Properties

Label 4998.2.a.ci
Level $4998$
Weight $2$
Character orbit 4998.a
Self dual yes
Analytic conductor $39.909$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.469.1
Defining polynomial: \(x^{3} - x^{2} - 5 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} -\beta_{2} q^{5} - q^{6} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} - q^{3} + q^{4} -\beta_{2} q^{5} - q^{6} + q^{8} + q^{9} -\beta_{2} q^{10} + ( 1 + \beta_{1} ) q^{11} - q^{12} + ( -1 + \beta_{1} ) q^{13} + \beta_{2} q^{15} + q^{16} + q^{17} + q^{18} + ( 1 - \beta_{1} - \beta_{2} ) q^{19} -\beta_{2} q^{20} + ( 1 + \beta_{1} ) q^{22} + 2 q^{23} - q^{24} + ( 3 - 2 \beta_{1} + \beta_{2} ) q^{25} + ( -1 + \beta_{1} ) q^{26} - q^{27} + ( 1 + \beta_{1} - \beta_{2} ) q^{29} + \beta_{2} q^{30} + ( -2 + 2 \beta_{1} ) q^{31} + q^{32} + ( -1 - \beta_{1} ) q^{33} + q^{34} + q^{36} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{37} + ( 1 - \beta_{1} - \beta_{2} ) q^{38} + ( 1 - \beta_{1} ) q^{39} -\beta_{2} q^{40} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{41} + ( 2 - \beta_{2} ) q^{43} + ( 1 + \beta_{1} ) q^{44} -\beta_{2} q^{45} + 2 q^{46} -2 \beta_{2} q^{47} - q^{48} + ( 3 - 2 \beta_{1} + \beta_{2} ) q^{50} - q^{51} + ( -1 + \beta_{1} ) q^{52} + ( 4 - \beta_{2} ) q^{53} - q^{54} + ( -2 + \beta_{2} ) q^{55} + ( -1 + \beta_{1} + \beta_{2} ) q^{57} + ( 1 + \beta_{1} - \beta_{2} ) q^{58} + ( 3 - \beta_{1} + \beta_{2} ) q^{59} + \beta_{2} q^{60} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{61} + ( -2 + 2 \beta_{1} ) q^{62} + q^{64} + ( -2 + 3 \beta_{2} ) q^{65} + ( -1 - \beta_{1} ) q^{66} + ( 8 + 2 \beta_{1} - \beta_{2} ) q^{67} + q^{68} -2 q^{69} + ( 4 - 2 \beta_{1} - 2 \beta_{2} ) q^{71} + q^{72} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{73} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{74} + ( -3 + 2 \beta_{1} - \beta_{2} ) q^{75} + ( 1 - \beta_{1} - \beta_{2} ) q^{76} + ( 1 - \beta_{1} ) q^{78} + ( 2 - 4 \beta_{1} - \beta_{2} ) q^{79} -\beta_{2} q^{80} + q^{81} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{82} + ( -1 - 3 \beta_{1} + 2 \beta_{2} ) q^{83} -\beta_{2} q^{85} + ( 2 - \beta_{2} ) q^{86} + ( -1 - \beta_{1} + \beta_{2} ) q^{87} + ( 1 + \beta_{1} ) q^{88} + ( -4 + 3 \beta_{2} ) q^{89} -\beta_{2} q^{90} + 2 q^{92} + ( 2 - 2 \beta_{1} ) q^{93} -2 \beta_{2} q^{94} + ( 10 - 2 \beta_{1} - 2 \beta_{2} ) q^{95} - q^{96} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{97} + ( 1 + \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - q^{5} - 3q^{6} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - q^{5} - 3q^{6} + 3q^{8} + 3q^{9} - q^{10} + 3q^{11} - 3q^{12} - 3q^{13} + q^{15} + 3q^{16} + 3q^{17} + 3q^{18} + 2q^{19} - q^{20} + 3q^{22} + 6q^{23} - 3q^{24} + 10q^{25} - 3q^{26} - 3q^{27} + 2q^{29} + q^{30} - 6q^{31} + 3q^{32} - 3q^{33} + 3q^{34} + 3q^{36} + q^{37} + 2q^{38} + 3q^{39} - q^{40} - 10q^{41} + 5q^{43} + 3q^{44} - q^{45} + 6q^{46} - 2q^{47} - 3q^{48} + 10q^{50} - 3q^{51} - 3q^{52} + 11q^{53} - 3q^{54} - 5q^{55} - 2q^{57} + 2q^{58} + 10q^{59} + q^{60} - 2q^{61} - 6q^{62} + 3q^{64} - 3q^{65} - 3q^{66} + 23q^{67} + 3q^{68} - 6q^{69} + 10q^{71} + 3q^{72} + 7q^{73} + q^{74} - 10q^{75} + 2q^{76} + 3q^{78} + 5q^{79} - q^{80} + 3q^{81} - 10q^{82} - q^{83} - q^{85} + 5q^{86} - 2q^{87} + 3q^{88} - 9q^{89} - q^{90} + 6q^{92} + 6q^{93} - 2q^{94} + 28q^{95} - 3q^{96} - 5q^{97} + 3q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 5 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + \nu - 4 \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 7\)\()/2\)

