Properties

Label 4830.2.a.bu
Level $4830$
Weight $2$
Character orbit 4830.a
Self dual yes
Analytic conductor $38.568$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4830.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.5677441763\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - q^{7} + q^{8} + q^{9} - q^{10} + ( 3 + \beta ) q^{11} - q^{12} - q^{14} + q^{15} + q^{16} + ( -3 - \beta ) q^{17} + q^{18} + ( -5 - \beta ) q^{19} - q^{20} + q^{21} + ( 3 + \beta ) q^{22} + q^{23} - q^{24} + q^{25} - q^{27} - q^{28} + ( -3 - 3 \beta ) q^{29} + q^{30} + ( -1 + 3 \beta ) q^{31} + q^{32} + ( -3 - \beta ) q^{33} + ( -3 - \beta ) q^{34} + q^{35} + q^{36} + ( 4 + 2 \beta ) q^{37} + ( -5 - \beta ) q^{38} - q^{40} + ( -4 + 2 \beta ) q^{41} + q^{42} + ( 1 - 5 \beta ) q^{43} + ( 3 + \beta ) q^{44} - q^{45} + q^{46} + ( 3 - \beta ) q^{47} - q^{48} + q^{49} + q^{50} + ( 3 + \beta ) q^{51} + 6 \beta q^{53} - q^{54} + ( -3 - \beta ) q^{55} - q^{56} + ( 5 + \beta ) q^{57} + ( -3 - 3 \beta ) q^{58} + ( -6 - 2 \beta ) q^{59} + q^{60} + ( -2 - 4 \beta ) q^{61} + ( -1 + 3 \beta ) q^{62} - q^{63} + q^{64} + ( -3 - \beta ) q^{66} + ( -7 + 3 \beta ) q^{67} + ( -3 - \beta ) q^{68} - q^{69} + q^{70} + ( -3 - \beta ) q^{71} + q^{72} -10 q^{73} + ( 4 + 2 \beta ) q^{74} - q^{75} + ( -5 - \beta ) q^{76} + ( -3 - \beta ) q^{77} - q^{80} + q^{81} + ( -4 + 2 \beta ) q^{82} + ( 1 + \beta ) q^{83} + q^{84} + ( 3 + \beta ) q^{85} + ( 1 - 5 \beta ) q^{86} + ( 3 + 3 \beta ) q^{87} + ( 3 + \beta ) q^{88} + ( -10 + 2 \beta ) q^{89} - q^{90} + q^{92} + ( 1 - 3 \beta ) q^{93} + ( 3 - \beta ) q^{94} + ( 5 + \beta ) q^{95} - q^{96} + 12 q^{97} + q^{98} + ( 3 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} - 2q^{10} + 6q^{11} - 2q^{12} - 2q^{14} + 2q^{15} + 2q^{16} - 6q^{17} + 2q^{18} - 10q^{19} - 2q^{20} + 2q^{21} + 6q^{22} + 2q^{23} - 2q^{24} + 2q^{25} - 2q^{27} - 2q^{28} - 6q^{29} + 2q^{30} - 2q^{31} + 2q^{32} - 6q^{33} - 6q^{34} + 2q^{35} + 2q^{36} + 8q^{37} - 10q^{38} - 2q^{40} - 8q^{41} + 2q^{42} + 2q^{43} + 6q^{44} - 2q^{45} + 2q^{46} + 6q^{47} - 2q^{48} + 2q^{49} + 2q^{50} + 6q^{51} - 2q^{54} - 6q^{55} - 2q^{56} + 10q^{57} - 6q^{58} - 12q^{59} + 2q^{60} - 4q^{61} - 2q^{62} - 2q^{63} + 2q^{64} - 6q^{66} - 14q^{67} - 6q^{68} - 2q^{69} + 2q^{70} - 6q^{71} + 2q^{72} - 20q^{73} + 8q^{74} - 2q^{75} - 10q^{76} - 6q^{77} - 2q^{80} + 2q^{81} - 8q^{82} + 2q^{83} + 2q^{84} + 6q^{85} + 2q^{86} + 6q^{87} + 6q^{88} - 20q^{89} - 2q^{90} + 2q^{92} + 2q^{93} + 6q^{94} + 10q^{95} - 2q^{96} + 24q^{97} + 2q^{98} + 6q^{99} + O(q^{100}) \)

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
−0.618034
1.61803
1.00000 −1.00000 1.00000 −1.00000 −1.00000 −1.00000 1.00000 1.00000 −1.00000
1.2 1.00000 −1.00000 1.00000 −1.00000 −1.00000 −1.00000 1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)
\(7\) \(1\)
\(23\) \(-1\)

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 4830.2.a.bu 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4830.2.a.bu 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4830))\):

\( T_{11}^{2} - 6 T_{11} + 4 \)
\( T_{13} \)
\( T_{17}^{2} + 6 T_{17} + 4 \)
\( T_{19}^{2} + 10 T_{19} + 20 \)
\( T_{29}^{2} + 6 T_{29} - 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( 4 - 6 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( 4 + 6 T + T^{2} \)
$19$ \( 20 + 10 T + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( -36 + 6 T + T^{2} \)
$31$ \( -44 + 2 T + T^{2} \)
$37$ \( -4 - 8 T + T^{2} \)
$41$ \( -4 + 8 T + T^{2} \)
$43$ \( -124 - 2 T + T^{2} \)
$47$ \( 4 - 6 T + T^{2} \)
$53$ \( -180 + T^{2} \)
$59$ \( 16 + 12 T + T^{2} \)
$61$ \( -76 + 4 T + T^{2} \)
$67$ \( 4 + 14 T + T^{2} \)
$71$ \( 4 + 6 T + T^{2} \)
$73$ \( ( 10 + T )^{2} \)
$79$ \( T^{2} \)
$83$ \( -4 - 2 T + T^{2} \)
$89$ \( 80 + 20 T + T^{2} \)
$97$ \( ( -12 + T )^{2} \)
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