Properties

Label 4598.2.a.bu
Level $4598$
Weight $2$
Character orbit 4598.a
Self dual yes
Analytic conductor $36.715$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7152148494\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.33452.1
Defining polynomial: \(x^{4} - x^{3} - 8 x^{2} + 7 x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + q^{2} + \beta_{1} q^{3} + q^{4} + ( 1 + \beta_{2} + \beta_{3} ) q^{5} + \beta_{1} q^{6} + \beta_{3} q^{7} + q^{8} + ( 1 + \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})\) \( q + q^{2} + \beta_{1} q^{3} + q^{4} + ( 1 + \beta_{2} + \beta_{3} ) q^{5} + \beta_{1} q^{6} + \beta_{3} q^{7} + q^{8} + ( 1 + \beta_{2} - \beta_{3} ) q^{9} + ( 1 + \beta_{2} + \beta_{3} ) q^{10} + \beta_{1} q^{12} + ( -2 + \beta_{2} + \beta_{3} ) q^{13} + \beta_{3} q^{14} + ( 1 + \beta_{1} + \beta_{2} ) q^{15} + q^{16} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{17} + ( 1 + \beta_{2} - \beta_{3} ) q^{18} - q^{19} + ( 1 + \beta_{2} + \beta_{3} ) q^{20} + ( 1 - 2 \beta_{1} ) q^{21} + ( -1 + \beta_{1} - \beta_{3} ) q^{23} + \beta_{1} q^{24} + ( 4 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{25} + ( -2 + \beta_{2} + \beta_{3} ) q^{26} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{27} + \beta_{3} q^{28} + ( 3 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{29} + ( 1 + \beta_{1} + \beta_{2} ) q^{30} + ( 7 - \beta_{1} - \beta_{2} ) q^{31} + q^{32} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{34} + ( 4 - \beta_{1} + 3 \beta_{3} ) q^{35} + ( 1 + \beta_{2} - \beta_{3} ) q^{36} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{37} - q^{38} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{39} + ( 1 + \beta_{2} + \beta_{3} ) q^{40} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{41} + ( 1 - 2 \beta_{1} ) q^{42} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{43} + ( 1 + 3 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{45} + ( -1 + \beta_{1} - \beta_{3} ) q^{46} + ( 6 + 2 \beta_{2} + 2 \beta_{3} ) q^{47} + \beta_{1} q^{48} + ( -1 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{49} + ( 4 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{50} + ( 3 - \beta_{1} - \beta_{3} ) q^{51} + ( -2 + \beta_{2} + \beta_{3} ) q^{52} + ( 1 - 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{53} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{54} + \beta_{3} q^{56} -\beta_{1} q^{57} + ( 3 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{58} + ( -1 + 3 \beta_{1} - \beta_{2} - \beta_{3} ) q^{59} + ( 1 + \beta_{1} + \beta_{2} ) q^{60} + ( 2 + 2 \beta_{3} ) q^{61} + ( 7 - \beta_{1} - \beta_{2} ) q^{62} + ( -8 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{63} + q^{64} + ( 6 + \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{65} + ( 4 - 2 \beta_{2} - \beta_{3} ) q^{67} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{68} + ( 3 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{69} + ( 4 - \beta_{1} + 3 \beta_{3} ) q^{70} + ( -5 + \beta_{2} - \beta_{3} ) q^{71} + ( 1 + \beta_{2} - \beta_{3} ) q^{72} + ( 5 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{73} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{74} + ( 6 - 2 \beta_{1} - \beta_{3} ) q^{75} - q^{76} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{78} + ( 4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{79} + ( 1 + \beta_{2} + \beta_{3} ) q^{80} + ( 5 + \beta_{1} + \beta_{3} ) q^{81} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{82} + ( -7 + 2 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{83} + ( 1 - 2 \beta_{1} ) q^{84} + ( -8 + 2 \beta_{2} - 2 \beta_{3} ) q^{85} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{86} + ( 10 + 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{87} + ( 4 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{89} + ( 1 + 3 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{90} + ( 4 - \beta_{1} ) q^{91} + ( -1 + \beta_{1} - \beta_{3} ) q^{92} + ( -4 + 5 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{93} + ( 6 + 2 \beta_{2} + 2 \beta_{3} ) q^{94} + ( -1 - \beta_{2} - \beta_{3} ) q^{95} + \beta_{1} q^{96} + ( 4 - 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{97} + ( -1 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + q^{3} + 4q^{4} + 3q^{5} + q^{6} - q^{7} + 4q^{8} + 5q^{9} + O(q^{10}) \) \( 4q + 4q^{2} + q^{3} + 4q^{4} + 3q^{5} + q^{6} - q^{7} + 4q^{8} + 5q^{9} + 3q^{10} + q^{12} - 9q^{13} - q^{14} + 5q^{15} + 4q^{16} - 2q^{17} + 5q^{18} - 4q^{19} + 3q^{20} + 2q^{21} - 2q^{23} + q^{24} + 15q^{25} - 9q^{26} - 2q^{27} - q^{28} + 5q^{29} + 5q^{30} + 27q^{31} + 4q^{32} - 2q^{34} + 12q^{35} + 5q^{36} + 6q^{37} - 4q^{38} + 2q^{39} + 3q^{40} - 5q^{41} + 2q^{42} + q^{43} + 11q^{45} - 2q^{46} + 22q^{47} + q^{48} - 7q^{49} + 15q^{50} + 12q^{51} - 9q^{52} - 2q^{54} - q^{56} - q^{57} + 5q^{58} + 5q^{60} + 6q^{61} + 27q^{62} - 30q^{63} + 4q^{64} + 26q^{65} + 17q^{67} - 2q^{68} + 14q^{69} + 12q^{70} - 19q^{71} + 5q^{72} + 20q^{73} + 6q^{74} + 23q^{75} - 4q^{76} + 2q^{78} + 12q^{79} + 3q^{80} + 20q^{81} - 5q^{82} - 23q^{83} + 2q^{84} - 30q^{85} + q^{86} + 45q^{87} + 16q^{89} + 11q^{90} + 15q^{91} - 2q^{92} - 12q^{93} + 22q^{94} - 3q^{95} + q^{96} + 8q^{97} - 7q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 8 x^{2} + 7 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - 8 \nu + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 8 \nu + 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{3} + \beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{2} + 8 \beta_{1} - 1\)

