Properties

Label 4998.2.a.bw
Level $4998$
Weight $2$
Character orbit 4998.a
Self dual yes
Analytic conductor $39.909$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 714)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−2.44949
2.44949
1.00000 −1.00000 1.00000 −2.00000 −1.00000 0 1.00000 1.00000 −2.00000
1.2 1.00000 −1.00000 1.00000 −2.00000 −1.00000 0 1.00000 1.00000 −2.00000
\(n\): 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.bw 2
7.b odd 2 1 714.2.a.m 2
21.c even 2 1 2142.2.a.v 2
28.d even 2 1 5712.2.a.bm 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.a.m 2 7.b odd 2 1
2142.2.a.v 2 21.c even 2 1
4998.2.a.bw 2 1.a even 1 1 trivial
5712.2.a.bm 2 28.d even 2 1

Hecke kernels

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

\( T_{5} + 2 \)
\( T_{11}^{2} - 24 \)
\( T_{13}^{2} - 4 T_{13} - 20 \)
\( T_{23} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 2 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( -24 + T^{2} \)
$13$ \( -20 - 4 T + T^{2} \)
$17$ \( ( -1 + T )^{2} \)
$19$ \( -24 + T^{2} \)
$23$ \( ( 4 + T )^{2} \)
$29$ \( -20 - 4 T + T^{2} \)
$31$ \( -96 + T^{2} \)
$37$ \( -20 + 4 T + T^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( ( -8 + T )^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( -8 - 8 T + T^{2} \)
$61$ \( ( 10 + T )^{2} \)
$67$ \( -80 + 8 T + T^{2} \)
$71$ \( -80 - 8 T + T^{2} \)
$73$ \( ( -2 + T )^{2} \)
$79$ \( -96 + T^{2} \)
$83$ \( 120 - 24 T + T^{2} \)
$89$ \( -92 - 4 T + T^{2} \)
$97$ \( ( 6 + T )^{2} \)
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