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: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + q^{2} + (\beta_1 - 1) q^{3} + q^{4} - \beta_1 q^{5} + (\beta_1 - 1) q^{6} + ( - \beta_{2} - 1) q^{7} + q^{8} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + (\beta_1 - 1) q^{3} + q^{4} - \beta_1 q^{5} + (\beta_1 - 1) q^{6} + ( - \beta_{2} - 1) q^{7} + q^{8} + (\beta_{2} + 2) q^{9} - \beta_1 q^{10} + (\beta_1 - 1) q^{12} + \beta_{2} q^{13} + ( - \beta_{2} - 1) q^{14} + ( - \beta_{2} - \beta_1 - 4) q^{15} + q^{16} + ( - \beta_{2} - 2) q^{17} + (\beta_{2} + 2) q^{18} - q^{19} - \beta_1 q^{20} + (2 \beta_{2} - \beta_1 + 3) q^{21} + (\beta_{2} + 1) q^{23} + (\beta_1 - 1) q^{24} + (\beta_{2} + 2 \beta_1 - 1) q^{25} + \beta_{2} q^{26} + ( - 2 \beta_{2} - \beta_1 - 1) q^{27} + ( - \beta_{2} - 1) q^{28} + (\beta_{2} - \beta_1 + 1) q^{29} + ( - \beta_{2} - \beta_1 - 4) q^{30} + ( - 2 \beta_{2} - 2 \beta_1) q^{31} + q^{32} + ( - \beta_{2} - 2) q^{34} + ( - \beta_{2} + \beta_1 - 2) q^{35} + (\beta_{2} + 2) q^{36} + (\beta_{2} - \beta_1 - 7) q^{37} - q^{38} + ( - 2 \beta_{2} - 2) q^{39} - \beta_1 q^{40} + (2 \beta_{2} - \beta_1 - 2) q^{41} + (2 \beta_{2} - \beta_1 + 3) q^{42} + (2 \beta_{2} + 2) q^{43} + (\beta_{2} - 2 \beta_1 + 2) q^{45} + (\beta_{2} + 1) q^{46} + (\beta_{2} + 4 \beta_1 - 3) q^{47} + (\beta_1 - 1) q^{48} + (\beta_{2} - 2 \beta_1) q^{49} + (\beta_{2} + 2 \beta_1 - 1) q^{50} + (2 \beta_{2} - 2 \beta_1 + 4) q^{51} + \beta_{2} q^{52} + ( - 3 \beta_{2} - \beta_1 + 1) q^{53} + ( - 2 \beta_{2} - \beta_1 - 1) q^{54} + ( - \beta_{2} - 1) q^{56} + ( - \beta_1 + 1) q^{57} + (\beta_{2} - \beta_1 + 1) q^{58} + ( - 2 \beta_{2} + \beta_1 - 7) q^{59} + ( - \beta_{2} - \beta_1 - 4) q^{60} + (3 \beta_{2} + 3 \beta_1 + 2) q^{61} + ( - 2 \beta_{2} - 2 \beta_1) q^{62} + ( - 2 \beta_{2} + 2 \beta_1 - 8) q^{63} + q^{64} + (\beta_{2} + 2) q^{65} + ( - 4 \beta_{2} - 4) q^{67} + ( - \beta_{2} - 2) q^{68} + ( - 2 \beta_{2} + \beta_1 - 3) q^{69} + ( - \beta_{2} + \beta_1 - 2) q^{70} + ( - \beta_{2} - \beta_1) q^{71} + (\beta_{2} + 2) q^{72} + ( - 2 \beta_{2} - 4 \beta_1 + 4) q^{73} + (\beta_{2} - \beta_1 - 7) q^{74} + (\beta_1 + 7) q^{75} - q^{76} + ( - 2 \beta_{2} - 2) q^{78} + ( - \beta_{2} + \beta_1 - 2) q^{79} - \beta_1 q^{80} + ( - 2 \beta_1 - 5) q^{81} + (2 \beta_{2} - \beta_1 - 2) q^{82} + (\beta_{2} - \beta_1 + 2) q^{83} + (2 \beta_{2} - \beta_1 + 3) q^{84} + ( - \beta_{2} + 2 \beta_1 - 2) q^{85} + (2 \beta_{2} + 2) q^{86} + ( - 3 \beta_{2} - 7) q^{87} + (4 \beta_{2} - \beta_1 + 4) q^{89} + (\beta_{2} - 2 \beta_1 + 2) q^{90} + (2 \beta_1 - 6) q^{91} + (\beta_{2} + 1) q^{92} + (2 \beta_{2} - 2 \beta_1 - 4) q^{93} + (\beta_{2} + 4 \beta_1 - 3) q^{94} + \beta_1 q^{95} + (\beta_1 - 1) q^{96} + (\beta_{2} + 2 \beta_1 + 2) q^{97} + (\beta_{2} - 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 2 q^{3} + 3 q^{4} - q^{5} - 2 q^{6} - 2 q^{7} + 3 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - 2 q^{3} + 3 q^{4} - q^{5} - 2 q^{6} - 2 q^{7} + 3 q^{8} + 5 q^{9} - q^{10} - 2 q^{12} - q^{13} - 2 q^{14} - 12 q^{15} + 3 q^{16} - 5 q^{17} + 5 q^{18} - 3 q^{19} - q^{20} + 6 q^{21} + 2 q^{23} - 2 q^{24} - 2 q^{25} - q^{26} - 2 q^{27} - 2 q^{28} + q^{29} - 12 q^{30} + 3 q^{32} - 5 q^{34} - 4 q^{35} + 5 q^{36} - 23 q^{37} - 3 q^{38} - 4 q^{39} - q^{40} - 9 q^{41} + 6 q^{42} + 4 q^{43} + 3 q^{45} + 2 q^{46} - 6 q^{47} - 2 q^{48} - 3 q^{49} - 2 q^{50} + 8 q^{51} - q^{52} + 5 q^{53} - 2 q^{54} - 2 q^{56} + 2 q^{57} + q^{58} - 18 q^{59} - 12 q^{60} + 6 q^{61} - 20 q^{63} + 3 q^{64} + 5 q^{65} - 8 q^{67} - 5 q^{68} - 6 q^{69} - 4 q^{70} + 5 q^{72} + 10 q^{73} - 23 q^{74} + 22 q^{75} - 3 q^{76} - 4 q^{78} - 4 q^{79} - q^{80} - 17 q^{81} - 9 q^{82} + 4 q^{83} + 6 q^{84} - 3 q^{85} + 4 q^{86} - 18 q^{87} + 7 q^{89} + 3 q^{90} - 16 q^{91} + 2 q^{92} - 16 q^{93} - 6 q^{94} + q^{95} - 2 q^{96} + 7 q^{97} - 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 4 \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\):

\( T_{3}^{3} + 2T_{3}^{2} - 5T_{3} - 8 \) Copy content Toggle raw display
\( T_{5}^{3} + T_{5}^{2} - 6T_{5} + 2 \) Copy content Toggle raw display
\( T_{7}^{3} + 2T_{7}^{2} - 7T_{7} - 4 \) Copy content Toggle raw display
\( T_{13}^{3} + T_{13}^{2} - 8T_{13} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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