Properties

Label 4998.2.a.cl
Level $4998$
Weight $2$
Character orbit 4998.a
Self dual yes
Analytic conductor $39.909$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.31808.1
Defining polynomial: \(x^{4} - 2 x^{3} - 11 x^{2} + 12 x + 28\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 11 x^{2} + 12 x + 28\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - \nu - 6 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - \nu^{2} - 8 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2} + \beta_{1} + 6\)
\(\nu^{3}\)\(=\)\(2 \beta_{3} + 2 \beta_{2} + 9 \beta_{1} + 6\)

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
−2.51304
−1.34975
2.34975
3.51304
−1.00000 −1.00000 1.00000 −2.51304 1.00000 0 −1.00000 1.00000 2.51304
1.2 −1.00000 −1.00000 1.00000 −1.34975 1.00000 0 −1.00000 1.00000 1.34975
1.3 −1.00000 −1.00000 1.00000 2.34975 1.00000 0 −1.00000 1.00000 −2.34975
1.4 −1.00000 −1.00000 1.00000 3.51304 1.00000 0 −1.00000 1.00000 −3.51304
\(n\): 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.cl 4
7.b odd 2 1 4998.2.a.cm yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4998.2.a.cl 4 1.a even 1 1 trivial
4998.2.a.cm yes 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4998))\):

\( T_{5}^{4} - 2 T_{5}^{3} - 11 T_{5}^{2} + 12 T_{5} + 28 \)
\( T_{11}^{4} + 2 T_{11}^{3} - 21 T_{11}^{2} + 4 T_{11} + 28 \)
\( T_{13}^{4} + 6 T_{13}^{3} + T_{13}^{2} - 24 T_{13} + 8 \)
\( T_{23}^{4} + 8 T_{23}^{3} - 2 T_{23}^{2} - 56 T_{23} + 56 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( 28 + 12 T - 11 T^{2} - 2 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 28 + 4 T - 21 T^{2} + 2 T^{3} + T^{4} \)
$13$ \( 8 - 24 T + T^{2} + 6 T^{3} + T^{4} \)
$17$ \( ( -1 + T )^{4} \)
$19$ \( 8 - 56 T - 10 T^{2} + 8 T^{3} + T^{4} \)
$23$ \( 56 - 56 T - 2 T^{2} + 8 T^{3} + T^{4} \)
$29$ \( 112 + 16 T - 26 T^{2} + T^{4} \)
$31$ \( -112 + 32 T + 70 T^{2} + 16 T^{3} + T^{4} \)
$37$ \( -542 + 120 T + 43 T^{2} - 14 T^{3} + T^{4} \)
$41$ \( -256 + 256 T - 46 T^{2} - 4 T^{3} + T^{4} \)
$43$ \( -752 - 472 T - 33 T^{2} + 10 T^{3} + T^{4} \)
$47$ \( -1264 + 448 T + 22 T^{2} - 16 T^{3} + T^{4} \)
$53$ \( 568 - 528 T + 169 T^{2} - 22 T^{3} + T^{4} \)
$59$ \( -32 - 80 T - 50 T^{2} + T^{4} \)
$61$ \( -224 - 384 T - 116 T^{2} - 4 T^{3} + T^{4} \)
$67$ \( -568 + 704 T - 25 T^{2} - 14 T^{3} + T^{4} \)
$71$ \( 668 + 24 T - 144 T^{2} + 4 T^{3} + T^{4} \)
$73$ \( -356 - 428 T - 3 T^{2} + 14 T^{3} + T^{4} \)
$79$ \( 8 + 48 T - 49 T^{2} + 6 T^{3} + T^{4} \)
$83$ \( -46 - 52 T - 3 T^{2} + 6 T^{3} + T^{4} \)
$89$ \( -2268 + 1620 T - 243 T^{2} - 6 T^{3} + T^{4} \)
$97$ \( 1372 - 28 T - 83 T^{2} + 2 T^{3} + T^{4} \)
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