Properties

Label 4730.2.a.p
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: \(\Q(\sqrt{2}) \)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.41421
1.41421
1.00000 −2.41421 1.00000 −1.00000 −2.41421 0.414214 1.00000 2.82843 −1.00000
1.2 1.00000 0.414214 1.00000 −1.00000 0.414214 −2.41421 1.00000 −2.82843 −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.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.p 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4730))\):

\( T_{3}^{2} + 2 T_{3} - 1 \)
\( T_{7}^{2} + 2 T_{7} - 1 \)
\( T_{13}^{2} - 4 T_{13} + 2 \)