Properties

Label 4009.2.a.a
Level 4009
Weight 2
Character orbit 4009.a
Self dual yes
Analytic conductor 32.012
Analytic rank 0
Dimension 1
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 4009 = 19 \cdot 211 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4009.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0120261703\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q - q^{2} + 2q^{3} - q^{4} + 3q^{5} - 2q^{6} + 3q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + 2q^{3} - q^{4} + 3q^{5} - 2q^{6} + 3q^{8} + q^{9} - 3q^{10} - 2q^{12} + 4q^{13} + 6q^{15} - q^{16} - q^{18} - q^{19} - 3q^{20} - q^{23} + 6q^{24} + 4q^{25} - 4q^{26} - 4q^{27} + 3q^{29} - 6q^{30} + 4q^{31} - 5q^{32} - q^{36} + q^{38} + 8q^{39} + 9q^{40} + 10q^{41} + 6q^{43} + 3q^{45} + q^{46} - 2q^{48} - 7q^{49} - 4q^{50} - 4q^{52} + 6q^{53} + 4q^{54} - 2q^{57} - 3q^{58} + 11q^{59} - 6q^{60} - 4q^{62} + 7q^{64} + 12q^{65} + 2q^{67} - 2q^{69} - 9q^{71} + 3q^{72} - 15q^{73} + 8q^{75} + q^{76} - 8q^{78} + 5q^{79} - 3q^{80} - 11q^{81} - 10q^{82} - 6q^{86} + 6q^{87} + 9q^{89} - 3q^{90} + q^{92} + 8q^{93} - 3q^{95} - 10q^{96} + 11q^{97} + 7q^{98} + O(q^{100}) \)

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
0
−1.00000 2.00000 −1.00000 3.00000 −2.00000 0 3.00000 1.00000 −3.00000
\(n\): 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 4009.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4009.2.a.a 1 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(19\) \(1\)
\(211\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4009))\).