Properties

Label 162.4.c.f
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: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
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.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 2) q^{2} - 4 \zeta_{6} q^{4} - 6 \zeta_{6} q^{5} + ( - 16 \zeta_{6} + 16) q^{7} - 8 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 2) q^{2} - 4 \zeta_{6} q^{4} - 6 \zeta_{6} q^{5} + ( - 16 \zeta_{6} + 16) q^{7} - 8 q^{8} - 12 q^{10} + (12 \zeta_{6} - 12) q^{11} - 38 \zeta_{6} q^{13} - 32 \zeta_{6} q^{14} + (16 \zeta_{6} - 16) q^{16} - 126 q^{17} + 20 q^{19} + (24 \zeta_{6} - 24) q^{20} + 24 \zeta_{6} q^{22} - 168 \zeta_{6} q^{23} + ( - 89 \zeta_{6} + 89) q^{25} - 76 q^{26} - 64 q^{28} + (30 \zeta_{6} - 30) q^{29} + 88 \zeta_{6} q^{31} + 32 \zeta_{6} q^{32} + (252 \zeta_{6} - 252) q^{34} - 96 q^{35} + 254 q^{37} + ( - 40 \zeta_{6} + 40) q^{38} + 48 \zeta_{6} q^{40} - 42 \zeta_{6} q^{41} + ( - 52 \zeta_{6} + 52) q^{43} + 48 q^{44} - 336 q^{46} + ( - 96 \zeta_{6} + 96) q^{47} + 87 \zeta_{6} q^{49} - 178 \zeta_{6} q^{50} + (152 \zeta_{6} - 152) q^{52} + 198 q^{53} + 72 q^{55} + (128 \zeta_{6} - 128) q^{56} + 60 \zeta_{6} q^{58} + 660 \zeta_{6} q^{59} + ( - 538 \zeta_{6} + 538) q^{61} + 176 q^{62} + 64 q^{64} + (228 \zeta_{6} - 228) q^{65} - 884 \zeta_{6} q^{67} + 504 \zeta_{6} q^{68} + (192 \zeta_{6} - 192) q^{70} + 792 q^{71} + 218 q^{73} + ( - 508 \zeta_{6} + 508) q^{74} - 80 \zeta_{6} q^{76} + 192 \zeta_{6} q^{77} + ( - 520 \zeta_{6} + 520) q^{79} + 96 q^{80} - 84 q^{82} + ( - 492 \zeta_{6} + 492) q^{83} + 756 \zeta_{6} q^{85} - 104 \zeta_{6} q^{86} + ( - 96 \zeta_{6} + 96) q^{88} + 810 q^{89} - 608 q^{91} + (672 \zeta_{6} - 672) q^{92} - 192 \zeta_{6} q^{94} - 120 \zeta_{6} q^{95} + (1154 \zeta_{6} - 1154) q^{97} + 174 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 4 q^{4} - 6 q^{5} + 16 q^{7} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 4 q^{4} - 6 q^{5} + 16 q^{7} - 16 q^{8} - 24 q^{10} - 12 q^{11} - 38 q^{13} - 32 q^{14} - 16 q^{16} - 252 q^{17} + 40 q^{19} - 24 q^{20} + 24 q^{22} - 168 q^{23} + 89 q^{25} - 152 q^{26} - 128 q^{28} - 30 q^{29} + 88 q^{31} + 32 q^{32} - 252 q^{34} - 192 q^{35} + 508 q^{37} + 40 q^{38} + 48 q^{40} - 42 q^{41} + 52 q^{43} + 96 q^{44} - 672 q^{46} + 96 q^{47} + 87 q^{49} - 178 q^{50} - 152 q^{52} + 396 q^{53} + 144 q^{55} - 128 q^{56} + 60 q^{58} + 660 q^{59} + 538 q^{61} + 352 q^{62} + 128 q^{64} - 228 q^{65} - 884 q^{67} + 504 q^{68} - 192 q^{70} + 1584 q^{71} + 436 q^{73} + 508 q^{74} - 80 q^{76} + 192 q^{77} + 520 q^{79} + 192 q^{80} - 168 q^{82} + 492 q^{83} + 756 q^{85} - 104 q^{86} + 96 q^{88} + 1620 q^{89} - 1216 q^{91} - 672 q^{92} - 192 q^{94} - 120 q^{95} - 1154 q^{97} + 348 q^{98}+O(q^{100}) \) 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 −3.00000 5.19615i 0 8.00000 13.8564i −8.00000 0 −12.0000
109.1 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −3.00000 + 5.19615i 0 8.00000 + 13.8564i −8.00000 0 −12.0000
