Properties

Label 4598.2.a.bn
Level $4598$
Weight $2$
Character orbit 4598.a
Self dual yes
Analytic conductor $36.715$
Analytic rank $1$
Dimension $3$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 6 x - 2\):

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

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
−1.76156
−0.363328
3.12489
1.00000 −2.76156 1.00000 1.76156 −2.76156 −3.62620 1.00000 4.62620 1.76156
1.2 1.00000 −1.36333 1.00000 0.363328 −1.36333 2.14134 1.00000 −1.14134 0.363328
1.3 1.00000 2.12489 1.00000 −3.12489 2.12489 −0.515138 1.00000 1.51514 −3.12489
\(n\): 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.bn yes 3
11.b odd 2 1 4598.2.a.bk 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4598.2.a.bk 3 11.b odd 2 1
4598.2.a.bn yes 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\):

\( T_{3}^{3} + 2 T_{3}^{2} - 5 T_{3} - 8 \)
\( T_{5}^{3} + T_{5}^{2} - 6 T_{5} + 2 \)
\( T_{7}^{3} + 2 T_{7}^{2} - 7 T_{7} - 4 \)
\( T_{13}^{3} + T_{13}^{2} - 8 T_{13} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( -8 - 5 T + 2 T^{2} + T^{3} \)
$5$ \( 2 - 6 T + T^{2} + T^{3} \)
$7$ \( -4 - 7 T + 2 T^{2} + T^{3} \)
$11$ \( T^{3} \)
$13$ \( -4 - 8 T + T^{2} + T^{3} \)
$17$ \( -8 + 5 T^{2} + T^{3} \)
$19$ \( ( 1 + T )^{3} \)
$23$ \( 4 - 7 T - 2 T^{2} + T^{3} \)
$29$ \( -25 - 19 T - T^{2} + T^{3} \)
$31$ \( -64 - 40 T + T^{3} \)
$37$ \( 271 + 157 T + 23 T^{2} + T^{3} \)
$41$ \( -242 - 22 T + 9 T^{2} + T^{3} \)
$43$ \( 32 - 28 T - 4 T^{2} + T^{3} \)
$47$ \( -508 - 79 T + 6 T^{2} + T^{3} \)
$53$ \( -37 - 59 T - 5 T^{2} + T^{3} \)
$59$ \( 44 + 59 T + 18 T^{2} + T^{3} \)
$61$ \( 388 - 78 T - 6 T^{2} + T^{3} \)
$67$ \( -256 - 112 T + 8 T^{2} + T^{3} \)
$71$ \( -8 - 10 T + T^{3} \)
$73$ \( 512 - 64 T - 10 T^{2} + T^{3} \)
$79$ \( 8 - 14 T + 4 T^{2} + T^{3} \)
$83$ \( -8 - 14 T - 4 T^{2} + T^{3} \)
$89$ \( -142 - 142 T - 7 T^{2} + T^{3} \)
$97$ \( 16 - 8 T - 7 T^{2} + T^{3} \)
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