Properties

Label 5290.2.a.l
Level $5290$
Weight $2$
Character orbit 5290.a
Self dual yes
Analytic conductor $42.241$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.41421
1.41421
1.00000 −2.41421 1.00000 1.00000 −2.41421 3.41421 1.00000 2.82843 1.00000
1.2 1.00000 0.414214 1.00000 1.00000 0.414214 0.585786 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.l yes 2
23.b odd 2 1 5290.2.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5290.2.a.k 2 23.b odd 2 1
5290.2.a.l 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5290))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( -1 + 2 T + T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( 2 - 4 T + T^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( -32 + T^{2} \)
$17$ \( -49 + 2 T + T^{2} \)
$19$ \( ( 3 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( -46 - 4 T + T^{2} \)
$31$ \( 62 - 16 T + T^{2} \)
$37$ \( -4 - 4 T + T^{2} \)
$41$ \( -32 + T^{2} \)
$43$ \( -9 - 6 T + T^{2} \)
$47$ \( 18 + 12 T + T^{2} \)
$53$ \( 62 - 16 T + T^{2} \)
$59$ \( -71 - 2 T + T^{2} \)
$61$ \( -32 + T^{2} \)
$67$ \( -25 + 10 T + T^{2} \)
$71$ \( -50 + T^{2} \)
$73$ \( -25 + 10 T + T^{2} \)
$79$ \( 34 - 12 T + T^{2} \)
$83$ \( -73 - 10 T + T^{2} \)
$89$ \( 136 - 24 T + T^{2} \)
$97$ \( 28 - 12 T + T^{2} \)
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