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
−2.16425
2.39138
0.772866
1.00000 −1.00000 1.00000 −3.84822 −1.00000 0 1.00000 1.00000 −3.84822
1.2 1.00000 −1.00000 1.00000 −0.327327 −1.00000 0 1.00000 1.00000 −0.327327
1.3 1.00000 −1.00000 1.00000 3.17554 −1.00000 0 1.00000 1.00000 3.17554
\(n\): 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.ci 3
7.b odd 2 1 4998.2.a.cj yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4998.2.a.ci 3 1.a even 1 1 trivial
4998.2.a.cj yes 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4998))\):

\( T_{5}^{3} + T_{5}^{2} - 12 T_{5} - 4 \)
\( T_{11}^{3} - 3 T_{11}^{2} - 10 T_{11} - 4 \)
\( T_{13}^{3} + 3 T_{13}^{2} - 10 T_{13} - 28 \)
\( T_{23} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( -4 - 12 T + T^{2} + T^{3} \)
$7$ \( T^{3} \)
$11$ \( -4 - 10 T - 3 T^{2} + T^{3} \)
$13$ \( -28 - 10 T + 3 T^{2} + T^{3} \)
$17$ \( ( -1 + T )^{3} \)
$19$ \( -32 - 28 T - 2 T^{2} + T^{3} \)
$23$ \( ( -2 + T )^{3} \)
$29$ \( 32 - 20 T - 2 T^{2} + T^{3} \)
$31$ \( -224 - 40 T + 6 T^{2} + T^{3} \)
$37$ \( 172 - 54 T - T^{2} + T^{3} \)
$41$ \( -392 - 52 T + 10 T^{2} + T^{3} \)
$43$ \( 16 - 4 T - 5 T^{2} + T^{3} \)
$47$ \( -32 - 48 T + 2 T^{2} + T^{3} \)
$53$ \( -4 + 28 T - 11 T^{2} + T^{3} \)
$59$ \( 16 + 12 T - 10 T^{2} + T^{3} \)
$61$ \( 88 - 84 T + 2 T^{2} + T^{3} \)
$67$ \( -112 + 120 T - 23 T^{2} + T^{3} \)
$71$ \( -56 - 84 T - 10 T^{2} + T^{3} \)
$73$ \( 212 - 40 T - 7 T^{2} + T^{3} \)
$79$ \( 944 - 228 T - 5 T^{2} + T^{3} \)
$83$ \( 76 - 142 T + T^{2} + T^{3} \)
$89$ \( -308 - 84 T + 9 T^{2} + T^{3} \)
$97$ \( -76 - 64 T + 5 T^{2} + T^{3} \)
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