Properties

Label 4730.2.a.h
Level 4730
Weight 2
Character orbit 4730.a
Self dual yes
Analytic conductor 37.769
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

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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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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
0
1.00000 1.00000 1.00000 −1.00000 1.00000 2.00000 1.00000 −2.00000 −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.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.h 1 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4730))\):

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T \)
$3$ \( 1 - T + 3 T^{2} \)
$5$ \( 1 + T \)
$7$ \( 1 - 2 T + 7 T^{2} \)
$11$ \( 1 + T \)
$13$ \( 1 + 4 T + 13 T^{2} \)
$17$ \( 1 + 17 T^{2} \)
$19$ \( 1 - 2 T + 19 T^{2} \)
$23$ \( 1 + 6 T + 23 T^{2} \)
$29$ \( 1 + 9 T + 29 T^{2} \)
$31$ \( 1 + 4 T + 31 T^{2} \)
$37$ \( 1 + 4 T + 37 T^{2} \)
$41$ \( 1 + 41 T^{2} \)
$43$ \( 1 - T \)
$47$ \( 1 + 6 T + 47 T^{2} \)
$53$ \( 1 - 3 T + 53 T^{2} \)
$59$ \( 1 + 59 T^{2} \)
$61$ \( 1 + T + 61 T^{2} \)
$67$ \( 1 - 2 T + 67 T^{2} \)
$71$ \( 1 - 12 T + 71 T^{2} \)
$73$ \( 1 - 11 T + 73 T^{2} \)
$79$ \( 1 + 7 T + 79 T^{2} \)
$83$ \( 1 - 15 T + 83 T^{2} \)
$89$ \( 1 + 89 T^{2} \)
$97$ \( 1 + 13 T + 97 T^{2} \)
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