Properties

Label 4598.2.a.be
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: \( x^{2} - x - 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + q^{2} - 2 \beta q^{3} + q^{4} + (3 \beta - 1) q^{5} - 2 \beta q^{6} + ( - 3 \beta + 1) q^{7} + q^{8} + (4 \beta + 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - 2 \beta q^{3} + q^{4} + (3 \beta - 1) q^{5} - 2 \beta q^{6} + ( - 3 \beta + 1) q^{7} + q^{8} + (4 \beta + 1) q^{9} + (3 \beta - 1) q^{10} - 2 \beta q^{12} - 4 q^{13} + ( - 3 \beta + 1) q^{14} + ( - 4 \beta - 6) q^{15} + q^{16} + \beta q^{17} + (4 \beta + 1) q^{18} + q^{19} + (3 \beta - 1) q^{20} + (4 \beta + 6) q^{21} + (3 \beta - 4) q^{23} - 2 \beta q^{24} + (3 \beta + 5) q^{25} - 4 q^{26} + ( - 4 \beta - 8) q^{27} + ( - 3 \beta + 1) q^{28} + ( - 6 \beta + 4) q^{29} + ( - 4 \beta - 6) q^{30} + (2 \beta - 6) q^{31} + q^{32} + \beta q^{34} + ( - 3 \beta - 10) q^{35} + (4 \beta + 1) q^{36} - 2 \beta q^{37} + q^{38} + 8 \beta q^{39} + (3 \beta - 1) q^{40} + (4 \beta - 6) q^{41} + (4 \beta + 6) q^{42} + ( - \beta - 6) q^{43} + (11 \beta + 11) q^{45} + (3 \beta - 4) q^{46} + ( - 3 \beta + 4) q^{47} - 2 \beta q^{48} + (3 \beta + 3) q^{49} + (3 \beta + 5) q^{50} + ( - 2 \beta - 2) q^{51} - 4 q^{52} + (2 \beta + 8) q^{53} + ( - 4 \beta - 8) q^{54} + ( - 3 \beta + 1) q^{56} - 2 \beta q^{57} + ( - 6 \beta + 4) q^{58} + (8 \beta - 6) q^{59} + ( - 4 \beta - 6) q^{60} + (3 \beta - 6) q^{61} + (2 \beta - 6) q^{62} + ( - 11 \beta - 11) q^{63} + q^{64} + ( - 12 \beta + 4) q^{65} + ( - 2 \beta - 6) q^{67} + \beta q^{68} + (2 \beta - 6) q^{69} + ( - 3 \beta - 10) q^{70} + (2 \beta - 6) q^{71} + (4 \beta + 1) q^{72} + ( - 8 \beta + 2) q^{73} - 2 \beta q^{74} + ( - 16 \beta - 6) q^{75} + q^{76} + 8 \beta q^{78} + ( - 2 \beta - 6) q^{79} + (3 \beta - 1) q^{80} + (12 \beta + 5) q^{81} + (4 \beta - 6) q^{82} + ( - 3 \beta + 7) q^{83} + (4 \beta + 6) q^{84} + (2 \beta + 3) q^{85} + ( - \beta - 6) q^{86} + (4 \beta + 12) q^{87} + ( - 10 \beta + 2) q^{89} + (11 \beta + 11) q^{90} + (12 \beta - 4) q^{91} + (3 \beta - 4) q^{92} + (8 \beta - 4) q^{93} + ( - 3 \beta + 4) q^{94} + (3 \beta - 1) q^{95} - 2 \beta q^{96} + (8 \beta + 4) q^{97} + (3 \beta + 3) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} - q^{7} + 2 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} - q^{7} + 2 q^{8} + 6 q^{9} + q^{10} - 2 q^{12} - 8 q^{13} - q^{14} - 16 q^{15} + 2 q^{16} + q^{17} + 6 q^{18} + 2 q^{19} + q^{20} + 16 q^{21} - 5 q^{23} - 2 q^{24} + 13 q^{25} - 8 q^{26} - 20 q^{27} - q^{28} + 2 q^{29} - 16 q^{30} - 10 q^{31} + 2 q^{32} + q^{34} - 23 q^{35} + 6 q^{36} - 2 q^{37} + 2 q^{38} + 8 q^{39} + q^{40} - 8 q^{41} + 16 q^{42} - 13 q^{43} + 33 q^{45} - 5 q^{46} + 5 q^{47} - 2 q^{48} + 9 q^{49} + 13 q^{50} - 6 q^{51} - 8 q^{52} + 18 q^{53} - 20 q^{54} - q^{56} - 2 q^{57} + 2 q^{58} - 4 q^{59} - 16 q^{60} - 9 q^{61} - 10 q^{62} - 33 q^{63} + 2 q^{64} - 4 q^{65} - 14 q^{67} + q^{68} - 10 q^{69} - 23 q^{70} - 10 q^{71} + 6 q^{72} - 4 q^{73} - 2 q^{74} - 28 q^{75} + 2 q^{76} + 8 q^{78} - 14 q^{79} + q^{80} + 22 q^{81} - 8 q^{82} + 11 q^{83} + 16 q^{84} + 8 q^{85} - 13 q^{86} + 28 q^{87} - 6 q^{89} + 33 q^{90} + 4 q^{91} - 5 q^{92} + 5 q^{94} + q^{95} - 2 q^{96} + 16 q^{97} + 9 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 −3.23607 1.00000 3.85410 −3.23607 −3.85410 1.00000 7.47214 3.85410
1.2 1.00000 1.23607 1.00000 −2.85410 1.23607 2.85410 1.00000 −1.47214 −2.85410
\(n\): 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.be 2
11.b odd 2 1 4598.2.a.v 2
11.c even 5 2 418.2.f.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.f.a 4 11.c even 5 2
4598.2.a.v 2 11.b odd 2 1
4598.2.a.be 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\):

\( T_{3}^{2} + 2T_{3} - 4 \) Copy content Toggle raw display
\( T_{5}^{2} - T_{5} - 11 \) Copy content Toggle raw display
\( T_{7}^{2} + T_{7} - 11 \) Copy content Toggle raw display
\( T_{13} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$5$ \( T^{2} - T - 11 \) Copy content Toggle raw display
$7$ \( T^{2} + T - 11 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 5T - 5 \) Copy content Toggle raw display
$29$ \( T^{2} - 2T - 44 \) Copy content Toggle raw display
$31$ \( T^{2} + 10T + 20 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$41$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$43$ \( T^{2} + 13T + 41 \) Copy content Toggle raw display
$47$ \( T^{2} - 5T - 5 \) Copy content Toggle raw display
$53$ \( T^{2} - 18T + 76 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T - 76 \) Copy content Toggle raw display
$61$ \( T^{2} + 9T + 9 \) Copy content Toggle raw display
$67$ \( T^{2} + 14T + 44 \) Copy content Toggle raw display
$71$ \( T^{2} + 10T + 20 \) Copy content Toggle raw display
$73$ \( T^{2} + 4T - 76 \) Copy content Toggle raw display
$79$ \( T^{2} + 14T + 44 \) Copy content Toggle raw display
$83$ \( T^{2} - 11T + 19 \) Copy content Toggle raw display
$89$ \( T^{2} + 6T - 116 \) Copy content Toggle raw display
$97$ \( T^{2} - 16T - 16 \) Copy content Toggle raw display
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