Properties

Label 162.4.c.b
Level $162$
Weight $4$
Character orbit 162.c
Analytic conductor $9.558$
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] = [162,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.55");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(9.55830942093\)
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: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} - 3 \zeta_{6} q^{5} + (29 \zeta_{6} - 29) q^{7} + 8 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 2) q^{2} - 4 \zeta_{6} q^{4} - 3 \zeta_{6} q^{5} + (29 \zeta_{6} - 29) q^{7} + 8 q^{8} + 6 q^{10} + ( - 57 \zeta_{6} + 57) q^{11} - 20 \zeta_{6} q^{13} - 58 \zeta_{6} q^{14} + (16 \zeta_{6} - 16) q^{16} - 72 q^{17} - 106 q^{19} + (12 \zeta_{6} - 12) q^{20} + 114 \zeta_{6} q^{22} - 174 \zeta_{6} q^{23} + ( - 116 \zeta_{6} + 116) q^{25} + 40 q^{26} + 116 q^{28} + ( - 210 \zeta_{6} + 210) q^{29} - 47 \zeta_{6} q^{31} - 32 \zeta_{6} q^{32} + ( - 144 \zeta_{6} + 144) q^{34} + 87 q^{35} + 2 q^{37} + ( - 212 \zeta_{6} + 212) q^{38} - 24 \zeta_{6} q^{40} + 6 \zeta_{6} q^{41} + (218 \zeta_{6} - 218) q^{43} - 228 q^{44} + 348 q^{46} + (474 \zeta_{6} - 474) q^{47} - 498 \zeta_{6} q^{49} + 232 \zeta_{6} q^{50} + (80 \zeta_{6} - 80) q^{52} + 81 q^{53} - 171 q^{55} + (232 \zeta_{6} - 232) q^{56} + 420 \zeta_{6} q^{58} - 84 \zeta_{6} q^{59} + (56 \zeta_{6} - 56) q^{61} + 94 q^{62} + 64 q^{64} + (60 \zeta_{6} - 60) q^{65} + 142 \zeta_{6} q^{67} + 288 \zeta_{6} q^{68} + (174 \zeta_{6} - 174) q^{70} + 360 q^{71} - 1159 q^{73} + (4 \zeta_{6} - 4) q^{74} + 424 \zeta_{6} q^{76} + 1653 \zeta_{6} q^{77} + ( - 160 \zeta_{6} + 160) q^{79} + 48 q^{80} - 12 q^{82} + (735 \zeta_{6} - 735) q^{83} + 216 \zeta_{6} q^{85} - 436 \zeta_{6} q^{86} + ( - 456 \zeta_{6} + 456) q^{88} - 954 q^{89} + 580 q^{91} + (696 \zeta_{6} - 696) q^{92} - 948 \zeta_{6} q^{94} + 318 \zeta_{6} q^{95} + (191 \zeta_{6} - 191) q^{97} + 996 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{4} - 3 q^{5} - 29 q^{7} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 4 q^{4} - 3 q^{5} - 29 q^{7} + 16 q^{8} + 12 q^{10} + 57 q^{11} - 20 q^{13} - 58 q^{14} - 16 q^{16} - 144 q^{17} - 212 q^{19} - 12 q^{20} + 114 q^{22} - 174 q^{23} + 116 q^{25} + 80 q^{26} + 232 q^{28} + 210 q^{29} - 47 q^{31} - 32 q^{32} + 144 q^{34} + 174 q^{35} + 4 q^{37} + 212 q^{38} - 24 q^{40} + 6 q^{41} - 218 q^{43} - 456 q^{44} + 696 q^{46} - 474 q^{47} - 498 q^{49} + 232 q^{50} - 80 q^{52} + 162 q^{53} - 342 q^{55} - 232 q^{56} + 420 q^{58} - 84 q^{59} - 56 q^{61} + 188 q^{62} + 128 q^{64} - 60 q^{65} + 142 q^{67} + 288 q^{68} - 174 q^{70} + 720 q^{71} - 2318 q^{73} - 4 q^{74} + 424 q^{76} + 1653 q^{77} + 160 q^{79} + 96 q^{80} - 24 q^{82} - 735 q^{83} + 216 q^{85} - 436 q^{86} + 456 q^{88} - 1908 q^{89} + 1160 q^{91} - 696 q^{92} - 948 q^{94} + 318 q^{95} - 191 q^{97} + 1992 q^{98}+O(q^{100}) \) 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
