Properties

Label 99.2.a.b
Level 99
Weight 2
Character orbit 99.a
Self dual Yes
Analytic conductor 0.791
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 99.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(0.790518980011\)
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

\(f(q)\) \(=\) \( q - q^{2} - q^{4} + 2q^{5} + 4q^{7} + 3q^{8} + O(q^{10}) \) \( q - q^{2} - q^{4} + 2q^{5} + 4q^{7} + 3q^{8} - 2q^{10} - q^{11} - 2q^{13} - 4q^{14} - q^{16} + 2q^{17} - 2q^{20} + q^{22} - 8q^{23} - q^{25} + 2q^{26} - 4q^{28} + 6q^{29} - 8q^{31} - 5q^{32} - 2q^{34} + 8q^{35} + 6q^{37} + 6q^{40} + 2q^{41} + q^{44} + 8q^{46} - 8q^{47} + 9q^{49} + q^{50} + 2q^{52} - 6q^{53} - 2q^{55} + 12q^{56} - 6q^{58} + 4q^{59} + 6q^{61} + 8q^{62} + 7q^{64} - 4q^{65} - 4q^{67} - 2q^{68} - 8q^{70} - 14q^{73} - 6q^{74} - 4q^{77} - 4q^{79} - 2q^{80} - 2q^{82} - 12q^{83} + 4q^{85} - 3q^{88} + 6q^{89} - 8q^{91} + 8q^{92} + 8q^{94} + 2q^{97} - 9q^{98} + O(q^{100}) \)

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 0 −1.00000 2.00000 0 4.00000 3.00000 0 −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
\(3\) \(-1\)
\(11\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(99))\):

\( T_{2} + 1 \)
\( T_{5} - 2 \)