Properties

Label 4598.2.a.bg
Level $4598$
Weight $2$
Character orbit 4598.a
Self dual yes
Analytic conductor $36.715$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.61803
−0.618034
1.00000 −1.61803 1.00000 −1.61803 −1.61803 3.85410 1.00000 −0.381966 −1.61803
1.2 1.00000 0.618034 1.00000 0.618034 0.618034 −2.85410 1.00000 −2.61803 0.618034
\(n\): 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.bg yes 2
11.b odd 2 1 4598.2.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4598.2.a.x 2 11.b odd 2 1
4598.2.a.bg yes 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\):

\( T_{3}^{2} + T_{3} - 1 \)
\( T_{5}^{2} + T_{5} - 1 \)
\( T_{7}^{2} - T_{7} - 11 \)
\( T_{13}^{2} - T_{13} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( -1 + T + T^{2} \)
$5$ \( -1 + T + T^{2} \)
$7$ \( -11 - T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( -1 - T + T^{2} \)
$17$ \( -4 + 2 T + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( ( 2 + T )^{2} \)
$29$ \( -5 + 5 T + T^{2} \)
$31$ \( 1 + 3 T + T^{2} \)
$37$ \( -4 + 2 T + T^{2} \)
$41$ \( -1 + T + T^{2} \)
$43$ \( 11 + 13 T + T^{2} \)
$47$ \( 4 + 6 T + T^{2} \)
$53$ \( 16 + 12 T + T^{2} \)
$59$ \( -64 - 8 T + T^{2} \)
$61$ \( -44 - 2 T + T^{2} \)
$67$ \( -31 + T + T^{2} \)
$71$ \( 79 - 19 T + T^{2} \)
$73$ \( 16 + 12 T + T^{2} \)
$79$ \( -20 + T^{2} \)
$83$ \( 11 - 7 T + T^{2} \)
$89$ \( -76 + 4 T + T^{2} \)
$97$ \( 176 + 28 T + T^{2} \)
show more
show less