Properties

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

Related objects

Downloads

Learn more

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: \(4\)
Coefficient field: 4.4.7488.1
Defining polynomial: \(x^{4} - 2 x^{3} - 4 x^{2} + 2 x + 1\)
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.

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

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

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

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
−0.326909
−1.43091
3.05896
0.698857
1.00000 −3.05896 1.00000 −1.32691 −3.05896 −1.00000 1.00000 6.35723 −1.32691
1.2 1.00000 −0.698857 1.00000 −2.43091 −0.698857 −1.00000 1.00000 −2.51160 −2.43091
1.3 1.00000 0.326909 1.00000 2.05896 0.326909 −1.00000 1.00000 −2.89313 2.05896
1.4 1.00000 1.43091 1.00000 −0.301143 1.43091 −1.00000 1.00000 −0.952503 −0.301143
\(n\): 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.bt yes 4
11.b odd 2 1 4598.2.a.bq 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4598.2.a.bq 4 11.b odd 2 1
4598.2.a.bt 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\):

\( T_{3}^{4} + 2 T_{3}^{3} - 4 T_{3}^{2} - 2 T_{3} + 1 \)
\( T_{5}^{4} + 2 T_{5}^{3} - 4 T_{5}^{2} - 8 T_{5} - 2 \)
\( T_{7} + 1 \)
\( T_{13}^{4} + 4 T_{13}^{3} - 16 T_{13}^{2} - 16 T_{13} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( 1 - 2 T - 4 T^{2} + 2 T^{3} + T^{4} \)
$5$ \( -2 - 8 T - 4 T^{2} + 2 T^{3} + T^{4} \)
$7$ \( ( 1 + T )^{4} \)
$11$ \( T^{4} \)
$13$ \( 16 - 16 T - 16 T^{2} + 4 T^{3} + T^{4} \)
$17$ \( 4 - 16 T - 4 T^{2} + 4 T^{3} + T^{4} \)
$19$ \( ( -1 + T )^{4} \)
$23$ \( -587 - 328 T - 30 T^{2} + 8 T^{3} + T^{4} \)
$29$ \( -11 - 62 T - 72 T^{2} - 2 T^{3} + T^{4} \)
$31$ \( 1336 + 72 T - 88 T^{2} + T^{4} \)
$37$ \( 1 + 2 T - 16 T^{2} + 10 T^{3} + T^{4} \)
$41$ \( 1606 - 140 T - 88 T^{2} + 2 T^{3} + T^{4} \)
$43$ \( -104 - 104 T + 80 T^{2} - 16 T^{3} + T^{4} \)
$47$ \( 97 + 96 T - 46 T^{2} + T^{4} \)
$53$ \( -11 - 42 T - 28 T^{2} + 6 T^{3} + T^{4} \)
$59$ \( -107 + 166 T + 104 T^{2} - 22 T^{3} + T^{4} \)
$61$ \( 1126 + 560 T - 168 T^{2} + 2 T^{3} + T^{4} \)
$67$ \( -6512 - 1328 T + 32 T^{2} + 20 T^{3} + T^{4} \)
$71$ \( 4174 - 608 T - 120 T^{2} + 10 T^{3} + T^{4} \)
$73$ \( 1924 - 24 T - 100 T^{2} + T^{4} \)
$79$ \( -2882 + 1704 T - 208 T^{2} - 6 T^{3} + T^{4} \)
$83$ \( -5402 + 344 T + 332 T^{2} + 34 T^{3} + T^{4} \)
$89$ \( 1678 + 76 T - 160 T^{2} + 2 T^{3} + T^{4} \)
$97$ \( -4232 - 3128 T - 280 T^{2} + 8 T^{3} + T^{4} \)
show more
show less