Properties

Label 5586.2.a.bm
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: \( 1 \)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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 −3.73205 1.00000 0 1.00000 1.00000 −3.73205
1.2 1.00000 1.00000 1.00000 −0.267949 1.00000 0 1.00000 1.00000 −0.267949
\(n\): 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.bm 2
7.b odd 2 1 5586.2.a.bl 2
7.d odd 6 2 798.2.j.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.j.g 4 7.d odd 6 2
5586.2.a.bl 2 7.b odd 2 1
5586.2.a.bm 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5586))\):

\( T_{5}^{2} + 4 T_{5} + 1 \)
\( T_{11} + 1 \)
\( T_{13}^{2} + 4 T_{13} - 8 \)
\( T_{17}^{2} + 6 T_{17} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( 1 + 4 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( -8 + 4 T + T^{2} \)
$17$ \( 6 + 6 T + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( -2 + 2 T + T^{2} \)
$29$ \( ( -1 + T )^{2} \)
$31$ \( -11 + 2 T + T^{2} \)
$37$ \( -26 - 2 T + T^{2} \)
$41$ \( 46 + 14 T + T^{2} \)
$43$ \( -2 + 10 T + T^{2} \)
$47$ \( 52 + 16 T + T^{2} \)
$53$ \( 13 - 10 T + T^{2} \)
$59$ \( -3 + T^{2} \)
$61$ \( 52 + 16 T + T^{2} \)
$67$ \( -12 - 12 T + T^{2} \)
$71$ \( 22 + 14 T + T^{2} \)
$73$ \( -48 + T^{2} \)
$79$ \( -107 + 2 T + T^{2} \)
$83$ \( ( -3 + T )^{2} \)
$89$ \( 6 + 18 T + T^{2} \)
$97$ \( 13 + 8 T + T^{2} \)
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