Properties

Label 4598.2.a.bo
Level $4598$
Weight $2$
Character orbit 4598.a
Self dual yes
Analytic conductor $36.715$
Analytic rank $0$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Defining polynomial: \(x^{3} - 6 x - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
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.

\(f(q)\) \(=\) \( q + q^{2} + \beta_{1} q^{3} + q^{4} + ( -1 - \beta_{1} - \beta_{2} ) q^{5} + \beta_{1} q^{6} + ( 2 + \beta_{1} - \beta_{2} ) q^{7} + q^{8} + ( 1 + \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})\) \( q + q^{2} + \beta_{1} q^{3} + q^{4} + ( -1 - \beta_{1} - \beta_{2} ) q^{5} + \beta_{1} q^{6} + ( 2 + \beta_{1} - \beta_{2} ) q^{7} + q^{8} + ( 1 + \beta_{1} + \beta_{2} ) q^{9} + ( -1 - \beta_{1} - \beta_{2} ) q^{10} + \beta_{1} q^{12} + ( \beta_{1} - \beta_{2} ) q^{13} + ( 2 + \beta_{1} - \beta_{2} ) q^{14} + ( -3 - 3 \beta_{1} ) q^{15} + q^{16} + ( 3 + \beta_{2} ) q^{17} + ( 1 + \beta_{1} + \beta_{2} ) q^{18} + q^{19} + ( -1 - \beta_{1} - \beta_{2} ) q^{20} + ( 5 + 2 \beta_{1} + 2 \beta_{2} ) q^{21} + ( -5 - \beta_{2} ) q^{23} + \beta_{1} q^{24} + ( 4 + 3 \beta_{1} ) q^{25} + ( \beta_{1} - \beta_{2} ) q^{26} + 3 q^{27} + ( 2 + \beta_{1} - \beta_{2} ) q^{28} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{29} + ( -3 - 3 \beta_{1} ) q^{30} + ( -1 + 3 \beta_{1} ) q^{31} + q^{32} + ( 3 + \beta_{2} ) q^{34} + ( -6 \beta_{1} - 3 \beta_{2} ) q^{35} + ( 1 + \beta_{1} + \beta_{2} ) q^{36} + ( -2 - 2 \beta_{2} ) q^{37} + q^{38} + ( 5 + 2 \beta_{2} ) q^{39} + ( -1 - \beta_{1} - \beta_{2} ) q^{40} + ( 3 + 3 \beta_{1} + 2 \beta_{2} ) q^{41} + ( 5 + 2 \beta_{1} + 2 \beta_{2} ) q^{42} + ( 7 - 3 \beta_{1} - \beta_{2} ) q^{43} + ( -9 - 3 \beta_{1} ) q^{45} + ( -5 - \beta_{2} ) q^{46} + ( -4 + 4 \beta_{1} + 2 \beta_{2} ) q^{47} + \beta_{1} q^{48} + ( 9 + \beta_{1} - 2 \beta_{2} ) q^{49} + ( 4 + 3 \beta_{1} ) q^{50} + ( -1 + 4 \beta_{1} - \beta_{2} ) q^{51} + ( \beta_{1} - \beta_{2} ) q^{52} + ( -3 - 4 \beta_{1} + \beta_{2} ) q^{53} + 3 q^{54} + ( 2 + \beta_{1} - \beta_{2} ) q^{56} + \beta_{1} q^{57} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{58} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{59} + ( -3 - 3 \beta_{1} ) q^{60} + ( -8 - 2 \beta_{2} ) q^{61} + ( -1 + 3 \beta_{1} ) q^{62} + ( 6 \beta_{1} + 3 \beta_{2} ) q^{63} + q^{64} + ( 2 - 4 \beta_{1} - \beta_{2} ) q^{65} + ( \beta_{1} - 3 \beta_{2} ) q^{67} + ( 3 + \beta_{2} ) q^{68} + ( 1 - 6 \beta_{1} + \beta_{2} ) q^{69} + ( -6 \beta_{1} - 3 \beta_{2} ) q^{70} + ( 7 - 5 \beta_{1} - \beta_{2} ) q^{71} + ( 1 + \beta_{1} + \beta_{2} ) q^{72} + ( 1 + \beta_{2} ) q^{73} + ( -2 - 2 \beta_{2} ) q^{74} + ( 12 + 7 \beta_{1} + 3 \beta_{2} ) q^{75} + q^{76} + ( 5 + 2 \beta_{2} ) q^{78} + ( 2 + 2 \beta_{1} + 4 \beta_{2} ) q^{79} + ( -1 - \beta_{1} - \beta_{2} ) q^{80} + ( -3 - 3 \beta_{2} ) q^{81} + ( 3 + 3 \beta_{1} + 2 \beta_{2} ) q^{82} + ( 11 + \beta_{1} - \beta_{2} ) q^{83} + ( 5 + 2 \beta_{1} + 2 \beta_{2} ) q^{84} + ( -8 - 2 \beta_{1} - 2 \beta_{2} ) q^{85} + ( 7 - 3 \beta_{1} - \beta_{2} ) q^{86} + ( -2 - 5 \beta_{1} + \beta_{2} ) q^{87} + ( 2 + 2 \beta_{1} + 2 \beta_{2} ) q^{89} + ( -9 - 3 \beta_{1} ) q^{90} + ( 12 - \beta_{1} ) q^{91} + ( -5 - \beta_{2} ) q^{92} + ( 12 + 2 \beta_{1} + 3 \beta_{2} ) q^{93} + ( -4 + 4 \beta_{1} + 2 \beta_{2} ) q^{94} + ( -1 - \beta_{1} - \beta_{2} ) q^{95} + \beta_{1} q^{96} + ( 4 - 4 \beta_{2} ) q^{97} + ( 9 + \beta_{1} - 2 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{4} - 3q^{5} + 6q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{4} - 3q^{5} + 6q^{7} + 3q^{8} + 3q^{9} - 3q^{10} + 6q^{14} - 9q^{15} + 3q^{16} + 9q^{17} + 3q^{18} + 3q^{19} - 3q^{20} + 15q^{21} - 15q^{23} + 12q^{25} + 9q^{27} + 6q^{28} - 6q^{29} - 9q^{30} - 3q^{31} + 3q^{32} + 9q^{34} + 3q^{36} - 6q^{37} + 3q^{38} + 15q^{39} - 3q^{40} + 9q^{41} + 15q^{42} + 21q^{43} - 27q^{45} - 15q^{46} - 12q^{47} + 27q^{49} + 12q^{50} - 3q^{51} - 9q^{53} + 9q^{54} + 6q^{56} - 6q^{58} - 3q^{59} - 9q^{60} - 24q^{61} - 3q^{62} + 3q^{64} + 6q^{65} + 9q^{68} + 3q^{69} + 21q^{71} + 3q^{72} + 3q^{73} - 6q^{74} + 36q^{75} + 3q^{76} + 15q^{78} + 6q^{79} - 3q^{80} - 9q^{81} + 9q^{82} + 33q^{83} + 15q^{84} - 24q^{85} + 21q^{86} - 6q^{87} + 6q^{89} - 27q^{90} + 36q^{91} - 15q^{92} + 36q^{93} - 12q^{94} - 3q^{95} + 12q^{97} + 27q^{98} + O(q^{100}) \)

