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 $1$

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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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + q^{2} - \beta q^{3} + q^{4} - \beta q^{5} - \beta q^{6} + (3 \beta - 1) q^{7} + q^{8} + (\beta - 2) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - \beta q^{3} + q^{4} - \beta q^{5} - \beta q^{6} + (3 \beta - 1) q^{7} + q^{8} + (\beta - 2) q^{9} - \beta q^{10} - \beta q^{12} + ( - \beta + 1) q^{13} + (3 \beta - 1) q^{14} + (\beta + 1) q^{15} + q^{16} - 2 \beta q^{17} + (\beta - 2) q^{18} + q^{19} - \beta q^{20} + ( - 2 \beta - 3) q^{21} - 2 q^{23} - \beta q^{24} + (\beta - 4) q^{25} + ( - \beta + 1) q^{26} + (4 \beta - 1) q^{27} + (3 \beta - 1) q^{28} + (3 \beta - 4) q^{29} + (\beta + 1) q^{30} + ( - \beta - 1) q^{31} + q^{32} - 2 \beta q^{34} + ( - 2 \beta - 3) q^{35} + (\beta - 2) q^{36} + (2 \beta - 2) q^{37} + q^{38} + q^{39} - \beta q^{40} + (\beta - 1) q^{41} + ( - 2 \beta - 3) q^{42} + ( - 5 \beta - 4) q^{43} + (\beta - 1) q^{45} - 2 q^{46} + (2 \beta - 4) q^{47} - \beta q^{48} + (3 \beta + 3) q^{49} + (\beta - 4) q^{50} + (2 \beta + 2) q^{51} + ( - \beta + 1) q^{52} + ( - 4 \beta - 4) q^{53} + (4 \beta - 1) q^{54} + (3 \beta - 1) q^{56} - \beta q^{57} + (3 \beta - 4) q^{58} + 8 \beta q^{59} + (\beta + 1) q^{60} + ( - 6 \beta + 4) q^{61} + ( - \beta - 1) q^{62} + ( - 4 \beta + 5) q^{63} + q^{64} + q^{65} + (5 \beta - 3) q^{67} - 2 \beta q^{68} + 2 \beta q^{69} + ( - 2 \beta - 3) q^{70} + (3 \beta + 8) q^{71} + (\beta - 2) q^{72} + ( - 4 \beta - 4) q^{73} + (2 \beta - 2) q^{74} + (3 \beta - 1) q^{75} + q^{76} + q^{78} + (4 \beta - 2) q^{79} - \beta q^{80} + ( - 6 \beta + 2) q^{81} + (\beta - 1) q^{82} + ( - \beta + 4) q^{83} + ( - 2 \beta - 3) q^{84} + (2 \beta + 2) q^{85} + ( - 5 \beta - 4) q^{86} + (\beta - 3) q^{87} + (8 \beta - 6) q^{89} + (\beta - 1) q^{90} + (\beta - 4) q^{91} - 2 q^{92} + (2 \beta + 1) q^{93} + (2 \beta - 4) q^{94} - \beta q^{95} - \beta q^{96} + ( - 4 \beta - 12) q^{97} + (3 \beta + 3) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - q^{5} - q^{6} + q^{7} + 2 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - q^{5} - q^{6} + q^{7} + 2 q^{8} - 3 q^{9} - q^{10} - q^{12} + q^{13} + q^{14} + 3 q^{15} + 2 q^{16} - 2 q^{17} - 3 q^{18} + 2 q^{19} - q^{20} - 8 q^{21} - 4 q^{23} - q^{24} - 7 q^{25} + q^{26} + 2 q^{27} + q^{28} - 5 q^{29} + 3 q^{30} - 3 q^{31} + 2 q^{32} - 2 q^{34} - 8 q^{35} - 3 q^{36} - 2 q^{37} + 2 q^{38} + 2 q^{39} - q^{40} - q^{41} - 8 q^{42} - 13 q^{43} - q^{45} - 4 q^{46} - 6 q^{47} - q^{48} + 9 q^{49} - 7 q^{50} + 6 q^{51} + q^{52} - 12 q^{53} + 2 q^{54} + q^{56} - q^{57} - 5 q^{58} + 8 q^{59} + 3 q^{60} + 2 q^{61} - 3 q^{62} + 6 q^{63} + 2 q^{64} + 2 q^{65} - q^{67} - 2 q^{68} + 2 q^{69} - 8 q^{70} + 19 q^{71} - 3 q^{72} - 12 q^{73} - 2 q^{74} + q^{75} + 2 q^{76} + 2 q^{78} - q^{80} - 2 q^{81} - q^{82} + 7 q^{83} - 8 q^{84} + 6 q^{85} - 13 q^{86} - 5 q^{87} - 4 q^{89} - q^{90} - 7 q^{91} - 4 q^{92} + 4 q^{93} - 6 q^{94} - q^{95} - q^{96} - 28 q^{97} + 9 q^{98}+O(q^{100}) \) 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 −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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\):

\( T_{3}^{2} + T_{3} - 1 \) Copy content Toggle raw display
\( T_{5}^{2} + T_{5} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} - T_{7} - 11 \) Copy content Toggle raw display
\( T_{13}^{2} - T_{13} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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