Properties

Label 4998.2.a.z
Level $4998$
Weight $2$
Character orbit 4998.a
Self dual yes
Analytic conductor $39.909$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4998 = 2 \cdot 3 \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4998.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(39.9092309302\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} + q^{9} - q^{10} - 5q^{11} - q^{12} + 5q^{13} + q^{15} + q^{16} - q^{17} + q^{18} + 4q^{19} - q^{20} - 5q^{22} - 6q^{23} - q^{24} - 4q^{25} + 5q^{26} - q^{27} + q^{30} + 2q^{31} + q^{32} + 5q^{33} - q^{34} + q^{36} + q^{37} + 4q^{38} - 5q^{39} - q^{40} - 2q^{41} + q^{43} - 5q^{44} - q^{45} - 6q^{46} + 6q^{47} - q^{48} - 4q^{50} + q^{51} + 5q^{52} - 9q^{53} - q^{54} + 5q^{55} - 4q^{57} - 4q^{59} + q^{60} - 6q^{61} + 2q^{62} + q^{64} - 5q^{65} + 5q^{66} - q^{67} - q^{68} + 6q^{69} - 6q^{71} + q^{72} + 7q^{73} + q^{74} + 4q^{75} + 4q^{76} - 5q^{78} - q^{79} - q^{80} + q^{81} - 2q^{82} - q^{83} + q^{85} + q^{86} - 5q^{88} - 7q^{89} - q^{90} - 6q^{92} - 2q^{93} + 6q^{94} - 4q^{95} - q^{96} + 3q^{97} - 5q^{99} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)
\(17\) \(1\)

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 4998.2.a.z 1
7.b odd 2 1 4998.2.a.bo yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4998.2.a.z 1 1.a even 1 1 trivial
4998.2.a.bo yes 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4998))\):

\( T_{5} + 1 \)
\( T_{11} + 5 \)
\( T_{13} - 5 \)
\( T_{23} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( 1 + T \)
$5$ \( 1 + T \)
$7$ \( T \)
$11$ \( 5 + T \)
$13$ \( -5 + T \)
$17$ \( 1 + T \)
$19$ \( -4 + T \)
$23$ \( 6 + T \)
$29$ \( T \)
$31$ \( -2 + T \)
$37$ \( -1 + T \)
$41$ \( 2 + T \)
$43$ \( -1 + T \)
$47$ \( -6 + T \)
$53$ \( 9 + T \)
$59$ \( 4 + T \)
$61$ \( 6 + T \)
$67$ \( 1 + T \)
$71$ \( 6 + T \)
$73$ \( -7 + T \)
$79$ \( 1 + T \)
$83$ \( 1 + T \)
$89$ \( 7 + T \)
$97$ \( -3 + T \)
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