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
−2.79836
0.180986
0.669883
2.94749
1.00000 −2.79836 1.00000 0.116104 −2.79836 −2.35735 1.00000 4.83081 0.116104
1.2 1.00000 0.180986 1.00000 4.08332 0.180986 3.52528 1.00000 −2.96724 4.08332
1.3 1.00000 0.669883 1.00000 −3.56566 0.669883 −0.507202 1.00000 −2.55126 −3.56566
1.4 1.00000 2.94749 1.00000 2.36624 2.94749 −1.66073 1.00000 5.68769 2.36624
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(1\)
\(19\) \(1\)

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 4598.2.a.bu yes 4
11.b odd 2 1 4598.2.a.br 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4598.2.a.br 4 11.b odd 2 1
4598.2.a.bu yes 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\):

\( T_{3}^{4} - T_{3}^{3} - 8 T_{3}^{2} + 7 T_{3} - 1 \)
\( T_{5}^{4} - 3 T_{5}^{3} - 13 T_{5}^{2} + 36 T_{5} - 4 \)
\( T_{7}^{4} + T_{7}^{3} - 10 T_{7}^{2} - 19 T_{7} - 7 \)
\( T_{13}^{4} + 9 T_{13}^{3} + 14 T_{13}^{2} - 15 T_{13} - 13 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( -1 + 7 T - 8 T^{2} - T^{3} + T^{4} \)
$5$ \( -4 + 36 T - 13 T^{2} - 3 T^{3} + T^{4} \)
$7$ \( -7 - 19 T - 10 T^{2} + T^{3} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( -13 - 15 T + 14 T^{2} + 9 T^{3} + T^{4} \)
$17$ \( 28 - 38 T - 19 T^{2} + 2 T^{3} + T^{4} \)
$19$ \( ( 1 + T )^{4} \)
$23$ \( 4 - 20 T - 15 T^{2} + 2 T^{3} + T^{4} \)
$29$ \( 2023 + 219 T - 86 T^{2} - 5 T^{3} + T^{4} \)
$31$ \( 644 - 856 T + 249 T^{2} - 27 T^{3} + T^{4} \)
$37$ \( -128 + 224 T - 68 T^{2} - 6 T^{3} + T^{4} \)
$41$ \( 796 - 152 T - 73 T^{2} + 5 T^{3} + T^{4} \)
$43$ \( -392 + 308 T - 65 T^{2} - T^{3} + T^{4} \)
$47$ \( -1408 + 160 T + 116 T^{2} - 22 T^{3} + T^{4} \)
$53$ \( -784 + 564 T - 107 T^{2} + T^{4} \)
$59$ \( 1088 + 72 T - 79 T^{2} + T^{4} \)
$61$ \( 32 - 28 T^{2} - 6 T^{3} + T^{4} \)
$67$ \( -23 - 33 T + 58 T^{2} - 17 T^{3} + T^{4} \)
$71$ \( 28 + 120 T + 103 T^{2} + 19 T^{3} + T^{4} \)
$73$ \( 112 - 212 T + 121 T^{2} - 20 T^{3} + T^{4} \)
$79$ \( -2464 + 1520 T - 108 T^{2} - 12 T^{3} + T^{4} \)
$83$ \( -4048 - 1136 T + 43 T^{2} + 23 T^{3} + T^{4} \)
$89$ \( -736 + 1456 T - 68 T^{2} - 16 T^{3} + T^{4} \)
$97$ \( 10624 + 608 T - 216 T^{2} - 8 T^{3} + T^{4} \)
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