Properties

Label 4730.2.a.q
Level 4730
Weight 2
Character orbit 4730.a
Self dual yes
Analytic conductor 37.769
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4730.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.7692401561\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
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.

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

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 −1.61803 1.00000 1.00000 −1.61803 −0.381966 1.00000 −0.381966 1.00000
1.2 1.00000 0.618034 1.00000 1.00000 0.618034 −2.61803 1.00000 −2.61803 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 4730.2.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.q 2 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(11\) \(-1\)
\(43\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4730))\):

\( T_{3}^{2} + T_{3} - 1 \)
\( T_{7}^{2} + 3 T_{7} + 1 \)
\( T_{13}^{2} - 20 \)