Properties

Label 2005.2.a.a
Level 2005
Weight 2
Character orbit 2005.a
Self dual Yes
Analytic conductor 16.010
Analytic rank 1
Dimension 1
CM No
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 2005 = 5 \cdot 401 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 2005.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(16.0100056053\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - q^{4} + q^{5} + 3q^{8} - 3q^{9} + O(q^{10}) \) \( q - q^{2} - q^{4} + q^{5} + 3q^{8} - 3q^{9} - q^{10} - 4q^{11} + 4q^{13} - q^{16} + 4q^{17} + 3q^{18} - 6q^{19} - q^{20} + 4q^{22} + 8q^{23} + q^{25} - 4q^{26} - 6q^{29} - 2q^{31} - 5q^{32} - 4q^{34} + 3q^{36} + 4q^{37} + 6q^{38} + 3q^{40} + 2q^{41} - 4q^{43} + 4q^{44} - 3q^{45} - 8q^{46} - 7q^{49} - q^{50} - 4q^{52} - 4q^{53} - 4q^{55} + 6q^{58} - 6q^{59} + 14q^{61} + 2q^{62} + 7q^{64} + 4q^{65} - 12q^{67} - 4q^{68} - 10q^{71} - 9q^{72} - 6q^{73} - 4q^{74} + 6q^{76} + 6q^{79} - q^{80} + 9q^{81} - 2q^{82} + 4q^{83} + 4q^{85} + 4q^{86} - 12q^{88} + 6q^{89} + 3q^{90} - 8q^{92} - 6q^{95} + 7q^{98} + 12q^{99} + O(q^{100}) \)

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 0 −1.00000 1.00000 0 0 3.00000 −3.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(401\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2005))\):

\( T_{2} + 1 \)
\( T_{11} + 4 \)