Properties

Label 5586.2.a.bd
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: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.73205
1.73205
−1.00000 1.00000 1.00000 −3.46410 −1.00000 0 −1.00000 1.00000 3.46410
1.2 −1.00000 1.00000 1.00000 3.46410 −1.00000 0 −1.00000 1.00000 −3.46410
\(n\): 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.bd 2
7.b odd 2 1 798.2.a.k 2
21.c even 2 1 2394.2.a.x 2
28.d even 2 1 6384.2.a.br 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.a.k 2 7.b odd 2 1
2394.2.a.x 2 21.c even 2 1
5586.2.a.bd 2 1.a even 1 1 trivial
6384.2.a.br 2 28.d even 2 1

Hecke kernels

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

\( T_{5}^{2} - 12 \)
\( T_{11}^{2} + 4 T_{11} - 8 \)
\( T_{13}^{2} - 12 \)
\( T_{17}^{2} + 8 T_{17} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( -12 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -8 + 4 T + T^{2} \)
$13$ \( -12 + T^{2} \)
$17$ \( 4 + 8 T + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( -8 + 4 T + T^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( -48 + T^{2} \)
$37$ \( ( -10 + T )^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( -48 + T^{2} \)
$47$ \( -32 + 8 T + T^{2} \)
$53$ \( ( -2 + T )^{2} \)
$59$ \( ( 4 + T )^{2} \)
$61$ \( -44 + 4 T + T^{2} \)
$67$ \( 24 - 12 T + T^{2} \)
$71$ \( -32 + 8 T + T^{2} \)
$73$ \( ( 10 + T )^{2} \)
$79$ \( -104 + 4 T + T^{2} \)
$83$ \( ( 8 + T )^{2} \)
$89$ \( ( -2 + T )^{2} \)
$97$ \( 52 + 16 T + T^{2} \)
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