Properties

Label 4830.2.a.bq
Level $4830$
Weight $2$
Character orbit 4830.a
Self dual yes
Analytic conductor $38.568$
Analytic rank $0$
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: \(0\)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{7} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{7} - q^{8} + q^{9} - q^{10} + ( -1 - \beta ) q^{11} + q^{12} + ( 1 + \beta ) q^{13} + q^{14} + q^{15} + q^{16} + ( 2 - 2 \beta ) q^{17} - q^{18} -2 \beta q^{19} + q^{20} - q^{21} + ( 1 + \beta ) q^{22} + q^{23} - q^{24} + q^{25} + ( -1 - \beta ) q^{26} + q^{27} - q^{28} + ( 1 + 3 \beta ) q^{29} - q^{30} -2 \beta q^{31} - q^{32} + ( -1 - \beta ) q^{33} + ( -2 + 2 \beta ) q^{34} - q^{35} + q^{36} + ( 4 - 2 \beta ) q^{37} + 2 \beta q^{38} + ( 1 + \beta ) q^{39} - q^{40} + ( 4 + 2 \beta ) q^{41} + q^{42} + ( -3 + \beta ) q^{43} + ( -1 - \beta ) q^{44} + q^{45} - q^{46} -2 q^{47} + q^{48} + q^{49} - q^{50} + ( 2 - 2 \beta ) q^{51} + ( 1 + \beta ) q^{52} + 10 q^{53} - q^{54} + ( -1 - \beta ) q^{55} + q^{56} -2 \beta q^{57} + ( -1 - 3 \beta ) q^{58} + 4 \beta q^{59} + q^{60} -2 q^{61} + 2 \beta q^{62} - q^{63} + q^{64} + ( 1 + \beta ) q^{65} + ( 1 + \beta ) q^{66} + ( -11 + \beta ) q^{67} + ( 2 - 2 \beta ) q^{68} + q^{69} + q^{70} + ( 3 + 3 \beta ) q^{71} - q^{72} + ( 6 + 4 \beta ) q^{73} + ( -4 + 2 \beta ) q^{74} + q^{75} -2 \beta q^{76} + ( 1 + \beta ) q^{77} + ( -1 - \beta ) q^{78} -4 q^{79} + q^{80} + q^{81} + ( -4 - 2 \beta ) q^{82} + ( 4 - 2 \beta ) q^{83} - q^{84} + ( 2 - 2 \beta ) q^{85} + ( 3 - \beta ) q^{86} + ( 1 + 3 \beta ) q^{87} + ( 1 + \beta ) q^{88} + ( 5 + \beta ) q^{89} - q^{90} + ( -1 - \beta ) q^{91} + q^{92} -2 \beta q^{93} + 2 q^{94} -2 \beta q^{95} - q^{96} + ( -1 + 3 \beta ) q^{97} - q^{98} + ( -1 - \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} - 2q^{10} - 2q^{11} + 2q^{12} + 2q^{13} + 2q^{14} + 2q^{15} + 2q^{16} + 4q^{17} - 2q^{18} + 2q^{20} - 2q^{21} + 2q^{22} + 2q^{23} - 2q^{24} + 2q^{25} - 2q^{26} + 2q^{27} - 2q^{28} + 2q^{29} - 2q^{30} - 2q^{32} - 2q^{33} - 4q^{34} - 2q^{35} + 2q^{36} + 8q^{37} + 2q^{39} - 2q^{40} + 8q^{41} + 2q^{42} - 6q^{43} - 2q^{44} + 2q^{45} - 2q^{46} - 4q^{47} + 2q^{48} + 2q^{49} - 2q^{50} + 4q^{51} + 2q^{52} + 20q^{53} - 2q^{54} - 2q^{55} + 2q^{56} - 2q^{58} + 2q^{60} - 4q^{61} - 2q^{63} + 2q^{64} + 2q^{65} + 2q^{66} - 22q^{67} + 4q^{68} + 2q^{69} + 2q^{70} + 6q^{71} - 2q^{72} + 12q^{73} - 8q^{74} + 2q^{75} + 2q^{77} - 2q^{78} - 8q^{79} + 2q^{80} + 2q^{81} - 8q^{82} + 8q^{83} - 2q^{84} + 4q^{85} + 6q^{86} + 2q^{87} + 2q^{88} + 10q^{89} - 2q^{90} - 2q^{91} + 2q^{92} + 4q^{94} - 2q^{96} - 2q^{97} - 2q^{98} - 2q^{99} + O(q^{100}) \)

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
1.61803
−0.618034
−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.bq 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4830.2.a.bq 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4830))\):

\( T_{11}^{2} + 2 T_{11} - 4 \)
\( T_{13}^{2} - 2 T_{13} - 4 \)
\( T_{17}^{2} - 4 T_{17} - 16 \)
\( T_{19}^{2} - 20 \)
\( T_{29}^{2} - 2 T_{29} - 44 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( -4 + 2 T + T^{2} \)
$13$ \( -4 - 2 T + T^{2} \)
$17$ \( -16 - 4 T + T^{2} \)
$19$ \( -20 + T^{2} \)
$23$ \( ( -1 + T )^{2} \)
$29$ \( -44 - 2 T + T^{2} \)
$31$ \( -20 + T^{2} \)
$37$ \( -4 - 8 T + T^{2} \)
$41$ \( -4 - 8 T + T^{2} \)
$43$ \( 4 + 6 T + T^{2} \)
$47$ \( ( 2 + T )^{2} \)
$53$ \( ( -10 + T )^{2} \)
$59$ \( -80 + T^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( 116 + 22 T + T^{2} \)
$71$ \( -36 - 6 T + T^{2} \)
$73$ \( -44 - 12 T + T^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( -4 - 8 T + T^{2} \)
$89$ \( 20 - 10 T + T^{2} \)
$97$ \( -44 + 2 T + T^{2} \)
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