Properties

Label 4730.2.a.l
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: \(\Q(\sqrt{17}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

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

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
2.56155
−1.56155
−1.00000 −2.56155 1.00000 1.00000 2.56155 −3.00000 −1.00000 3.56155 −1.00000
1.2 −1.00000 1.56155 1.00000 1.00000 −1.56155 −3.00000 −1.00000 −0.561553 −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.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.l 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4730))\):

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