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

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

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
−2.14510
−0.523976
2.66908
1.00000 −2.14510 1.00000 −1.60147 −2.14510 −2.89167 1.00000 1.60147 −1.60147
1.2 1.00000 −0.523976 1.00000 2.72545 −0.523976 4.67750 1.00000 −2.72545 2.72545
1.3 1.00000 2.66908 1.00000 −4.12398 2.66908 4.21417 1.00000 4.12398 −4.12398
\(n\): 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.bo 3
11.b odd 2 1 418.2.a.g 3
33.d even 2 1 3762.2.a.bg 3
44.c even 2 1 3344.2.a.q 3
209.d even 2 1 7942.2.a.bi 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.a.g 3 11.b odd 2 1
3344.2.a.q 3 44.c even 2 1
3762.2.a.bg 3 33.d even 2 1
4598.2.a.bo 3 1.a even 1 1 trivial
7942.2.a.bi 3 209.d even 2 1

Hecke kernels

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

\( T_{3}^{3} - 6 T_{3} - 3 \)
\( T_{5}^{3} + 3 T_{5}^{2} - 9 T_{5} - 18 \)
\( T_{7}^{3} - 6 T_{7}^{2} - 6 T_{7} + 57 \)
\( T_{13}^{3} - 18 T_{13} + 29 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( -3 - 6 T + T^{3} \)
$5$ \( -18 - 9 T + 3 T^{2} + T^{3} \)
$7$ \( 57 - 6 T - 6 T^{2} + T^{3} \)
$11$ \( T^{3} \)
$13$ \( 29 - 18 T + T^{3} \)
$17$ \( 4 + 18 T - 9 T^{2} + T^{3} \)
$19$ \( ( -1 + T )^{3} \)
$23$ \( 76 + 66 T + 15 T^{2} + T^{3} \)
$29$ \( -147 - 24 T + 6 T^{2} + T^{3} \)
$31$ \( -134 - 51 T + 3 T^{2} + T^{3} \)
$37$ \( -96 - 24 T + 6 T^{2} + T^{3} \)
$41$ \( 122 - 45 T - 9 T^{2} + T^{3} \)
$43$ \( 184 + 93 T - 21 T^{2} + T^{3} \)
$47$ \( -672 - 60 T + 12 T^{2} + T^{3} \)
$53$ \( -452 - 90 T + 9 T^{2} + T^{3} \)
$59$ \( -64 - 24 T + 3 T^{2} + T^{3} \)
$61$ \( 192 + 156 T + 24 T^{2} + T^{3} \)
$67$ \( 123 - 96 T + T^{3} \)
$71$ \( 1306 + 3 T - 21 T^{2} + T^{3} \)
$73$ \( 12 - 6 T - 3 T^{2} + T^{3} \)
$79$ \( 944 - 132 T - 6 T^{2} + T^{3} \)
$83$ \( -1104 + 345 T - 33 T^{2} + T^{3} \)
$89$ \( 144 - 36 T - 6 T^{2} + T^{3} \)
$97$ \( 256 - 96 T - 12 T^{2} + T^{3} \)
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