Properties

Label 5390.2.a.r
Level $5390$
Weight $2$
Character orbit 5390.a
Self dual yes
Analytic conductor $43.039$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5390 = 2 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5390.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.0393666895\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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
0
−1.00000 2.00000 1.00000 1.00000 −2.00000 0 −1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(7\) \(-1\)
\(11\) \(1\)

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 5390.2.a.r 1
7.b odd 2 1 770.2.a.a 1
21.c even 2 1 6930.2.a.bm 1
28.d even 2 1 6160.2.a.k 1
35.c odd 2 1 3850.2.a.ba 1
35.f even 4 2 3850.2.c.o 2
77.b even 2 1 8470.2.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.a 1 7.b odd 2 1
3850.2.a.ba 1 35.c odd 2 1
3850.2.c.o 2 35.f even 4 2
5390.2.a.r 1 1.a even 1 1 trivial
6160.2.a.k 1 28.d even 2 1
6930.2.a.bm 1 21.c even 2 1
8470.2.a.r 1 77.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5390))\):

\( T_{3} - 2 \)
\( T_{13} + 2 \)
\( T_{17} - 6 \)
\( T_{19} + 2 \)
\( T_{31} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( -2 + T \)
$5$ \( -1 + T \)
$7$ \( T \)
$11$ \( 1 + T \)
$13$ \( 2 + T \)
$17$ \( -6 + T \)
$19$ \( 2 + T \)
$23$ \( 6 + T \)
$29$ \( T \)
$31$ \( 8 + T \)
$37$ \( 4 + T \)
$41$ \( 12 + T \)
$43$ \( 4 + T \)
$47$ \( 12 + T \)
$53$ \( T \)
$59$ \( T \)
$61$ \( 2 + T \)
$67$ \( -8 + T \)
$71$ \( -12 + T \)
$73$ \( 2 + T \)
$79$ \( -14 + T \)
$83$ \( 12 + T \)
$89$ \( 6 + T \)
$97$ \( 8 + T \)
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