Properties

Label 162.2.c.b
Level $162$
Weight $2$
Character orbit 162.c
Analytic conductor $1.294$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [162,2,Mod(55,162)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(162, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.55");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 162.c (of order \(3\), degree \(2\), not minimal)

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: no
Analytic conductor: \(1.29357651274\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 54)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + 3 \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} + q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + 3 \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} + q^{8} - 3 q^{10} + (3 \zeta_{6} - 3) q^{11} + 4 \zeta_{6} q^{13} + \zeta_{6} q^{14} + (\zeta_{6} - 1) q^{16} + 2 q^{19} + ( - 3 \zeta_{6} + 3) q^{20} - 3 \zeta_{6} q^{22} - 6 \zeta_{6} q^{23} + (4 \zeta_{6} - 4) q^{25} - 4 q^{26} - q^{28} + ( - 6 \zeta_{6} + 6) q^{29} - 5 \zeta_{6} q^{31} - \zeta_{6} q^{32} + 3 q^{35} + 2 q^{37} + (2 \zeta_{6} - 2) q^{38} + 3 \zeta_{6} q^{40} - 6 \zeta_{6} q^{41} + ( - 10 \zeta_{6} + 10) q^{43} + 3 q^{44} + 6 q^{46} + ( - 6 \zeta_{6} + 6) q^{47} + 6 \zeta_{6} q^{49} - 4 \zeta_{6} q^{50} + ( - 4 \zeta_{6} + 4) q^{52} - 9 q^{53} - 9 q^{55} + ( - \zeta_{6} + 1) q^{56} + 6 \zeta_{6} q^{58} + 12 \zeta_{6} q^{59} + (8 \zeta_{6} - 8) q^{61} + 5 q^{62} + q^{64} + (12 \zeta_{6} - 12) q^{65} - 14 \zeta_{6} q^{67} + (3 \zeta_{6} - 3) q^{70} - 7 q^{73} + (2 \zeta_{6} - 2) q^{74} - 2 \zeta_{6} q^{76} + 3 \zeta_{6} q^{77} + (8 \zeta_{6} - 8) q^{79} - 3 q^{80} + 6 q^{82} + (3 \zeta_{6} - 3) q^{83} + 10 \zeta_{6} q^{86} + (3 \zeta_{6} - 3) q^{88} + 18 q^{89} + 4 q^{91} + (6 \zeta_{6} - 6) q^{92} + 6 \zeta_{6} q^{94} + 6 \zeta_{6} q^{95} + ( - \zeta_{6} + 1) q^{97} - 6 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} + 3 q^{5} + q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{4} + 3 q^{5} + q^{7} + 2 q^{8} - 6 q^{10} - 3 q^{11} + 4 q^{13} + q^{14} - q^{16} + 4 q^{19} + 3 q^{20} - 3 q^{22} - 6 q^{23} - 4 q^{25} - 8 q^{26} - 2 q^{28} + 6 q^{29} - 5 q^{31} - q^{32} + 6 q^{35} + 4 q^{37} - 2 q^{38} + 3 q^{40} - 6 q^{41} + 10 q^{43} + 6 q^{44} + 12 q^{46} + 6 q^{47} + 6 q^{49} - 4 q^{50} + 4 q^{52} - 18 q^{53} - 18 q^{55} + q^{56} + 6 q^{58} + 12 q^{59} - 8 q^{61} + 10 q^{62} + 2 q^{64} - 12 q^{65} - 14 q^{67} - 3 q^{70} - 14 q^{73} - 2 q^{74} - 2 q^{76} + 3 q^{77} - 8 q^{79} - 6 q^{80} + 12 q^{82} - 3 q^{83} + 10 q^{86} - 3 q^{88} + 36 q^{89} + 8 q^{91} - 6 q^{92} + 6 q^{94} + 6 q^{95} + q^{97} - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 1.50000 + 2.59808i 0 0.500000 0.866025i 1.00000 0 −3.00000
109.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.50000 2.59808i 0 0.500000 + 0.866025i 1.00000 0 −3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.2.c.b 2
3.b odd 2 1 162.2.c.c 2
4.b odd 2 1 1296.2.i.o 2
9.c even 3 1 54.2.a.b yes 1
9.c even 3 1 inner 162.2.c.b 2
9.d odd 6 1 54.2.a.a 1
9.d odd 6 1 162.2.c.c 2
12.b even 2 1 1296.2.i.c 2
36.f odd 6 1 432.2.a.b 1
36.f odd 6 1 1296.2.i.o 2
36.h even 6 1 432.2.a.g 1
36.h even 6 1 1296.2.i.c 2
45.h odd 6 1 1350.2.a.r 1
45.j even 6 1 1350.2.a.h 1
45.k odd 12 2 1350.2.c.k 2
45.l even 12 2 1350.2.c.b 2
63.l odd 6 1 2646.2.a.bd 1
63.o even 6 1 2646.2.a.a 1
72.j odd 6 1 1728.2.a.c 1
72.l even 6 1 1728.2.a.d 1
72.n even 6 1 1728.2.a.y 1
72.p odd 6 1 1728.2.a.z 1
99.g even 6 1 6534.2.a.bc 1
99.h odd 6 1 6534.2.a.b 1
117.n odd 6 1 9126.2.a.u 1
117.t even 6 1 9126.2.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.2.a.a 1 9.d odd 6 1
54.2.a.b yes 1 9.c even 3 1
162.2.c.b 2 1.a even 1 1 trivial
162.2.c.b 2 9.c even 3 1 inner
162.2.c.c 2 3.b odd 2 1
162.2.c.c 2 9.d odd 6 1
432.2.a.b 1 36.f odd 6 1
432.2.a.g 1 36.h even 6 1
1296.2.i.c 2 12.b even 2 1
1296.2.i.c 2 36.h even 6 1
1296.2.i.o 2 4.b odd 2 1
1296.2.i.o 2 36.f odd 6 1
1350.2.a.h 1 45.j even 6 1
1350.2.a.r 1 45.h odd 6 1
1350.2.c.b 2 45.l even 12 2
1350.2.c.k 2 45.k odd 12 2
1728.2.a.c 1 72.j odd 6 1
1728.2.a.d 1 72.l even 6 1
1728.2.a.y 1 72.n even 6 1
1728.2.a.z 1 72.p odd 6 1
2646.2.a.a 1 63.o even 6 1
2646.2.a.bd 1 63.l odd 6 1
6534.2.a.b 1 99.h odd 6 1
6534.2.a.bc 1 99.g even 6 1
9126.2.a.r 1 117.t even 6 1
9126.2.a.u 1 117.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(162, [\chi])\):

\( T_{5}^{2} - 3T_{5} + 9 \) Copy content Toggle raw display
\( T_{7}^{2} - T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T - 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$31$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$43$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$47$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$53$ \( (T + 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$67$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 7)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$83$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$89$ \( (T - 18)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - T + 1 \) Copy content Toggle raw display
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