Properties

Label 5290.2.a.v
Level $5290$
Weight $2$
Character orbit 5290.a
Self dual yes
Analytic conductor $42.241$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + ( 1 + \beta_{1} + \beta_{3} ) q^{3} + q^{4} + q^{5} + ( -1 - \beta_{1} - \beta_{3} ) q^{6} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} - q^{8} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q - q^{2} + ( 1 + \beta_{1} + \beta_{3} ) q^{3} + q^{4} + q^{5} + ( -1 - \beta_{1} - \beta_{3} ) q^{6} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} - q^{8} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{9} - q^{10} + ( \beta_{1} + 3 \beta_{3} ) q^{11} + ( 1 + \beta_{1} + \beta_{3} ) q^{12} + ( -2 - 2 \beta_{3} ) q^{13} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{14} + ( 1 + \beta_{1} + \beta_{3} ) q^{15} + q^{16} + ( 3 + \beta_{1} - \beta_{2} ) q^{17} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{18} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{19} + q^{20} + ( -1 + \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{21} + ( -\beta_{1} - 3 \beta_{3} ) q^{22} + ( -1 - \beta_{1} - \beta_{3} ) q^{24} + q^{25} + ( 2 + 2 \beta_{3} ) q^{26} + ( 1 - \beta_{1} - \beta_{3} ) q^{27} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{28} + ( -3 + 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{29} + ( -1 - \beta_{1} - \beta_{3} ) q^{30} + ( 7 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{31} - q^{32} + ( 4 + \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{33} + ( -3 - \beta_{1} + \beta_{2} ) q^{34} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{35} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{36} + ( 2 + 2 \beta_{1} + 2 \beta_{3} ) q^{37} + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{38} + ( -4 - 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{39} - q^{40} + ( -2 - 4 \beta_{1} + 2 \beta_{2} ) q^{41} + ( 1 - \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{42} + ( -3 - \beta_{2} - \beta_{3} ) q^{43} + ( \beta_{1} + 3 \beta_{3} ) q^{44} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{45} + ( -1 + 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{47} + ( 1 + \beta_{1} + \beta_{3} ) q^{48} + ( 3 + 4 \beta_{1} - 2 \beta_{2} + 8 \beta_{3} ) q^{49} - q^{50} + ( 4 + 3 \beta_{1} + 4 \beta_{3} ) q^{51} + ( -2 - 2 \beta_{3} ) q^{52} + ( 5 + 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{53} + ( -1 + \beta_{1} + \beta_{3} ) q^{54} + ( \beta_{1} + 3 \beta_{3} ) q^{55} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{56} + ( -3 - 5 \beta_{1} - 3 \beta_{2} ) q^{57} + ( 3 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{58} + 3 q^{59} + ( 1 + \beta_{1} + \beta_{3} ) q^{60} + ( 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{61} + ( -7 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{62} + ( 4 \beta_{2} - 4 \beta_{3} ) q^{63} + q^{64} + ( -2 - 2 \beta_{3} ) q^{65} + ( -4 - \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{66} + ( 3 + 5 \beta_{1} + 3 \beta_{2} + 8 \beta_{3} ) q^{67} + ( 3 + \beta_{1} - \beta_{2} ) q^{68} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{70} + ( 5 + \beta_{1} + \beta_{2} - 5 \beta_{3} ) q^{71} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{72} + ( -4 - \beta_{1} + \beta_{2} + 5 \beta_{3} ) q^{73} + ( -2 - 2 \beta_{1} - 2 \beta_{3} ) q^{74} + ( 1 + \beta_{1} + \beta_{3} ) q^{75} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{76} + ( -6 - 2 \beta_{1} + 4 \beta_{2} - 8 \beta_{3} ) q^{77} + ( 4 + 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{78} + ( -1 - \beta_{1} - 7 \beta_{2} + 3 \beta_{3} ) q^{79} + q^{80} + ( -1 - 6 \beta_{1} - 6 \beta_{3} ) q^{81} + ( 2 + 4 \beta_{1} - 2 \beta_{2} ) q^{82} + ( 3 + 3 \beta_{1} + 3 \beta_{2} ) q^{83} + ( -1 + \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{84} + ( 3 + \beta_{1} - \beta_{2} ) q^{85} + ( 3 + \beta_{2} + \beta_{3} ) q^{86} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{87} + ( -\beta_{1} - 3 \beta_{3} ) q^{88} + ( 10 + 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{89} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{90} + ( 8 - 4 \beta_{2} + 8 \beta_{3} ) q^{91} + ( 9 + 9 \beta_{1} + \beta_{2} + 7 \beta_{3} ) q^{93} + ( 1 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{94} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{95} + ( -1 - \beta_{1} - \beta_{3} ) q^{96} + ( 4 + \beta_{1} + 6 \beta_{2} + 3 \beta_{3} ) q^{97} + ( -3 - 4 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} ) q^{98} + ( 8 - 4 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} + 4q^{3} + 4q^{4} + 4q^{5} - 4q^{6} - 4q^{7} - 4q^{8} + O(q^{10}) \) \( 4q - 4q^{2} + 4q^{3} + 4q^{4} + 4q^{5} - 4q^{6} - 4q^{7} - 4q^{8} - 4q^{10} + 4q^{12} - 8q^{13} + 4q^{14} + 4q^{15} + 4q^{16} + 12q^{17} - 8q^{19} + 4q^{20} - 4q^{21} - 4q^{24} + 4q^{25} + 8q^{26} + 4q^{27} - 4q^{28} - 12q^{29} - 4q^{30} + 28q^{31} - 4q^{32} + 16q^{33} - 12q^{34} - 4q^{35} + 8q^{37} + 8q^{38} - 16q^{39} - 4q^{40} - 8q^{41} + 4q^{42} - 12q^{43} - 4q^{47} + 4q^{48} + 12q^{49} - 4q^{50} + 16q^{51} - 8q^{52} + 20q^{53} - 4q^{54} + 4q^{56} - 12q^{57} + 12q^{58} + 12q^{59} + 4q^{60} - 28q^{62} + 4q^{64} - 8q^{65} - 16q^{66} + 12q^{67} + 12q^{68} + 4q^{70} + 20q^{71} - 16q^{73} - 8q^{74} + 4q^{75} - 8q^{76} - 24q^{77} + 16q^{78} - 4q^{79} + 4q^{80} - 4q^{81} + 8q^{82} + 12q^{83} - 4q^{84} + 12q^{85} + 12q^{86} + 4q^{87} + 40q^{89} + 32q^{91} + 36q^{93} + 4q^{94} - 8q^{95} - 4q^{96} + 16q^{97} - 12q^{98} + 32q^{99} + O(q^{100}) \)

