Properties

Label 5290.2.a.z
Level $5290$
Weight $2$
Character orbit 5290.a
Self dual yes
Analytic conductor $42.241$
Analytic rank $1$
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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 1\)
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 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.

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

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 −1.93185 1.00000 1.00000 −1.93185 −0.482362 1.00000 0.732051 1.00000
1.2 1.00000 −0.517638 1.00000 1.00000 −0.517638 0.931852 1.00000 −2.73205 1.00000
1.3 1.00000 0.517638 1.00000 1.00000 0.517638 −2.93185 1.00000 −2.73205 1.00000
1.4 1.00000 1.93185 1.00000 1.00000 1.93185 −1.51764 1.00000 0.732051 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.z yes 4
23.b odd 2 1 5290.2.a.y 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5290.2.a.y 4 23.b odd 2 1
5290.2.a.z 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5290))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( 1 - 4 T^{2} + T^{4} \)
$5$ \( ( -1 + T )^{4} \)
$7$ \( -2 - 4 T + 2 T^{2} + 4 T^{3} + T^{4} \)
$11$ \( -32 - 32 T + 8 T^{2} + 8 T^{3} + T^{4} \)
$13$ \( -200 - 160 T - 28 T^{2} + 4 T^{3} + T^{4} \)
$17$ \( ( -5 + 2 T + T^{2} )^{2} \)
$19$ \( -23 - 28 T - 4 T^{2} + 4 T^{3} + T^{4} \)
$23$ \( T^{4} \)
$29$ \( -50 + 100 T - 46 T^{2} - 4 T^{3} + T^{4} \)
$31$ \( -2 - 4 T + 6 T^{2} + 8 T^{3} + T^{4} \)
$37$ \( -3056 - 832 T + 8 T^{2} + 16 T^{3} + T^{4} \)
$41$ \( 4 + 8 T - 16 T^{2} + 4 T^{3} + T^{4} \)
$43$ \( -239 - 276 T - 10 T^{2} + 12 T^{3} + T^{4} \)
$47$ \( -386 - 244 T - 18 T^{2} + 8 T^{3} + T^{4} \)
$53$ \( 46 + 60 T - 34 T^{2} + T^{4} \)
$59$ \( -2087 - 1300 T - 166 T^{2} + 4 T^{3} + T^{4} \)
$61$ \( 184 + 128 T - 76 T^{2} + 4 T^{3} + T^{4} \)
$67$ \( -743 + 576 T - 118 T^{2} + T^{4} \)
$71$ \( -6434 + 908 T + 162 T^{2} - 28 T^{3} + T^{4} \)
$73$ \( 1225 - 1260 T - 124 T^{2} + 12 T^{3} + T^{4} \)
$79$ \( 8878 + 4180 T + 674 T^{2} + 44 T^{3} + T^{4} \)
$83$ \( -7751 + 1628 T + 14 T^{2} - 20 T^{3} + T^{4} \)
$89$ \( -7664 + 800 T + 464 T^{2} + 40 T^{3} + T^{4} \)
$97$ \( 10468 - 2936 T - 160 T^{2} + 20 T^{3} + T^{4} \)
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