Properties

Label 99.2.a.c
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: 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:

Atkin-Lehner signs

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

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.c yes 1
3.b odd 2 1 99.2.a.a 1
4.b odd 2 1 1584.2.a.r 1
5.b even 2 1 2475.2.a.c 1
5.c odd 4 2 2475.2.c.g 2
7.b odd 2 1 4851.2.a.o 1
8.b even 2 1 6336.2.a.b 1
8.d odd 2 1 6336.2.a.f 1
9.c even 3 2 891.2.e.c 2
9.d odd 6 2 891.2.e.j 2
11.b odd 2 1 1089.2.a.d 1
12.b even 2 1 1584.2.a.b 1
15.d odd 2 1 2475.2.a.j 1
15.e even 4 2 2475.2.c.b 2
21.c even 2 1 4851.2.a.g 1
24.f even 2 1 6336.2.a.cm 1
24.h odd 2 1 6336.2.a.cl 1
33.d even 2 1 1089.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.a.a 1 3.b odd 2 1
99.2.a.c yes 1 1.a even 1 1 trivial
891.2.e.c 2 9.c even 3 2
891.2.e.j 2 9.d odd 6 2
1089.2.a.d 1 11.b odd 2 1
1089.2.a.h 1 33.d even 2 1
1584.2.a.b 1 12.b even 2 1
1584.2.a.r 1 4.b odd 2 1
2475.2.a.c 1 5.b even 2 1
2475.2.a.j 1 15.d odd 2 1
2475.2.c.b 2 15.e even 4 2
2475.2.c.g 2 5.c odd 4 2
4851.2.a.g 1 21.c even 2 1
4851.2.a.o 1 7.b odd 2 1
6336.2.a.b 1 8.b even 2 1
6336.2.a.f 1 8.d odd 2 1
6336.2.a.cl 1 24.h odd 2 1
6336.2.a.cm 1 24.f even 2 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

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