Properties

Label 4598.2.a.bf
Level $4598$
Weight $2$
Character orbit 4598.a
Self dual yes
Analytic conductor $36.715$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.73205
1.73205
1.00000 −1.00000 1.00000 −0.732051 −1.00000 −0.267949 1.00000 −2.00000 −0.732051
1.2 1.00000 −1.00000 1.00000 2.73205 −1.00000 −3.73205 1.00000 −2.00000 2.73205
\(n\): 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.bf yes 2
11.b odd 2 1 4598.2.a.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4598.2.a.w 2 11.b odd 2 1
4598.2.a.bf 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\):

\( T_{3} + 1 \)
\( T_{5}^{2} - 2 T_{5} - 2 \)
\( T_{7}^{2} + 4 T_{7} + 1 \)
\( T_{13}^{2} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( -2 - 2 T + T^{2} \)
$7$ \( 1 + 4 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( -12 + T^{2} \)
$17$ \( 4 - 8 T + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( -3 + T^{2} \)
$31$ \( -8 - 4 T + T^{2} \)
$37$ \( -3 + 6 T + T^{2} \)
$41$ \( 78 - 18 T + T^{2} \)
$43$ \( -44 - 4 T + T^{2} \)
$47$ \( 69 - 18 T + T^{2} \)
$53$ \( -39 + 6 T + T^{2} \)
$59$ \( -107 - 2 T + T^{2} \)
$61$ \( 6 + 6 T + T^{2} \)
$67$ \( ( -14 + T )^{2} \)
$71$ \( 54 - 18 T + T^{2} \)
$73$ \( ( 2 + T )^{2} \)
$79$ \( -18 + 6 T + T^{2} \)
$83$ \( 54 - 18 T + T^{2} \)
$89$ \( 22 - 14 T + T^{2} \)
$97$ \( 88 - 20 T + T^{2} \)
show more
show less