Properties

Label 162.6.a.b
Level $162$
Weight $6$
Character orbit 162.a
Self dual yes
Analytic conductor $25.982$
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] = [162,6,Mod(1,162)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(162, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 162.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: \(25.9821788097\)
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 4 q^{2} + 16 q^{4} + 21 q^{5} + 74 q^{7} + 64 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 16 q^{4} + 21 q^{5} + 74 q^{7} + 64 q^{8} + 84 q^{10} + 270 q^{11} - 115 q^{13} + 296 q^{14} + 256 q^{16} - 861 q^{17} + 1850 q^{19} + 336 q^{20} + 1080 q^{22} + 3618 q^{23} - 2684 q^{25} - 460 q^{26} + 1184 q^{28} + 1125 q^{29} + 5228 q^{31} + 1024 q^{32} - 3444 q^{34} + 1554 q^{35} + 9917 q^{37} + 7400 q^{38} + 1344 q^{40} + 10758 q^{41} - 19714 q^{43} + 4320 q^{44} + 14472 q^{46} + 9984 q^{47} - 11331 q^{49} - 10736 q^{50} - 1840 q^{52} + 36726 q^{53} + 5670 q^{55} + 4736 q^{56} + 4500 q^{58} + 26460 q^{59} - 53779 q^{61} + 20912 q^{62} + 4096 q^{64} - 2415 q^{65} - 12934 q^{67} - 13776 q^{68} + 6216 q^{70} - 4254 q^{71} - 17521 q^{73} + 39668 q^{74} + 29600 q^{76} + 19980 q^{77} - 36946 q^{79} + 5376 q^{80} + 43032 q^{82} - 76416 q^{83} - 18081 q^{85} - 78856 q^{86} + 17280 q^{88} - 45357 q^{89} - 8510 q^{91} + 57888 q^{92} + 39936 q^{94} + 38850 q^{95} + 127574 q^{97} - 45324 q^{98}+O(q^{100}) \) 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 21.0000 0 74.0000 64.0000 0 84.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 162.6.a.b yes 1
3.b odd 2 1 162.6.a.a 1
9.c even 3 2 162.6.c.d 2
9.d odd 6 2 162.6.c.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.6.a.a 1 3.b odd 2 1
162.6.a.b yes 1 1.a even 1 1 trivial
162.6.c.d 2 9.c even 3 2
162.6.c.i 2 9.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 21 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(162))\). 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
$5$ \( T - 21 \) Copy content Toggle raw display
$7$ \( T - 74 \) Copy content Toggle raw display
$11$ \( T - 270 \) Copy content Toggle raw display
$13$ \( T + 115 \) Copy content Toggle raw display
$17$ \( T + 861 \) Copy content Toggle raw display
$19$ \( T - 1850 \) Copy content Toggle raw display
$23$ \( T - 3618 \) Copy content Toggle raw display
$29$ \( T - 1125 \) Copy content Toggle raw display
$31$ \( T - 5228 \) Copy content Toggle raw display
$37$ \( T - 9917 \) Copy content Toggle raw display
$41$ \( T - 10758 \) Copy content Toggle raw display
$43$ \( T + 19714 \) Copy content Toggle raw display
$47$ \( T - 9984 \) Copy content Toggle raw display
$53$ \( T - 36726 \) Copy content Toggle raw display
$59$ \( T - 26460 \) Copy content Toggle raw display
$61$ \( T + 53779 \) Copy content Toggle raw display
$67$ \( T + 12934 \) Copy content Toggle raw display
$71$ \( T + 4254 \) Copy content Toggle raw display
$73$ \( T + 17521 \) Copy content Toggle raw display
$79$ \( T + 36946 \) Copy content Toggle raw display
$83$ \( T + 76416 \) Copy content Toggle raw display
$89$ \( T + 45357 \) Copy content Toggle raw display
$97$ \( T - 127574 \) Copy content Toggle raw display
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