Properties

Label 4598.2.a.w
Level $4598$
Weight $2$
Character orbit 4598.a
Self dual yes
Analytic conductor $36.715$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + (\beta + 1) q^{5} + q^{6} + (\beta + 2) q^{7} - q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} + (\beta + 1) q^{5} + q^{6} + (\beta + 2) q^{7} - q^{8} - 2 q^{9} + ( - \beta - 1) q^{10} - q^{12} + 2 \beta q^{13} + ( - \beta - 2) q^{14} + ( - \beta - 1) q^{15} + q^{16} + (2 \beta - 4) q^{17} + 2 q^{18} + q^{19} + (\beta + 1) q^{20} + ( - \beta - 2) q^{21} - q^{23} + q^{24} + (2 \beta - 1) q^{25} - 2 \beta q^{26} + 5 q^{27} + (\beta + 2) q^{28} - \beta q^{29} + (\beta + 1) q^{30} + (2 \beta + 2) q^{31} - q^{32} + ( - 2 \beta + 4) q^{34} + (3 \beta + 5) q^{35} - 2 q^{36} + (2 \beta - 3) q^{37} - q^{38} - 2 \beta q^{39} + ( - \beta - 1) q^{40} + (\beta - 9) q^{41} + (\beta + 2) q^{42} + ( - 4 \beta - 2) q^{43} + ( - 2 \beta - 2) q^{45} + q^{46} + ( - 2 \beta + 9) q^{47} - q^{48} + 4 \beta q^{49} + ( - 2 \beta + 1) q^{50} + ( - 2 \beta + 4) q^{51} + 2 \beta q^{52} + (4 \beta - 3) q^{53} - 5 q^{54} + ( - \beta - 2) q^{56} - q^{57} + \beta q^{58} + (6 \beta + 1) q^{59} + ( - \beta - 1) q^{60} + (\beta + 3) q^{61} + ( - 2 \beta - 2) q^{62} + ( - 2 \beta - 4) q^{63} + q^{64} + (2 \beta + 6) q^{65} + 14 q^{67} + (2 \beta - 4) q^{68} + q^{69} + ( - 3 \beta - 5) q^{70} + ( - 3 \beta + 9) q^{71} + 2 q^{72} + 2 q^{73} + ( - 2 \beta + 3) q^{74} + ( - 2 \beta + 1) q^{75} + q^{76} + 2 \beta q^{78} + (3 \beta + 3) q^{79} + (\beta + 1) q^{80} + q^{81} + ( - \beta + 9) q^{82} + ( - 3 \beta - 9) q^{83} + ( - \beta - 2) q^{84} + ( - 2 \beta + 2) q^{85} + (4 \beta + 2) q^{86} + \beta q^{87} + ( - 3 \beta + 7) q^{89} + (2 \beta + 2) q^{90} + (4 \beta + 6) q^{91} - q^{92} + ( - 2 \beta - 2) q^{93} + (2 \beta - 9) q^{94} + (\beta + 1) q^{95} + q^{96} + ( - 2 \beta + 10) q^{97} - 4 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 4 q^{7} - 2 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 4 q^{7} - 2 q^{8} - 4 q^{9} - 2 q^{10} - 2 q^{12} - 4 q^{14} - 2 q^{15} + 2 q^{16} - 8 q^{17} + 4 q^{18} + 2 q^{19} + 2 q^{20} - 4 q^{21} - 2 q^{23} + 2 q^{24} - 2 q^{25} + 10 q^{27} + 4 q^{28} + 2 q^{30} + 4 q^{31} - 2 q^{32} + 8 q^{34} + 10 q^{35} - 4 q^{36} - 6 q^{37} - 2 q^{38} - 2 q^{40} - 18 q^{41} + 4 q^{42} - 4 q^{43} - 4 q^{45} + 2 q^{46} + 18 q^{47} - 2 q^{48} + 2 q^{50} + 8 q^{51} - 6 q^{53} - 10 q^{54} - 4 q^{56} - 2 q^{57} + 2 q^{59} - 2 q^{60} + 6 q^{61} - 4 q^{62} - 8 q^{63} + 2 q^{64} + 12 q^{65} + 28 q^{67} - 8 q^{68} + 2 q^{69} - 10 q^{70} + 18 q^{71} + 4 q^{72} + 4 q^{73} + 6 q^{74} + 2 q^{75} + 2 q^{76} + 6 q^{79} + 2 q^{80} + 2 q^{81} + 18 q^{82} - 18 q^{83} - 4 q^{84} + 4 q^{85} + 4 q^{86} + 14 q^{89} + 4 q^{90} + 12 q^{91} - 2 q^{92} - 4 q^{93} - 18 q^{94} + 2 q^{95} + 2 q^{96} + 20 q^{97}+O(q^{100}) \) 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
−1.73205
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.w 2
11.b odd 2 1 4598.2.a.bf yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4598.2.a.w 2 1.a even 1 1 trivial
4598.2.a.bf yes 2 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\):

\( T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} - 2T_{5} - 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 4T_{7} + 1 \) Copy content Toggle raw display
\( T_{13}^{2} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

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