Properties

Label 99.2.a.d
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: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{2} + 2q^{4} - q^{5} - 2q^{7} + O(q^{10}) \) \( q + 2q^{2} + 2q^{4} - q^{5} - 2q^{7} - 2q^{10} - q^{11} + 4q^{13} - 4q^{14} - 4q^{16} + 2q^{17} - 2q^{20} - 2q^{22} + q^{23} - 4q^{25} + 8q^{26} - 4q^{28} + 7q^{31} - 8q^{32} + 4q^{34} + 2q^{35} + 3q^{37} + 8q^{41} - 6q^{43} - 2q^{44} + 2q^{46} - 8q^{47} - 3q^{49} - 8q^{50} + 8q^{52} + 6q^{53} + q^{55} - 5q^{59} + 12q^{61} + 14q^{62} - 8q^{64} - 4q^{65} - 7q^{67} + 4q^{68} + 4q^{70} + 3q^{71} + 4q^{73} + 6q^{74} + 2q^{77} - 10q^{79} + 4q^{80} + 16q^{82} + 6q^{83} - 2q^{85} - 12q^{86} - 15q^{89} - 8q^{91} + 2q^{92} - 16q^{94} - 7q^{97} - 6q^{98} + O(q^{100}) \)

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
2.00000 0 2.00000 −1.00000 0 −2.00000 0 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.d 1
3.b odd 2 1 11.2.a.a 1
4.b odd 2 1 1584.2.a.g 1
5.b even 2 1 2475.2.a.a 1
5.c odd 4 2 2475.2.c.a 2
7.b odd 2 1 4851.2.a.t 1
8.b even 2 1 6336.2.a.br 1
8.d odd 2 1 6336.2.a.bu 1
9.c even 3 2 891.2.e.b 2
9.d odd 6 2 891.2.e.k 2
11.b odd 2 1 1089.2.a.b 1
12.b even 2 1 176.2.a.b 1
15.d odd 2 1 275.2.a.b 1
15.e even 4 2 275.2.b.a 2
21.c even 2 1 539.2.a.a 1
21.g even 6 2 539.2.e.g 2
21.h odd 6 2 539.2.e.h 2
24.f even 2 1 704.2.a.c 1
24.h odd 2 1 704.2.a.h 1
33.d even 2 1 121.2.a.d 1
33.f even 10 4 121.2.c.a 4
33.h odd 10 4 121.2.c.e 4
39.d odd 2 1 1859.2.a.b 1
48.i odd 4 2 2816.2.c.j 2
48.k even 4 2 2816.2.c.f 2
51.c odd 2 1 3179.2.a.a 1
57.d even 2 1 3971.2.a.b 1
60.h even 2 1 4400.2.a.i 1
60.l odd 4 2 4400.2.b.h 2
69.c even 2 1 5819.2.a.a 1
84.h odd 2 1 8624.2.a.j 1
87.d odd 2 1 9251.2.a.d 1
132.d odd 2 1 1936.2.a.i 1
165.d even 2 1 3025.2.a.a 1
231.h odd 2 1 5929.2.a.h 1
264.m even 2 1 7744.2.a.x 1
264.p odd 2 1 7744.2.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.2.a.a 1 3.b odd 2 1
99.2.a.d 1 1.a even 1 1 trivial
121.2.a.d 1 33.d even 2 1
121.2.c.a 4 33.f even 10 4
121.2.c.e 4 33.h odd 10 4
176.2.a.b 1 12.b even 2 1
275.2.a.b 1 15.d odd 2 1
275.2.b.a 2 15.e even 4 2
539.2.a.a 1 21.c even 2 1
539.2.e.g 2 21.g even 6 2
539.2.e.h 2 21.h odd 6 2
704.2.a.c 1 24.f even 2 1
704.2.a.h 1 24.h odd 2 1
891.2.e.b 2 9.c even 3 2
891.2.e.k 2 9.d odd 6 2
1089.2.a.b 1 11.b odd 2 1
1584.2.a.g 1 4.b odd 2 1
1859.2.a.b 1 39.d odd 2 1
1936.2.a.i 1 132.d odd 2 1
2475.2.a.a 1 5.b even 2 1
2475.2.c.a 2 5.c odd 4 2
2816.2.c.f 2 48.k even 4 2
2816.2.c.j 2 48.i odd 4 2
3025.2.a.a 1 165.d even 2 1
3179.2.a.a 1 51.c odd 2 1
3971.2.a.b 1 57.d even 2 1
4400.2.a.i 1 60.h even 2 1
4400.2.b.h 2 60.l odd 4 2
4851.2.a.t 1 7.b odd 2 1
5819.2.a.a 1 69.c even 2 1
5929.2.a.h 1 231.h odd 2 1
6336.2.a.br 1 8.b even 2 1
6336.2.a.bu 1 8.d odd 2 1
7744.2.a.k 1 264.p odd 2 1
7744.2.a.x 1 264.m even 2 1
8624.2.a.j 1 84.h odd 2 1
9251.2.a.d 1 87.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(99))\):

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 2 T^{2} \)
$3$ 1
$5$ \( 1 + T + 5 T^{2} \)
$7$ \( 1 + 2 T + 7 T^{2} \)
$11$ \( 1 + T \)
$13$ \( 1 - 4 T + 13 T^{2} \)
$17$ \( 1 - 2 T + 17 T^{2} \)
$19$ \( 1 + 19 T^{2} \)
$23$ \( 1 - T + 23 T^{2} \)
$29$ \( 1 + 29 T^{2} \)
$31$ \( 1 - 7 T + 31 T^{2} \)
$37$ \( 1 - 3 T + 37 T^{2} \)
$41$ \( 1 - 8 T + 41 T^{2} \)
$43$ \( 1 + 6 T + 43 T^{2} \)
$47$ \( 1 + 8 T + 47 T^{2} \)
$53$ \( 1 - 6 T + 53 T^{2} \)
$59$ \( 1 + 5 T + 59 T^{2} \)
$61$ \( 1 - 12 T + 61 T^{2} \)
$67$ \( 1 + 7 T + 67 T^{2} \)
$71$ \( 1 - 3 T + 71 T^{2} \)
$73$ \( 1 - 4 T + 73 T^{2} \)
$79$ \( 1 + 10 T + 79 T^{2} \)
$83$ \( 1 - 6 T + 83 T^{2} \)
$89$ \( 1 + 15 T + 89 T^{2} \)
$97$ \( 1 + 7 T + 97 T^{2} \)
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