Properties

Label 5586.2.a.bo
Level $5586$
Weight $2$
Character orbit 5586.a
Self dual yes
Analytic conductor $44.604$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.41421
1.41421
1.00000 1.00000 1.00000 0 1.00000 0 1.00000 1.00000 0
1.2 1.00000 1.00000 1.00000 0 1.00000 0 1.00000 1.00000 0
\(n\): 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.bo yes 2
7.b odd 2 1 5586.2.a.bj 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5586.2.a.bj 2 7.b odd 2 1
5586.2.a.bo yes 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5586))\):

\( T_{5} \)
\( T_{11}^{2} + 8 T_{11} + 14 \)
\( T_{13}^{2} - 8 \)
\( T_{17}^{2} + 8 T_{17} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( 14 + 8 T + T^{2} \)
$13$ \( -8 + T^{2} \)
$17$ \( 8 + 8 T + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( -18 + T^{2} \)
$29$ \( ( 8 + T )^{2} \)
$31$ \( -28 + 4 T + T^{2} \)
$37$ \( ( 2 + T )^{2} \)
$41$ \( 14 + 8 T + T^{2} \)
$43$ \( 8 + 8 T + T^{2} \)
$47$ \( -28 + 4 T + T^{2} \)
$53$ \( ( 2 + T )^{2} \)
$59$ \( T^{2} \)
$61$ \( 14 - 8 T + T^{2} \)
$67$ \( -18 + T^{2} \)
$71$ \( -200 + T^{2} \)
$73$ \( 62 - 16 T + T^{2} \)
$79$ \( -82 + 8 T + T^{2} \)
$83$ \( -16 + 8 T + T^{2} \)
$89$ \( 14 - 8 T + T^{2} \)
$97$ \( 136 - 24 T + T^{2} \)
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