Properties

Label 4730.2.a.t
Level 4730
Weight 2
Character orbit 4730.a
Self dual yes
Analytic conductor 37.769
Analytic rank 1
Dimension 3
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: \(3\)
Coefficient field: 3.3.1373.1
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 8 x - 5\):

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

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
−2.43931
−0.661120
3.10043
−1.00000 −2.43931 1.00000 1.00000 2.43931 2.43931 −1.00000 2.95024 −1.00000
1.2 −1.00000 −0.661120 1.00000 1.00000 0.661120 0.661120 −1.00000 −2.56292 −1.00000
1.3 −1.00000 3.10043 1.00000 1.00000 −3.10043 −3.10043 −1.00000 6.61268 −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.t 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.t 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4730))\):

\( T_{3}^{3} - 8 T_{3} - 5 \)
\( T_{7}^{3} - 8 T_{7} + 5 \)
\( T_{13} + 4 \)