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

Related objects

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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: \( 1 \)
Twist minimal: no (minimal twist has level 33)
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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.2.a.b 1
3.b odd 2 1 33.2.a.a 1
4.b odd 2 1 1584.2.a.o 1
5.b even 2 1 2475.2.a.g 1
5.c odd 4 2 2475.2.c.d 2
7.b odd 2 1 4851.2.a.b 1
8.b even 2 1 6336.2.a.x 1
8.d odd 2 1 6336.2.a.n 1
9.c even 3 2 891.2.e.g 2
9.d odd 6 2 891.2.e.e 2
11.b odd 2 1 1089.2.a.j 1
12.b even 2 1 528.2.a.g 1
15.d odd 2 1 825.2.a.a 1
15.e even 4 2 825.2.c.a 2
21.c even 2 1 1617.2.a.j 1
24.f even 2 1 2112.2.a.j 1
24.h odd 2 1 2112.2.a.bb 1
33.d even 2 1 363.2.a.b 1
33.f even 10 4 363.2.e.g 4
33.h odd 10 4 363.2.e.e 4
39.d odd 2 1 5577.2.a.a 1
51.c odd 2 1 9537.2.a.m 1
132.d odd 2 1 5808.2.a.t 1
165.d even 2 1 9075.2.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.a.a 1 3.b odd 2 1
99.2.a.b 1 1.a even 1 1 trivial
363.2.a.b 1 33.d even 2 1
363.2.e.e 4 33.h odd 10 4
363.2.e.g 4 33.f even 10 4
528.2.a.g 1 12.b even 2 1
825.2.a.a 1 15.d odd 2 1
825.2.c.a 2 15.e even 4 2
891.2.e.e 2 9.d odd 6 2
891.2.e.g 2 9.c even 3 2
1089.2.a.j 1 11.b odd 2 1
1584.2.a.o 1 4.b odd 2 1
1617.2.a.j 1 21.c even 2 1
2112.2.a.j 1 24.f even 2 1
2112.2.a.bb 1 24.h odd 2 1
2475.2.a.g 1 5.b even 2 1
2475.2.c.d 2 5.c odd 4 2
4851.2.a.b 1 7.b odd 2 1
5577.2.a.a 1 39.d odd 2 1
5808.2.a.t 1 132.d odd 2 1
6336.2.a.n 1 8.d odd 2 1
6336.2.a.x 1 8.b even 2 1
9075.2.a.q 1 165.d even 2 1
9537.2.a.m 1 51.c odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(1\)

Hecke kernels

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

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