Properties

Label 198.6.a.i
Level $198$
Weight $6$
Character orbit 198.a
Self dual yes
Analytic conductor $31.756$
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] = [198,6,Mod(1,198)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(198, 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("198.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 198 = 2 \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 198.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: \(31.7559963230\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 22)
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)
 
\(f(q)\) \(=\) \( q + 4 q^{2} + 16 q^{4} + 51 q^{5} - 166 q^{7} + 64 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 16 q^{4} + 51 q^{5} - 166 q^{7} + 64 q^{8} + 204 q^{10} + 121 q^{11} + 692 q^{13} - 664 q^{14} + 256 q^{16} + 738 q^{17} + 1424 q^{19} + 816 q^{20} + 484 q^{22} + 1779 q^{23} - 524 q^{25} + 2768 q^{26} - 2656 q^{28} + 2064 q^{29} + 6245 q^{31} + 1024 q^{32} + 2952 q^{34} - 8466 q^{35} - 14785 q^{37} + 5696 q^{38} + 3264 q^{40} - 5304 q^{41} + 17798 q^{43} + 1936 q^{44} + 7116 q^{46} + 17184 q^{47} + 10749 q^{49} - 2096 q^{50} + 11072 q^{52} + 30726 q^{53} + 6171 q^{55} - 10624 q^{56} + 8256 q^{58} + 34989 q^{59} - 45940 q^{61} + 24980 q^{62} + 4096 q^{64} + 35292 q^{65} + 25343 q^{67} + 11808 q^{68} - 33864 q^{70} - 13311 q^{71} - 53260 q^{73} - 59140 q^{74} + 22784 q^{76} - 20086 q^{77} + 77234 q^{79} + 13056 q^{80} - 21216 q^{82} - 55014 q^{83} + 37638 q^{85} + 71192 q^{86} + 7744 q^{88} - 125415 q^{89} - 114872 q^{91} + 28464 q^{92} + 68736 q^{94} + 72624 q^{95} - 88807 q^{97} + 42996 q^{98}+O(q^{100}) \) 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
4.00000 0 16.0000 51.0000 0 −166.000 64.0000 0 204.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(11\) \(-1\)

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 198.6.a.i 1
3.b odd 2 1 22.6.a.b 1
12.b even 2 1 176.6.a.b 1
15.d odd 2 1 550.6.a.f 1
15.e even 4 2 550.6.b.f 2
21.c even 2 1 1078.6.a.a 1
24.f even 2 1 704.6.a.f 1
24.h odd 2 1 704.6.a.e 1
33.d even 2 1 242.6.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.6.a.b 1 3.b odd 2 1
176.6.a.b 1 12.b even 2 1
198.6.a.i 1 1.a even 1 1 trivial
242.6.a.d 1 33.d even 2 1
550.6.a.f 1 15.d odd 2 1
550.6.b.f 2 15.e even 4 2
704.6.a.e 1 24.h odd 2 1
704.6.a.f 1 24.f even 2 1
1078.6.a.a 1 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(198))\):

\( T_{5} - 51 \) Copy content Toggle raw display
\( T_{7} + 166 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 4 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 51 \) Copy content Toggle raw display
$7$ \( T + 166 \) Copy content Toggle raw display
$11$ \( T - 121 \) Copy content Toggle raw display
$13$ \( T - 692 \) Copy content Toggle raw display
$17$ \( T - 738 \) Copy content Toggle raw display
$19$ \( T - 1424 \) Copy content Toggle raw display
$23$ \( T - 1779 \) Copy content Toggle raw display
$29$ \( T - 2064 \) Copy content Toggle raw display
$31$ \( T - 6245 \) Copy content Toggle raw display
$37$ \( T + 14785 \) Copy content Toggle raw display
$41$ \( T + 5304 \) Copy content Toggle raw display
$43$ \( T - 17798 \) Copy content Toggle raw display
$47$ \( T - 17184 \) Copy content Toggle raw display
$53$ \( T - 30726 \) Copy content Toggle raw display
$59$ \( T - 34989 \) Copy content Toggle raw display
$61$ \( T + 45940 \) Copy content Toggle raw display
$67$ \( T - 25343 \) Copy content Toggle raw display
$71$ \( T + 13311 \) Copy content Toggle raw display
$73$ \( T + 53260 \) Copy content Toggle raw display
$79$ \( T - 77234 \) Copy content Toggle raw display
$83$ \( T + 55014 \) Copy content Toggle raw display
$89$ \( T + 125415 \) Copy content Toggle raw display
$97$ \( T + 88807 \) Copy content Toggle raw display
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