Properties

Label 99.6.a.b
Level $99$
Weight $6$
Character orbit 99.a
Self dual yes
Analytic conductor $15.878$
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] = [99,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(15.8779981615\)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{2} - 28 q^{4} - 46 q^{5} + 148 q^{7} - 120 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - 28 q^{4} - 46 q^{5} + 148 q^{7} - 120 q^{8} - 92 q^{10} - 121 q^{11} + 574 q^{13} + 296 q^{14} + 656 q^{16} + 722 q^{17} + 2160 q^{19} + 1288 q^{20} - 242 q^{22} + 2536 q^{23} - 1009 q^{25} + 1148 q^{26} - 4144 q^{28} - 4650 q^{29} + 5032 q^{31} + 5152 q^{32} + 1444 q^{34} - 6808 q^{35} + 8118 q^{37} + 4320 q^{38} + 5520 q^{40} + 5138 q^{41} + 8304 q^{43} + 3388 q^{44} + 5072 q^{46} - 24728 q^{47} + 5097 q^{49} - 2018 q^{50} - 16072 q^{52} + 28746 q^{53} + 5566 q^{55} - 17760 q^{56} - 9300 q^{58} + 5860 q^{59} - 53658 q^{61} + 10064 q^{62} - 10688 q^{64} - 26404 q^{65} + 30908 q^{67} - 20216 q^{68} - 13616 q^{70} + 69648 q^{71} - 18446 q^{73} + 16236 q^{74} - 60480 q^{76} - 17908 q^{77} - 25300 q^{79} - 30176 q^{80} + 10276 q^{82} + 17556 q^{83} - 33212 q^{85} + 16608 q^{86} + 14520 q^{88} - 132570 q^{89} + 84952 q^{91} - 71008 q^{92} - 49456 q^{94} - 99360 q^{95} + 70658 q^{97} + 10194 q^{98}+O(q^{100}) \) 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
2.00000 0 −28.0000 −46.0000 0 148.000 −120.000 0 −92.0000
\(n\): 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 99.6.a.b 1
3.b odd 2 1 33.6.a.a 1
11.b odd 2 1 1089.6.a.d 1
12.b even 2 1 528.6.a.i 1
15.d odd 2 1 825.6.a.b 1
33.d even 2 1 363.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.6.a.a 1 3.b odd 2 1
99.6.a.b 1 1.a even 1 1 trivial
363.6.a.c 1 33.d even 2 1
528.6.a.i 1 12.b even 2 1
825.6.a.b 1 15.d odd 2 1
1089.6.a.d 1 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 2 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(99))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 46 \) Copy content Toggle raw display
$7$ \( T - 148 \) Copy content Toggle raw display
$11$ \( T + 121 \) Copy content Toggle raw display
$13$ \( T - 574 \) Copy content Toggle raw display
$17$ \( T - 722 \) Copy content Toggle raw display
$19$ \( T - 2160 \) Copy content Toggle raw display
$23$ \( T - 2536 \) Copy content Toggle raw display
$29$ \( T + 4650 \) Copy content Toggle raw display
$31$ \( T - 5032 \) Copy content Toggle raw display
$37$ \( T - 8118 \) Copy content Toggle raw display
$41$ \( T - 5138 \) Copy content Toggle raw display
$43$ \( T - 8304 \) Copy content Toggle raw display
$47$ \( T + 24728 \) Copy content Toggle raw display
$53$ \( T - 28746 \) Copy content Toggle raw display
$59$ \( T - 5860 \) Copy content Toggle raw display
$61$ \( T + 53658 \) Copy content Toggle raw display
$67$ \( T - 30908 \) Copy content Toggle raw display
$71$ \( T - 69648 \) Copy content Toggle raw display
$73$ \( T + 18446 \) Copy content Toggle raw display
$79$ \( T + 25300 \) Copy content Toggle raw display
$83$ \( T - 17556 \) Copy content Toggle raw display
$89$ \( T + 132570 \) Copy content Toggle raw display
$97$ \( T - 70658 \) Copy content Toggle raw display
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