Properties

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

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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: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + ( 1 - 2 \beta ) q^{3} + q^{4} + ( -1 + \beta ) q^{5} + ( 1 - 2 \beta ) q^{6} + ( -2 + 3 \beta ) q^{7} + q^{8} + 2 q^{9} +O(q^{10})\) \( q + q^{2} + ( 1 - 2 \beta ) q^{3} + q^{4} + ( -1 + \beta ) q^{5} + ( 1 - 2 \beta ) q^{6} + ( -2 + 3 \beta ) q^{7} + q^{8} + 2 q^{9} + ( -1 + \beta ) q^{10} + ( 1 - 2 \beta ) q^{12} + \beta q^{13} + ( -2 + 3 \beta ) q^{14} + ( -3 + \beta ) q^{15} + q^{16} + ( -3 - 2 \beta ) q^{17} + 2 q^{18} + q^{19} + ( -1 + \beta ) q^{20} + ( -8 + \beta ) q^{21} + ( 3 - 2 \beta ) q^{23} + ( 1 - 2 \beta ) q^{24} + ( -3 - \beta ) q^{25} + \beta q^{26} + ( -1 + 2 \beta ) q^{27} + ( -2 + 3 \beta ) q^{28} + ( -4 - 4 \beta ) q^{29} + ( -3 + \beta ) q^{30} + ( -1 - 6 \beta ) q^{31} + q^{32} + ( -3 - 2 \beta ) q^{34} + ( 5 - 2 \beta ) q^{35} + 2 q^{36} + ( -3 - 3 \beta ) q^{37} + q^{38} + ( -2 - \beta ) q^{39} + ( -1 + \beta ) q^{40} + ( -4 + 5 \beta ) q^{41} + ( -8 + \beta ) q^{42} + ( -6 + 6 \beta ) q^{43} + ( -2 + 2 \beta ) q^{45} + ( 3 - 2 \beta ) q^{46} + ( 5 + 4 \beta ) q^{47} + ( 1 - 2 \beta ) q^{48} + ( 6 - 3 \beta ) q^{49} + ( -3 - \beta ) q^{50} + ( 1 + 8 \beta ) q^{51} + \beta q^{52} + ( -7 + 8 \beta ) q^{53} + ( -1 + 2 \beta ) q^{54} + ( -2 + 3 \beta ) q^{56} + ( 1 - 2 \beta ) q^{57} + ( -4 - 4 \beta ) q^{58} + ( -5 - 5 \beta ) q^{59} + ( -3 + \beta ) q^{60} + ( -6 + \beta ) q^{61} + ( -1 - 6 \beta ) q^{62} + ( -4 + 6 \beta ) q^{63} + q^{64} + q^{65} + ( 2 - 5 \beta ) q^{67} + ( -3 - 2 \beta ) q^{68} + ( 7 - 4 \beta ) q^{69} + ( 5 - 2 \beta ) q^{70} + ( -7 + 2 \beta ) q^{71} + 2 q^{72} + ( -10 + 7 \beta ) q^{73} + ( -3 - 3 \beta ) q^{74} + ( -1 + 7 \beta ) q^{75} + q^{76} + ( -2 - \beta ) q^{78} + 3 q^{79} + ( -1 + \beta ) q^{80} -11 q^{81} + ( -4 + 5 \beta ) q^{82} + ( -7 + 5 \beta ) q^{83} + ( -8 + \beta ) q^{84} + ( 1 - 3 \beta ) q^{85} + ( -6 + 6 \beta ) q^{86} + ( 4 + 12 \beta ) q^{87} + ( 7 - 10 \beta ) q^{89} + ( -2 + 2 \beta ) q^{90} + ( 3 + \beta ) q^{91} + ( 3 - 2 \beta ) q^{92} + ( 11 + 8 \beta ) q^{93} + ( 5 + 4 \beta ) q^{94} + ( -1 + \beta ) q^{95} + ( 1 - 2 \beta ) q^{96} + ( -3 + 12 \beta ) q^{97} + ( 6 - 3 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} - q^{5} - q^{7} + 2q^{8} + 4q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} - q^{5} - q^{7} + 2q^{8} + 4q^{9} - q^{10} + q^{13} - q^{14} - 5q^{15} + 2q^{16} - 8q^{17} + 4q^{18} + 2q^{19} - q^{20} - 15q^{21} + 4q^{23} - 7q^{25} + q^{26} - q^{28} - 12q^{29} - 5q^{30} - 8q^{31} + 2q^{32} - 8q^{34} + 8q^{35} + 4q^{36} - 9q^{37} + 2q^{38} - 5q^{39} - q^{40} - 3q^{41} - 15q^{42} - 6q^{43} - 2q^{45} + 4q^{46} + 14q^{47} + 9q^{49} - 7q^{50} + 10q^{51} + q^{52} - 6q^{53} - q^{56} - 12q^{58} - 15q^{59} - 5q^{60} - 11q^{61} - 8q^{62} - 2q^{63} + 2q^{64} + 2q^{65} - q^{67} - 8q^{68} + 10q^{69} + 8q^{70} - 12q^{71} + 4q^{72} - 13q^{73} - 9q^{74} + 5q^{75} + 2q^{76} - 5q^{78} + 6q^{79} - q^{80} - 22q^{81} - 3q^{82} - 9q^{83} - 15q^{84} - q^{85} - 6q^{86} + 20q^{87} + 4q^{89} - 2q^{90} + 7q^{91} + 4q^{92} + 30q^{93} + 14q^{94} - q^{95} + 6q^{97} + 9q^{98} + O(q^{100}) \)

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 −2.23607 1.00000 0.618034 −2.23607 2.85410 1.00000 2.00000 0.618034
1.2 1.00000 2.23607 1.00000 −1.61803 2.23607 −3.85410 1.00000 2.00000 −1.61803
\(n\): 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.bi 2
11.b odd 2 1 4598.2.a.ba 2
11.c even 5 2 418.2.f.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.f.e 4 11.c even 5 2
4598.2.a.ba 2 11.b odd 2 1
4598.2.a.bi 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( -5 + T^{2} \)
$5$ \( -1 + T + T^{2} \)
$7$ \( -11 + T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( -1 - T + T^{2} \)
$17$ \( 11 + 8 T + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( -1 - 4 T + T^{2} \)
$29$ \( 16 + 12 T + T^{2} \)
$31$ \( -29 + 8 T + T^{2} \)
$37$ \( 9 + 9 T + T^{2} \)
$41$ \( -29 + 3 T + T^{2} \)
$43$ \( -36 + 6 T + T^{2} \)
$47$ \( 29 - 14 T + T^{2} \)
$53$ \( -71 + 6 T + T^{2} \)
$59$ \( 25 + 15 T + T^{2} \)
$61$ \( 29 + 11 T + T^{2} \)
$67$ \( -31 + T + T^{2} \)
$71$ \( 31 + 12 T + T^{2} \)
$73$ \( -19 + 13 T + T^{2} \)
$79$ \( ( -3 + T )^{2} \)
$83$ \( -11 + 9 T + T^{2} \)
$89$ \( -121 - 4 T + T^{2} \)
$97$ \( -171 - 6 T + T^{2} \)
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