Properties

Label 99.2.a.a
Level 99
Weight 2
Character orbit 99.a
Self dual yes
Analytic conductor 0.791
Analytic rank 1
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - q^{4} - 4q^{5} - 2q^{7} + 3q^{8} + O(q^{10}) \) \( q - q^{2} - q^{4} - 4q^{5} - 2q^{7} + 3q^{8} + 4q^{10} - q^{11} - 2q^{13} + 2q^{14} - q^{16} + 2q^{17} - 6q^{19} + 4q^{20} + q^{22} + 4q^{23} + 11q^{25} + 2q^{26} + 2q^{28} - 6q^{29} + 4q^{31} - 5q^{32} - 2q^{34} + 8q^{35} - 6q^{37} + 6q^{38} - 12q^{40} - 10q^{41} + 6q^{43} + q^{44} - 4q^{46} - 8q^{47} - 3q^{49} - 11q^{50} + 2q^{52} + 4q^{55} - 6q^{56} + 6q^{58} + 4q^{59} - 6q^{61} - 4q^{62} + 7q^{64} + 8q^{65} + 8q^{67} - 2q^{68} - 8q^{70} - 2q^{73} + 6q^{74} + 6q^{76} + 2q^{77} - 10q^{79} + 4q^{80} + 10q^{82} + 12q^{83} - 8q^{85} - 6q^{86} - 3q^{88} + 4q^{91} - 4q^{92} + 8q^{94} + 24q^{95} + 2q^{97} + 3q^{98} + O(q^{100}) \)

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 −4.00000 0 −2.00000 3.00000 0 4.00000
\(n\): 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.a 1
3.b odd 2 1 99.2.a.c yes 1
4.b odd 2 1 1584.2.a.b 1
5.b even 2 1 2475.2.a.j 1
5.c odd 4 2 2475.2.c.b 2
7.b odd 2 1 4851.2.a.g 1
8.b even 2 1 6336.2.a.cl 1
8.d odd 2 1 6336.2.a.cm 1
9.c even 3 2 891.2.e.j 2
9.d odd 6 2 891.2.e.c 2
11.b odd 2 1 1089.2.a.h 1
12.b even 2 1 1584.2.a.r 1
15.d odd 2 1 2475.2.a.c 1
15.e even 4 2 2475.2.c.g 2
21.c even 2 1 4851.2.a.o 1
24.f even 2 1 6336.2.a.f 1
24.h odd 2 1 6336.2.a.b 1
33.d even 2 1 1089.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.a.a 1 1.a even 1 1 trivial
99.2.a.c yes 1 3.b odd 2 1
891.2.e.c 2 9.d odd 6 2
891.2.e.j 2 9.c even 3 2
1089.2.a.d 1 33.d even 2 1
1089.2.a.h 1 11.b odd 2 1
1584.2.a.b 1 4.b odd 2 1
1584.2.a.r 1 12.b even 2 1
2475.2.a.c 1 15.d odd 2 1
2475.2.a.j 1 5.b even 2 1
2475.2.c.b 2 5.c odd 4 2
2475.2.c.g 2 15.e even 4 2
4851.2.a.g 1 7.b odd 2 1
4851.2.a.o 1 21.c even 2 1
6336.2.a.b 1 24.h odd 2 1
6336.2.a.f 1 24.f even 2 1
6336.2.a.cl 1 8.b even 2 1
6336.2.a.cm 1 8.d 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(99))\):

\( T_{2} + 1 \)
\( T_{5} + 4 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 2 T^{2} \)
$3$ \( \)
$5$ \( 1 + 4 T + 5 T^{2} \)
$7$ \( 1 + 2 T + 7 T^{2} \)
$11$ \( 1 + T \)
$13$ \( 1 + 2 T + 13 T^{2} \)
$17$ \( 1 - 2 T + 17 T^{2} \)
$19$ \( 1 + 6 T + 19 T^{2} \)
$23$ \( 1 - 4 T + 23 T^{2} \)
$29$ \( 1 + 6 T + 29 T^{2} \)
$31$ \( 1 - 4 T + 31 T^{2} \)
$37$ \( 1 + 6 T + 37 T^{2} \)
$41$ \( 1 + 10 T + 41 T^{2} \)
$43$ \( 1 - 6 T + 43 T^{2} \)
$47$ \( 1 + 8 T + 47 T^{2} \)
$53$ \( 1 + 53 T^{2} \)
$59$ \( 1 - 4 T + 59 T^{2} \)
$61$ \( 1 + 6 T + 61 T^{2} \)
$67$ \( 1 - 8 T + 67 T^{2} \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 + 2 T + 73 T^{2} \)
$79$ \( 1 + 10 T + 79 T^{2} \)
$83$ \( 1 - 12 T + 83 T^{2} \)
$89$ \( 1 + 89 T^{2} \)
$97$ \( 1 - 2 T + 97 T^{2} \)
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