Properties

Label 4730.2.a.o
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: \(\Q(\sqrt{13}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.30278
2.30278
−1.00000 −1.30278 1.00000 −1.00000 1.30278 1.30278 −1.00000 −1.30278 1.00000
1.2 −1.00000 2.30278 1.00000 −1.00000 −2.30278 −2.30278 −1.00000 2.30278 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.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.o 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4730))\):

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