Properties

Label 4730.2.a.n
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: \(\Q(\sqrt{3}) \)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.73205
1.73205
−1.00000 −1.73205 1.00000 −1.00000 1.73205 1.73205 −1.00000 0 1.00000
1.2 −1.00000 1.73205 1.00000 −1.00000 −1.73205 −1.73205 −1.00000 0 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.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.n 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4730))\):

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