Properties

Label 5586.2.a.bf
Level $5586$
Weight $2$
Character orbit 5586.a
Self dual yes
Analytic conductor $44.604$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \(x^{2} - 7\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−2.64575
2.64575
−1.00000 1.00000 1.00000 −1.64575 −1.00000 0 −1.00000 1.00000 1.64575
1.2 −1.00000 1.00000 1.00000 3.64575 −1.00000 0 −1.00000 1.00000 −3.64575
\(n\): 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.bf yes 2
7.b odd 2 1 5586.2.a.bc 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5586.2.a.bc 2 7.b odd 2 1
5586.2.a.bf 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5586))\):

\( T_{5}^{2} - 2 T_{5} - 6 \)
\( T_{11} - 2 \)
\( T_{13}^{2} - 6 T_{13} + 2 \)
\( T_{17}^{2} - 4 T_{17} - 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( -6 - 2 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( 2 - 6 T + T^{2} \)
$17$ \( -24 - 4 T + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( -6 + 2 T + T^{2} \)
$29$ \( -12 - 8 T + T^{2} \)
$31$ \( -6 + 2 T + T^{2} \)
$37$ \( -6 + 2 T + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( -12 + 8 T + T^{2} \)
$47$ \( -6 - 2 T + T^{2} \)
$53$ \( -12 - 8 T + T^{2} \)
$59$ \( ( -2 + T )^{2} \)
$61$ \( -12 + 8 T + T^{2} \)
$67$ \( ( -10 + T )^{2} \)
$71$ \( -24 + 4 T + T^{2} \)
$73$ \( -76 + 12 T + T^{2} \)
$79$ \( -14 - 14 T + T^{2} \)
$83$ \( ( -12 + T )^{2} \)
$89$ \( ( 8 + T )^{2} \)
$97$ \( 72 + 20 T + T^{2} \)
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