Properties

Label 4730.2.a.u
Level 4730
Weight 2
Character orbit 4730.a
Self dual yes
Analytic conductor 37.769
Analytic rank 1
Dimension 4
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: \(4\)
Coefficient field: 4.4.23252.1
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 6 x^{2} + 2\):

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

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
−1.88474
−0.634868
0.565882
2.95372
1.00000 −2.88474 1.00000 1.00000 −2.88474 −2.43697 1.00000 5.32171 1.00000
1.2 1.00000 −1.63487 1.00000 1.00000 −1.63487 1.96207 1.00000 −0.327207 1.00000
1.3 1.00000 −0.434118 1.00000 1.00000 −0.434118 3.24566 1.00000 −2.81154 1.00000
1.4 1.00000 1.95372 1.00000 1.00000 1.95372 −2.77076 1.00000 0.817036 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.u 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.u 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4730))\):

\( T_{3}^{4} + 3 T_{3}^{3} - 3 T_{3}^{2} - 11 T_{3} - 4 \)
\( T_{7}^{4} - 14 T_{7}^{2} - 2 T_{7} + 43 \)
\( T_{13}^{4} + 9 T_{13}^{3} + 20 T_{13}^{2} - 6 T_{13} - 22 \)