Properties

Label 5290.2.a.g
Level $5290$
Weight $2$
Character orbit 5290.a
Self dual yes
Analytic conductor $42.241$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5290 = 2 \cdot 5 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5290.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(42.2408626693\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta q^{3} + q^{4} + q^{5} -\beta q^{6} + ( 1 + \beta ) q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + \beta q^{3} + q^{4} + q^{5} -\beta q^{6} + ( 1 + \beta ) q^{7} - q^{8} - q^{10} + ( -2 - 2 \beta ) q^{11} + \beta q^{12} + ( 2 + 2 \beta ) q^{13} + ( -1 - \beta ) q^{14} + \beta q^{15} + q^{16} + ( -3 - 2 \beta ) q^{17} + ( -2 - \beta ) q^{19} + q^{20} + ( 3 + \beta ) q^{21} + ( 2 + 2 \beta ) q^{22} -\beta q^{24} + q^{25} + ( -2 - 2 \beta ) q^{26} -3 \beta q^{27} + ( 1 + \beta ) q^{28} + ( -3 - \beta ) q^{29} -\beta q^{30} + ( -7 + \beta ) q^{31} - q^{32} + ( -6 - 2 \beta ) q^{33} + ( 3 + 2 \beta ) q^{34} + ( 1 + \beta ) q^{35} -2 \beta q^{37} + ( 2 + \beta ) q^{38} + ( 6 + 2 \beta ) q^{39} - q^{40} + ( 6 + 2 \beta ) q^{41} + ( -3 - \beta ) q^{42} + ( 1 - 4 \beta ) q^{43} + ( -2 - 2 \beta ) q^{44} + ( -1 - 7 \beta ) q^{47} + \beta q^{48} + ( -3 + 2 \beta ) q^{49} - q^{50} + ( -6 - 3 \beta ) q^{51} + ( 2 + 2 \beta ) q^{52} + ( 3 + \beta ) q^{53} + 3 \beta q^{54} + ( -2 - 2 \beta ) q^{55} + ( -1 - \beta ) q^{56} + ( -3 - 2 \beta ) q^{57} + ( 3 + \beta ) q^{58} -7 q^{59} + \beta q^{60} + ( -4 + 4 \beta ) q^{61} + ( 7 - \beta ) q^{62} + q^{64} + ( 2 + 2 \beta ) q^{65} + ( 6 + 2 \beta ) q^{66} + ( -1 - 4 \beta ) q^{67} + ( -3 - 2 \beta ) q^{68} + ( -1 - \beta ) q^{70} + ( 3 + 7 \beta ) q^{71} -\beta q^{73} + 2 \beta q^{74} + \beta q^{75} + ( -2 - \beta ) q^{76} + ( -8 - 4 \beta ) q^{77} + ( -6 - 2 \beta ) q^{78} + ( -3 + \beta ) q^{79} + q^{80} -9 q^{81} + ( -6 - 2 \beta ) q^{82} + ( -11 - 4 \beta ) q^{83} + ( 3 + \beta ) q^{84} + ( -3 - 2 \beta ) q^{85} + ( -1 + 4 \beta ) q^{86} + ( -3 - 3 \beta ) q^{87} + ( 2 + 2 \beta ) q^{88} + ( 8 - 2 \beta ) q^{89} + ( 8 + 4 \beta ) q^{91} + ( 3 - 7 \beta ) q^{93} + ( 1 + 7 \beta ) q^{94} + ( -2 - \beta ) q^{95} -\beta q^{96} + ( 8 + 4 \beta ) q^{97} + ( 3 - 2 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} + 2q^{5} + 2q^{7} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} + 2q^{5} + 2q^{7} - 2q^{8} - 2q^{10} - 4q^{11} + 4q^{13} - 2q^{14} + 2q^{16} - 6q^{17} - 4q^{19} + 2q^{20} + 6q^{21} + 4q^{22} + 2q^{25} - 4q^{26} + 2q^{28} - 6q^{29} - 14q^{31} - 2q^{32} - 12q^{33} + 6q^{34} + 2q^{35} + 4q^{38} + 12q^{39} - 2q^{40} + 12q^{41} - 6q^{42} + 2q^{43} - 4q^{44} - 2q^{47} - 6q^{49} - 2q^{50} - 12q^{51} + 4q^{52} + 6q^{53} - 4q^{55} - 2q^{56} - 6q^{57} + 6q^{58} - 14q^{59} - 8q^{61} + 14q^{62} + 2q^{64} + 4q^{65} + 12q^{66} - 2q^{67} - 6q^{68} - 2q^{70} + 6q^{71} - 4q^{76} - 16q^{77} - 12q^{78} - 6q^{79} + 2q^{80} - 18q^{81} - 12q^{82} - 22q^{83} + 6q^{84} - 6q^{85} - 2q^{86} - 6q^{87} + 4q^{88} + 16q^{89} + 16q^{91} + 6q^{93} + 2q^{94} - 4q^{95} + 16q^{97} + 6q^{98} + O(q^{100}) \)

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 −1.73205 1.00000 1.00000 1.73205 −0.732051 −1.00000 0 −1.00000
1.2 −1.00000 1.73205 1.00000 1.00000 −1.73205 2.73205 −1.00000 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(23\) \(-1\)

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 5290.2.a.g yes 2
23.b odd 2 1 5290.2.a.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5290.2.a.f 2 23.b odd 2 1
5290.2.a.g 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5290))\):

\( T_{3}^{2} - 3 \)
\( T_{7}^{2} - 2 T_{7} - 2 \)
\( T_{11}^{2} + 4 T_{11} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( -3 + T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( -2 - 2 T + T^{2} \)
$11$ \( -8 + 4 T + T^{2} \)
$13$ \( -8 - 4 T + T^{2} \)
$17$ \( -3 + 6 T + T^{2} \)
$19$ \( 1 + 4 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( 6 + 6 T + T^{2} \)
$31$ \( 46 + 14 T + T^{2} \)
$37$ \( -12 + T^{2} \)
$41$ \( 24 - 12 T + T^{2} \)
$43$ \( -47 - 2 T + T^{2} \)
$47$ \( -146 + 2 T + T^{2} \)
$53$ \( 6 - 6 T + T^{2} \)
$59$ \( ( 7 + T )^{2} \)
$61$ \( -32 + 8 T + T^{2} \)
$67$ \( -47 + 2 T + T^{2} \)
$71$ \( -138 - 6 T + T^{2} \)
$73$ \( -3 + T^{2} \)
$79$ \( 6 + 6 T + T^{2} \)
$83$ \( 73 + 22 T + T^{2} \)
$89$ \( 52 - 16 T + T^{2} \)
$97$ \( 16 - 16 T + T^{2} \)
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