Properties

Label 522.2.a.b
Level $522$
Weight $2$
Character orbit 522.a
Self dual yes
Analytic conductor $4.168$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.16819098551\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{5} - 2 q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} - q^{5} - 2 q^{7} - q^{8} + q^{10} + 3 q^{11} - q^{13} + 2 q^{14} + q^{16} - 8 q^{17} - q^{20} - 3 q^{22} - 4 q^{23} - 4 q^{25} + q^{26} - 2 q^{28} + q^{29} - 3 q^{31} - q^{32} + 8 q^{34} + 2 q^{35} + 8 q^{37} + q^{40} - 2 q^{41} - 11 q^{43} + 3 q^{44} + 4 q^{46} - 13 q^{47} - 3 q^{49} + 4 q^{50} - q^{52} + 11 q^{53} - 3 q^{55} + 2 q^{56} - q^{58} - 8 q^{61} + 3 q^{62} + q^{64} + q^{65} - 12 q^{67} - 8 q^{68} - 2 q^{70} - 2 q^{71} + 4 q^{73} - 8 q^{74} - 6 q^{77} + 15 q^{79} - q^{80} + 2 q^{82} - 4 q^{83} + 8 q^{85} + 11 q^{86} - 3 q^{88} + 10 q^{89} + 2 q^{91} - 4 q^{92} + 13 q^{94} - 2 q^{97} + 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(29\) \(-1\)

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 522.2.a.b 1
3.b odd 2 1 58.2.a.b 1
4.b odd 2 1 4176.2.a.n 1
12.b even 2 1 464.2.a.e 1
15.d odd 2 1 1450.2.a.c 1
15.e even 4 2 1450.2.b.b 2
21.c even 2 1 2842.2.a.e 1
24.f even 2 1 1856.2.a.f 1
24.h odd 2 1 1856.2.a.k 1
33.d even 2 1 7018.2.a.a 1
39.d odd 2 1 9802.2.a.a 1
87.d odd 2 1 1682.2.a.d 1
87.f even 4 2 1682.2.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.2.a.b 1 3.b odd 2 1
464.2.a.e 1 12.b even 2 1
522.2.a.b 1 1.a even 1 1 trivial
1450.2.a.c 1 15.d odd 2 1
1450.2.b.b 2 15.e even 4 2
1682.2.a.d 1 87.d odd 2 1
1682.2.b.a 2 87.f even 4 2
1856.2.a.f 1 24.f even 2 1
1856.2.a.k 1 24.h odd 2 1
2842.2.a.e 1 21.c even 2 1
4176.2.a.n 1 4.b odd 2 1
7018.2.a.a 1 33.d even 2 1
9802.2.a.a 1 39.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(522))\):

\( T_{5} + 1 \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T - 3 \) Copy content Toggle raw display
$13$ \( T + 1 \) Copy content Toggle raw display
$17$ \( T + 8 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T + 4 \) Copy content Toggle raw display
$29$ \( T - 1 \) Copy content Toggle raw display
$31$ \( T + 3 \) Copy content Toggle raw display
$37$ \( T - 8 \) Copy content Toggle raw display
$41$ \( T + 2 \) Copy content Toggle raw display
$43$ \( T + 11 \) Copy content Toggle raw display
$47$ \( T + 13 \) Copy content Toggle raw display
$53$ \( T - 11 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 8 \) Copy content Toggle raw display
$67$ \( T + 12 \) Copy content Toggle raw display
$71$ \( T + 2 \) Copy content Toggle raw display
$73$ \( T - 4 \) Copy content Toggle raw display
$79$ \( T - 15 \) Copy content Toggle raw display
$83$ \( T + 4 \) Copy content Toggle raw display
$89$ \( T - 10 \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
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