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
−1.93185
0.517638
−0.517638
1.93185
−1.00000 −0.414214 1.00000 1.00000 0.414214 −1.71744 −1.00000 −2.82843 −1.00000
1.2 −1.00000 −0.414214 1.00000 1.00000 0.414214 −0.282561 −1.00000 −2.82843 −1.00000
1.3 −1.00000 2.41421 1.00000 1.00000 −2.41421 −5.18154 −1.00000 2.82843 −1.00000
1.4 −1.00000 2.41421 1.00000 1.00000 −2.41421 3.18154 −1.00000 2.82843 −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.v yes 4
23.b odd 2 1 5290.2.a.u 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5290.2.a.u 4 23.b odd 2 1
5290.2.a.v yes 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5290))\):

\( T_{3}^{2} - 2 T_{3} - 1 \)
\( T_{7}^{4} + 4 T_{7}^{3} - 12 T_{7}^{2} - 32 T_{7} - 8 \)
\( T_{11}^{4} - 28 T_{11}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( ( -1 - 2 T + T^{2} )^{2} \)
$5$ \( ( -1 + T )^{4} \)
$7$ \( -8 - 32 T - 12 T^{2} + 4 T^{3} + T^{4} \)
$11$ \( 4 - 28 T^{2} + T^{4} \)
$13$ \( -32 - 32 T + 8 T^{2} + 8 T^{3} + T^{4} \)
$17$ \( -47 - 36 T + 44 T^{2} - 12 T^{3} + T^{4} \)
$19$ \( 49 - 56 T - 4 T^{2} + 8 T^{3} + T^{4} \)
$23$ \( T^{4} \)
$29$ \( -8 + 20 T^{2} + 12 T^{3} + T^{4} \)
$31$ \( 1912 - 1232 T + 284 T^{2} - 28 T^{3} + T^{4} \)
$37$ \( ( -4 - 4 T + T^{2} )^{2} \)
$41$ \( -1472 - 704 T - 64 T^{2} + 8 T^{3} + T^{4} \)
$43$ \( -47 + 36 T + 44 T^{2} + 12 T^{3} + T^{4} \)
$47$ \( -8 + 32 T - 28 T^{2} + 4 T^{3} + T^{4} \)
$53$ \( -968 + 176 T + 92 T^{2} - 20 T^{3} + T^{4} \)
$59$ \( ( -3 + T )^{4} \)
$61$ \( 1600 - 112 T^{2} + T^{4} \)
$67$ \( -5903 + 2796 T - 196 T^{2} - 12 T^{3} + T^{4} \)
$71$ \( -2312 + 1088 T + 20 T^{2} - 20 T^{3} + T^{4} \)
$73$ \( 1081 - 496 T - 34 T^{2} + 16 T^{3} + T^{4} \)
$79$ \( 13432 - 1360 T - 340 T^{2} + 4 T^{3} + T^{4} \)
$83$ \( 81 + 108 T - 36 T^{2} - 12 T^{3} + T^{4} \)
$89$ \( 4036 - 2800 T + 540 T^{2} - 40 T^{3} + T^{4} \)
$97$ \( 2692 + 2272 T - 148 T^{2} - 16 T^{3} + T^{4} \)
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