Properties

Label 4998.2.a.x
Level $4998$
Weight $2$
Character orbit 4998.a
Self dual yes
Analytic conductor $39.909$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 102)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + 4q^{5} - q^{6} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} + q^{4} + 4q^{5} - q^{6} - q^{8} + q^{9} - 4q^{10} + q^{12} + 6q^{13} + 4q^{15} + q^{16} + q^{17} - q^{18} - 4q^{19} + 4q^{20} + 6q^{23} - q^{24} + 11q^{25} - 6q^{26} + q^{27} - 4q^{29} - 4q^{30} + 6q^{31} - q^{32} - q^{34} + q^{36} - 4q^{37} + 4q^{38} + 6q^{39} - 4q^{40} + 10q^{41} - 4q^{43} + 4q^{45} - 6q^{46} - 4q^{47} + q^{48} - 11q^{50} + q^{51} + 6q^{52} - 2q^{53} - q^{54} - 4q^{57} + 4q^{58} - 12q^{59} + 4q^{60} + 4q^{61} - 6q^{62} + q^{64} + 24q^{65} - 12q^{67} + q^{68} + 6q^{69} - 6q^{71} - q^{72} - 2q^{73} + 4q^{74} + 11q^{75} - 4q^{76} - 6q^{78} + 10q^{79} + 4q^{80} + q^{81} - 10q^{82} + 12q^{83} + 4q^{85} + 4q^{86} - 4q^{87} + 2q^{89} - 4q^{90} + 6q^{92} + 6q^{93} + 4q^{94} - 16q^{95} - q^{96} - 6q^{97} + O(q^{100}) \)

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 4.00000 −1.00000 0 −1.00000 1.00000 −4.00000
\(n\): 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.x 1
7.b odd 2 1 102.2.a.a 1
21.c even 2 1 306.2.a.d 1
28.d even 2 1 816.2.a.h 1
35.c odd 2 1 2550.2.a.be 1
35.f even 4 2 2550.2.d.q 2
56.e even 2 1 3264.2.a.p 1
56.h odd 2 1 3264.2.a.bf 1
84.h odd 2 1 2448.2.a.t 1
105.g even 2 1 7650.2.a.z 1
119.d odd 2 1 1734.2.a.h 1
119.f odd 4 2 1734.2.b.d 2
119.l odd 8 4 1734.2.f.g 4
168.e odd 2 1 9792.2.a.b 1
168.i even 2 1 9792.2.a.a 1
357.c even 2 1 5202.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.2.a.a 1 7.b odd 2 1
306.2.a.d 1 21.c even 2 1
816.2.a.h 1 28.d even 2 1
1734.2.a.h 1 119.d odd 2 1
1734.2.b.d 2 119.f odd 4 2
1734.2.f.g 4 119.l odd 8 4
2448.2.a.t 1 84.h odd 2 1
2550.2.a.be 1 35.c odd 2 1
2550.2.d.q 2 35.f even 4 2
3264.2.a.p 1 56.e even 2 1
3264.2.a.bf 1 56.h odd 2 1
4998.2.a.x 1 1.a even 1 1 trivial
5202.2.a.g 1 357.c even 2 1
7650.2.a.z 1 105.g even 2 1
9792.2.a.a 1 168.i even 2 1
9792.2.a.b 1 168.e odd 2 1

Hecke kernels

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

\( T_{5} - 4 \)
\( T_{11} \)
\( T_{13} - 6 \)
\( T_{23} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( -1 + T \)
$5$ \( -4 + T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( -6 + T \)
$17$ \( -1 + T \)
$19$ \( 4 + T \)
$23$ \( -6 + T \)
$29$ \( 4 + T \)
$31$ \( -6 + T \)
$37$ \( 4 + T \)
$41$ \( -10 + T \)
$43$ \( 4 + T \)
$47$ \( 4 + T \)
$53$ \( 2 + T \)
$59$ \( 12 + T \)
$61$ \( -4 + T \)
$67$ \( 12 + T \)
$71$ \( 6 + T \)
$73$ \( 2 + T \)
$79$ \( -10 + T \)
$83$ \( -12 + T \)
$89$ \( -2 + T \)
$97$ \( 6 + T \)
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