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

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1001))\):

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