Properties

Label 1001.2.a.a
Level 1001
Weight 2
Character orbit 1001.a
Self dual Yes
Analytic conductor 7.993
Analytic rank 2
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1001 = 7 \cdot 11 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1001.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(7.99302524233\)
Analytic rank: \(2\)
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

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

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
−2.00000 −3.00000 2.00000 −3.00000 6.00000 −1.00000 0 6.00000 6.00000
\(n\): 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
\(7\) \(1\)
\(11\) \(-1\)
\(13\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1001))\):

\( T_{2} + 2 \)
\( T_{3} + 3 \)