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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [99,2,Mod(1,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 99.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{2} - q^{4} + 2 q^{5} + 4 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{4} + 2 q^{5} + 4 q^{7} + 3 q^{8} - 2 q^{10} - q^{11} - 2 q^{13} - 4 q^{14} - q^{16} + 2 q^{17} - 2 q^{20} + q^{22} - 8 q^{23} - q^{25} + 2 q^{26} - 4 q^{28} + 6 q^{29} - 8 q^{31} - 5 q^{32} - 2 q^{34} + 8 q^{35} + 6 q^{37} + 6 q^{40} + 2 q^{41} + q^{44} + 8 q^{46} - 8 q^{47} + 9 q^{49} + q^{50} + 2 q^{52} - 6 q^{53} - 2 q^{55} + 12 q^{56} - 6 q^{58} + 4 q^{59} + 6 q^{61} + 8 q^{62} + 7 q^{64} - 4 q^{65} - 4 q^{67} - 2 q^{68} - 8 q^{70} - 14 q^{73} - 6 q^{74} - 4 q^{77} - 4 q^{79} - 2 q^{80} - 2 q^{82} - 12 q^{83} + 4 q^{85} - 3 q^{88} + 6 q^{89} - 8 q^{91} + 8 q^{92} + 8 q^{94} + 2 q^{97} - 9 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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:

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.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

Hecke kernels

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

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{5} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T - 4 \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T + 8 \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T + 8 \) Copy content Toggle raw display
$37$ \( T - 6 \) Copy content Toggle raw display
$41$ \( T - 2 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T + 8 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T - 4 \) Copy content Toggle raw display
$61$ \( T - 6 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 14 \) Copy content Toggle raw display
$79$ \( T + 4 \) Copy content Toggle raw display
$83$ \( T + 12 \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T - 2 \) Copy content Toggle raw display
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