Properties

Label 4730.2.a.v
Level 4730
Weight 2
Character orbit 4730.a
Self dual yes
Analytic conductor 37.769
Analytic rank 1
Dimension 5
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: \(5\)
Coefficient field: 5.5.220036.1
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 a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 5 x^{3} + 11 x^{2} - 2 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - 5 \nu + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - \nu^{3} - 5 \nu^{2} + 6 \nu - 1 \)
\(\beta_{4}\)\(=\)\( -\nu^{4} + \nu^{3} + 6 \nu^{2} - 6 \nu - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{2} + 5 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(5 \beta_{4} + 6 \beta_{3} + \beta_{2} - \beta_{1} + 15\)

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
−2.32118
−0.217250
0.493132
1.75849
2.28680
1.00000 −2.32118 1.00000 −1.00000 −2.32118 −3.71898 1.00000 2.38786 −1.00000
1.2 1.00000 −0.217250 1.00000 −1.00000 −0.217250 −0.574201 1.00000 −2.95280 −1.00000
1.3 1.00000 0.493132 1.00000 −1.00000 0.493132 2.43893 1.00000 −2.75682 −1.00000
1.4 1.00000 1.75849 1.00000 −1.00000 1.75849 −2.87831 1.00000 0.0923014 −1.00000
1.5 1.00000 2.28680 1.00000 −1.00000 2.28680 −1.26744 1.00000 2.22946 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.v 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.v 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4730))\):

\( T_{3}^{5} - 2 T_{3}^{4} - 5 T_{3}^{3} + 11 T_{3}^{2} - 2 T_{3} - 1 \)
\( T_{7}^{5} + 6 T_{7}^{4} + 3 T_{7}^{3} - 33 T_{7}^{2} - 52 T_{7} - 19 \)
\( T_{13}^{5} + 12 T_{13}^{4} + 42 T_{13}^{3} + 30 T_{13}^{2} - 40 T_{13} + 8 \)