Properties

Label 4998.2.a.cb
Level $4998$
Weight $2$
Character orbit 4998.a
Self dual yes
Analytic conductor $39.909$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{8} + q^{9} + q^{10} + ( -3 + \beta ) q^{11} - q^{12} + ( -1 + 2 \beta ) q^{13} - q^{15} + q^{16} + q^{17} + q^{18} + ( 2 - 3 \beta ) q^{19} + q^{20} + ( -3 + \beta ) q^{22} + ( -6 - 2 \beta ) q^{23} - q^{24} -4 q^{25} + ( -1 + 2 \beta ) q^{26} - q^{27} + ( -4 - \beta ) q^{29} - q^{30} + ( 4 - 3 \beta ) q^{31} + q^{32} + ( 3 - \beta ) q^{33} + q^{34} + q^{36} + ( -5 - 4 \beta ) q^{37} + ( 2 - 3 \beta ) q^{38} + ( 1 - 2 \beta ) q^{39} + q^{40} + ( 2 + 3 \beta ) q^{41} + ( -5 - \beta ) q^{43} + ( -3 + \beta ) q^{44} + q^{45} + ( -6 - 2 \beta ) q^{46} + ( -8 - 3 \beta ) q^{47} - q^{48} -4 q^{50} - q^{51} + ( -1 + 2 \beta ) q^{52} + ( 1 + 2 \beta ) q^{53} - q^{54} + ( -3 + \beta ) q^{55} + ( -2 + 3 \beta ) q^{57} + ( -4 - \beta ) q^{58} -4 q^{59} - q^{60} + ( 6 + 4 \beta ) q^{61} + ( 4 - 3 \beta ) q^{62} + q^{64} + ( -1 + 2 \beta ) q^{65} + ( 3 - \beta ) q^{66} + ( -5 + 7 \beta ) q^{67} + q^{68} + ( 6 + 2 \beta ) q^{69} + ( 4 + 9 \beta ) q^{71} + q^{72} + ( 7 + 4 \beta ) q^{73} + ( -5 - 4 \beta ) q^{74} + 4 q^{75} + ( 2 - 3 \beta ) q^{76} + ( 1 - 2 \beta ) q^{78} + ( -5 + \beta ) q^{79} + q^{80} + q^{81} + ( 2 + 3 \beta ) q^{82} + ( -7 + \beta ) q^{83} + q^{85} + ( -5 - \beta ) q^{86} + ( 4 + \beta ) q^{87} + ( -3 + \beta ) q^{88} + ( -11 + 4 \beta ) q^{89} + q^{90} + ( -6 - 2 \beta ) q^{92} + ( -4 + 3 \beta ) q^{93} + ( -8 - 3 \beta ) q^{94} + ( 2 - 3 \beta ) q^{95} - q^{96} + ( -3 - 2 \beta ) q^{97} + ( -3 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 2q^{8} + 2q^{9} + 2q^{10} - 6q^{11} - 2q^{12} - 2q^{13} - 2q^{15} + 2q^{16} + 2q^{17} + 2q^{18} + 4q^{19} + 2q^{20} - 6q^{22} - 12q^{23} - 2q^{24} - 8q^{25} - 2q^{26} - 2q^{27} - 8q^{29} - 2q^{30} + 8q^{31} + 2q^{32} + 6q^{33} + 2q^{34} + 2q^{36} - 10q^{37} + 4q^{38} + 2q^{39} + 2q^{40} + 4q^{41} - 10q^{43} - 6q^{44} + 2q^{45} - 12q^{46} - 16q^{47} - 2q^{48} - 8q^{50} - 2q^{51} - 2q^{52} + 2q^{53} - 2q^{54} - 6q^{55} - 4q^{57} - 8q^{58} - 8q^{59} - 2q^{60} + 12q^{61} + 8q^{62} + 2q^{64} - 2q^{65} + 6q^{66} - 10q^{67} + 2q^{68} + 12q^{69} + 8q^{71} + 2q^{72} + 14q^{73} - 10q^{74} + 8q^{75} + 4q^{76} + 2q^{78} - 10q^{79} + 2q^{80} + 2q^{81} + 4q^{82} - 14q^{83} + 2q^{85} - 10q^{86} + 8q^{87} - 6q^{88} - 22q^{89} + 2q^{90} - 12q^{92} - 8q^{93} - 16q^{94} + 4q^{95} - 2q^{96} - 6q^{97} - 6q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
1.00000 −1.00000 1.00000 1.00000 −1.00000 0 1.00000 1.00000 1.00000
1.2 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.cb 2
7.b odd 2 1 4998.2.a.cc yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4998.2.a.cb 2 1.a even 1 1 trivial
4998.2.a.cc yes 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4998))\):

\( T_{5} - 1 \)
\( T_{11}^{2} + 6 T_{11} + 7 \)
\( T_{13}^{2} + 2 T_{13} - 7 \)
\( T_{23}^{2} + 12 T_{23} + 28 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( 7 + 6 T + T^{2} \)
$13$ \( -7 + 2 T + T^{2} \)
$17$ \( ( -1 + T )^{2} \)
$19$ \( -14 - 4 T + T^{2} \)
$23$ \( 28 + 12 T + T^{2} \)
$29$ \( 14 + 8 T + T^{2} \)
$31$ \( -2 - 8 T + T^{2} \)
$37$ \( -7 + 10 T + T^{2} \)
$41$ \( -14 - 4 T + T^{2} \)
$43$ \( 23 + 10 T + T^{2} \)
$47$ \( 46 + 16 T + T^{2} \)
$53$ \( -7 - 2 T + T^{2} \)
$59$ \( ( 4 + T )^{2} \)
$61$ \( 4 - 12 T + T^{2} \)
$67$ \( -73 + 10 T + T^{2} \)
$71$ \( -146 - 8 T + T^{2} \)
$73$ \( 17 - 14 T + T^{2} \)
$79$ \( 23 + 10 T + T^{2} \)
$83$ \( 47 + 14 T + T^{2} \)
$89$ \( 89 + 22 T + T^{2} \)
$97$ \( 1 + 6 T + T^{2} \)
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