Properties

Label 89.2.a.b
Level 89
Weight 2
Character orbit 89.a
Self dual Yes
Analytic conductor 0.711
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 89 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 89.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(0.710668577989\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q + q^{2} + 2q^{3} - q^{4} - 2q^{5} + 2q^{6} + 2q^{7} - 3q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} + 2q^{3} - q^{4} - 2q^{5} + 2q^{6} + 2q^{7} - 3q^{8} + q^{9} - 2q^{10} - 4q^{11} - 2q^{12} + 2q^{13} + 2q^{14} - 4q^{15} - q^{16} + 6q^{17} + q^{18} - 2q^{19} + 2q^{20} + 4q^{21} - 4q^{22} + 2q^{23} - 6q^{24} - q^{25} + 2q^{26} - 4q^{27} - 2q^{28} - 6q^{29} - 4q^{30} + 6q^{31} + 5q^{32} - 8q^{33} + 6q^{34} - 4q^{35} - q^{36} + 10q^{37} - 2q^{38} + 4q^{39} + 6q^{40} - 6q^{41} + 4q^{42} + 2q^{43} + 4q^{44} - 2q^{45} + 2q^{46} + 12q^{47} - 2q^{48} - 3q^{49} - q^{50} + 12q^{51} - 2q^{52} - 6q^{53} - 4q^{54} + 8q^{55} - 6q^{56} - 4q^{57} - 6q^{58} - 10q^{59} + 4q^{60} - 6q^{61} + 6q^{62} + 2q^{63} + 7q^{64} - 4q^{65} - 8q^{66} + 12q^{67} - 6q^{68} + 4q^{69} - 4q^{70} + 4q^{71} - 3q^{72} + 10q^{73} + 10q^{74} - 2q^{75} + 2q^{76} - 8q^{77} + 4q^{78} - 12q^{79} + 2q^{80} - 11q^{81} - 6q^{82} - 6q^{83} - 4q^{84} - 12q^{85} + 2q^{86} - 12q^{87} + 12q^{88} + q^{89} - 2q^{90} + 4q^{91} - 2q^{92} + 12q^{93} + 12q^{94} + 4q^{95} + 10q^{96} - 18q^{97} - 3q^{98} - 4q^{99} + O(q^{100}) \)

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 2.00000 −1.00000 −2.00000 2.00000 2.00000 −3.00000 1.00000 −2.00000
\(n\): 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
\(89\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(89))\).