 4
7.d odd 6 2 798.2.j.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.j.k 8 7.d odd 6 2
5586.2.a.ca 4 1.a even 1 1 trivial
5586.2.a.cb 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5586))\):

\( T_{5}^{4} + 2 T_{5}^{3} - 12 T_{5}^{2} - 12 T_{5} + 33 \)
\( T_{11}^{4} - 30 T_{11}^{2} + 12 T_{11} + 9 \)
\( T_{13} + 2 \)
\( T_{17}^{4} + 10 T_{17}^{3} + 24 T_{17}^{2} - 6 T_{17} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( 33 - 12 T - 12 T^{2} + 2 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 9 + 12 T - 30 T^{2} + T^{4} \)
$13$ \( ( 2 + T )^{4} \)
$17$ \( -12 - 6 T + 24 T^{2} + 10 T^{3} + T^{4} \)
$19$ \( ( 1 + T )^{4} \)
$23$ \( -894 + 492 T - 54 T^{2} - 7 T^{3} + T^{4} \)
$29$ \( 12 - 81 T - 21 T^{2} + 5 T^{3} + T^{4} \)
$31$ \( -168 + 335 T - 69 T^{2} - 3 T^{3} + T^{4} \)
$37$ \( 32 - 14 T - 12 T^{2} + 2 T^{3} + T^{4} \)
$41$ \( 1944 - 522 T - 54 T^{2} + 12 T^{3} + T^{4} \)
$43$ \( 14 - 86 T + 12 T^{2} + 11 T^{3} + T^{4} \)
$47$ \( 7056 + 1272 T - 192 T^{2} - 9 T^{3} + T^{4} \)
$53$ \( 1704 + 423 T - 69 T^{2} - 11 T^{3} + T^{4} \)
$59$ \( -18738 - 3825 T - 81 T^{2} + 21 T^{3} + T^{4} \)
$61$ \( 10264 - 1756 T - 222 T^{2} + 11 T^{3} + T^{4} \)
$67$ \( -9088 - 3680 T - 216 T^{2} + 14 T^{3} + T^{4} \)
$71$ \( 972 - 702 T + 18 T^{3} + T^{4} \)
$73$ \( -108 - 4 T + 54 T^{2} + 15 T^{3} + T^{4} \)
$79$ \( -356 - 311 T - 57 T^{2} + 7 T^{3} + T^{4} \)
$83$ \( 453 - 360 T + 18 T^{2} + 16 T^{3} + T^{4} \)
$89$ \( 84 + 18 T - 18 T^{2} - 2 T^{3} + T^{4} \)
$97$ \( 1556 - 679 T - 15 T^{2} + 19 T^{3} + T^{4} \)
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