Properties

Label 4998.2.a.co
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: 4.4.16448.2
Defining polynomial: \(x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 14\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 14\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 5 \beta_{1} + 4\)

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
2.18398
2.87996
−1.18398
−1.87996
1.00000 −1.00000 1.00000 −3.59819 −1.00000 0 1.00000 1.00000 −3.59819
1.2 1.00000 −1.00000 1.00000 −1.46575 −1.00000 0 1.00000 1.00000 −1.46575
1.3 1.00000 −1.00000 1.00000 −0.230234 −1.00000 0 1.00000 1.00000 −0.230234
1.4 1.00000 −1.00000 1.00000 3.29417 −1.00000 0 1.00000 1.00000 3.29417
\(n\): 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.co 4
7.b odd 2 1 4998.2.a.cp yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4998.2.a.co 4 1.a even 1 1 trivial
4998.2.a.cp 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4998))\):

\( T_{5}^{4} + 2 T_{5}^{3} - 11 T_{5}^{2} - 20 T_{5} - 4 \)
\( T_{11}^{4} - 10 T_{11}^{3} + 19 T_{11}^{2} + 52 T_{11} - 124 \)
\( T_{13}^{4} + 6 T_{13}^{3} + T_{13}^{2} - 32 T_{13} - 32 \)
\( T_{23}^{4} - 18 T_{23}^{2} + 8 T_{23} + 56 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( -4 - 20 T - 11 T^{2} + 2 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( -124 + 52 T + 19 T^{2} - 10 T^{3} + T^{4} \)
$13$ \( -32 - 32 T + T^{2} + 6 T^{3} + T^{4} \)
$17$ \( ( 1 + T )^{4} \)
$19$ \( 56 - 8 T - 18 T^{2} + T^{4} \)
$23$ \( 56 + 8 T - 18 T^{2} + T^{4} \)
$29$ \( -64 + 96 T - 18 T^{2} - 8 T^{3} + T^{4} \)
$31$ \( -16 + 48 T + 6 T^{2} - 8 T^{3} + T^{4} \)
$37$ \( 2 - 13 T^{2} - 6 T^{3} + T^{4} \)
$41$ \( -128 + 224 T - 62 T^{2} - 4 T^{3} + T^{4} \)
$43$ \( 6944 + 768 T - 185 T^{2} - 6 T^{3} + T^{4} \)
$47$ \( 3472 + 832 T - 138 T^{2} - 8 T^{3} + T^{4} \)
$53$ \( -1528 + 504 T + 25 T^{2} - 18 T^{3} + T^{4} \)
$59$ \( 32 + 400 T + 174 T^{2} + 24 T^{3} + T^{4} \)
$61$ \( 224 - 32 T - 68 T^{2} - 4 T^{3} + T^{4} \)
$67$ \( -8 + 8 T + 15 T^{2} - 14 T^{3} + T^{4} \)
$71$ \( 28 + 88 T + 16 T^{2} - 12 T^{3} + T^{4} \)
$73$ \( 124 + 652 T + 13 T^{2} - 18 T^{3} + T^{4} \)
$79$ \( -4088 + 1840 T - 145 T^{2} - 10 T^{3} + T^{4} \)
$83$ \( -686 + 524 T - 67 T^{2} - 14 T^{3} + T^{4} \)
$89$ \( -6044 - 196 T + 325 T^{2} - 34 T^{3} + T^{4} \)
$97$ \( -3556 + 1700 T - 195 T^{2} - 6 T^{3} + T^{4} \)
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