−1.00000 + 1.73205i 0 −2.00000 3.46410i −1.50000 2.59808i 0 −14.5000 + 25.1147i 8.00000 0 6.00000
109.1 −1.00000 1.73205i 0 −2.00000 + 3.46410i −1.50000 + 2.59808i 0 −14.5000 25.1147i 8.00000 0 6.00000
\(n\): 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 162.4.c.b 2
3.b odd 2 1 162.4.c.g 2
9.c even 3 1 54.4.a.c yes 1
9.c even 3 1 inner 162.4.c.b 2
9.d odd 6 1 54.4.a.b 1
9.d odd 6 1 162.4.c.g 2
36.f odd 6 1 432.4.a.j 1
36.h even 6 1 432.4.a.e 1
45.h odd 6 1 1350.4.a.o 1
45.j even 6 1 1350.4.a.a 1
45.k odd 12 2 1350.4.c.b 2
45.l even 12 2 1350.4.c.s 2
72.j odd 6 1 1728.4.a.v 1
72.l even 6 1 1728.4.a.u 1
72.n even 6 1 1728.4.a.l 1
72.p odd 6 1 1728.4.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.4.a.b 1 9.d odd 6 1
54.4.a.c yes 1 9.c even 3 1
162.4.c.b 2 1.a even 1 1 trivial
162.4.c.b 2 9.c even 3 1 inner
162.4.c.g 2 3.b odd 2 1
162.4.c.g 2 9.d odd 6 1
432.4.a.e 1 36.h even 6 1
432.4.a.j 1 36.f odd 6 1
1350.4.a.a 1 45.j even 6 1
1350.4.a.o 1 45.h odd 6 1
1350.4.c.b 2 45.k odd 12 2
1350.4.c.s 2 45.l even 12 2
1728.4.a.k 1 72.p odd 6 1
1728.4.a.l 1 72.n even 6 1
1728.4.a.u 1 72.l even 6 1
1728.4.a.v 1 72.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 3T_{5} + 9 \) acting on \(S_{4}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} + 29T + 841 \) Copy content Toggle raw display
$11$ \( T^{2} - 57T + 3249 \) Copy content Toggle raw display
$13$ \( T^{2} + 20T + 400 \) Copy content Toggle raw display
$17$ \( (T + 72)^{2} \) Copy content Toggle raw display
$19$ \( (T + 106)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 174T + 30276 \) Copy content Toggle raw display
$29$ \( T^{2} - 210T + 44100 \) Copy content Toggle raw display
$31$ \( T^{2} + 47T + 2209 \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$43$ \( T^{2} + 218T + 47524 \) Copy content Toggle raw display
$47$ \( T^{2} + 474T + 224676 \) Copy content Toggle raw display
$53$ \( (T - 81)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 84T + 7056 \) Copy content Toggle raw display
$61$ \( T^{2} + 56T + 3136 \) Copy content Toggle raw display
$67$ \( T^{2} - 142T + 20164 \) Copy content Toggle raw display
$71$ \( (T - 360)^{2} \) Copy content Toggle raw display
$73$ \( (T + 1159)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 160T + 25600 \) Copy content Toggle raw display
$83$ \( T^{2} + 735T + 540225 \) Copy content Toggle raw display
$89$ \( (T + 954)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 191T + 36481 \) Copy content Toggle raw display
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