\(n\): 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.f 2
3.b odd 2 1 162.4.c.c 2
9.c even 3 1 6.4.a.a 1
9.c even 3 1 inner 162.4.c.f 2
9.d odd 6 1 18.4.a.a 1
9.d odd 6 1 162.4.c.c 2
36.f odd 6 1 48.4.a.c 1
36.h even 6 1 144.4.a.c 1
45.h odd 6 1 450.4.a.h 1
45.j even 6 1 150.4.a.i 1
45.k odd 12 2 150.4.c.d 2
45.l even 12 2 450.4.c.e 2
63.g even 3 1 294.4.e.h 2
63.h even 3 1 294.4.e.h 2
63.i even 6 1 882.4.g.f 2
63.j odd 6 1 882.4.g.i 2
63.k odd 6 1 294.4.e.g 2
63.l odd 6 1 294.4.a.e 1
63.n odd 6 1 882.4.g.i 2
63.o even 6 1 882.4.a.n 1
63.s even 6 1 882.4.g.f 2
63.t odd 6 1 294.4.e.g 2
72.j odd 6 1 576.4.a.q 1
72.l even 6 1 576.4.a.r 1
72.n even 6 1 192.4.a.i 1
72.p odd 6 1 192.4.a.c 1
99.g even 6 1 2178.4.a.e 1
99.h odd 6 1 726.4.a.f 1
117.t even 6 1 1014.4.a.g 1
117.y odd 12 2 1014.4.b.d 2
144.v odd 12 2 768.4.d.c 2
144.x even 12 2 768.4.d.n 2
153.h even 6 1 1734.4.a.d 1
171.o odd 6 1 2166.4.a.i 1
180.p odd 6 1 1200.4.a.b 1
180.x even 12 2 1200.4.f.j 2
252.bi even 6 1 2352.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 9.c even 3 1
18.4.a.a 1 9.d odd 6 1
48.4.a.c 1 36.f odd 6 1
144.4.a.c 1 36.h even 6 1
150.4.a.i 1 45.j even 6 1
150.4.c.d 2 45.k odd 12 2
162.4.c.c 2 3.b odd 2 1
162.4.c.c 2 9.d odd 6 1
162.4.c.f 2 1.a even 1 1 trivial
162.4.c.f 2 9.c even 3 1 inner
192.4.a.c 1 72.p odd 6 1
192.4.a.i 1 72.n even 6 1
294.4.a.e 1 63.l odd 6 1
294.4.e.g 2 63.k odd 6 1
294.4.e.g 2 63.t odd 6 1
294.4.e.h 2 63.g even 3 1
294.4.e.h 2 63.h even 3 1
450.4.a.h 1 45.h odd 6 1
450.4.c.e 2 45.l even 12 2
576.4.a.q 1 72.j odd 6 1
576.4.a.r 1 72.l even 6 1
726.4.a.f 1 99.h odd 6 1
768.4.d.c 2 144.v odd 12 2
768.4.d.n 2 144.x even 12 2
882.4.a.n 1 63.o even 6 1
882.4.g.f 2 63.i even 6 1
882.4.g.f 2 63.s even 6 1
882.4.g.i 2 63.j odd 6 1
882.4.g.i 2 63.n odd 6 1
1014.4.a.g 1 117.t even 6 1
1014.4.b.d 2 117.y odd 12 2
1200.4.a.b 1 180.p odd 6 1
1200.4.f.j 2 180.x even 12 2
1734.4.a.d 1 153.h even 6 1
2166.4.a.i 1 171.o odd 6 1
2178.4.a.e 1 99.g even 6 1
2352.4.a.e 1 252.bi even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 6T_{5} + 36 \) 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
$5$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$7$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$11$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$13$ \( T^{2} + 38T + 1444 \) Copy content Toggle raw display
$17$ \( (T + 126)^{2} \) Copy content Toggle raw display
$19$ \( (T - 20)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 168T + 28224 \) Copy content Toggle raw display
$29$ \( T^{2} + 30T + 900 \) Copy content Toggle raw display
$31$ \( T^{2} - 88T + 7744 \) Copy content Toggle raw display
$37$ \( (T - 254)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 42T + 1764 \) Copy content Toggle raw display
$43$ \( T^{2} - 52T + 2704 \) Copy content Toggle raw display
$47$ \( T^{2} - 96T + 9216 \) Copy content Toggle raw display
$53$ \( (T - 198)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 660T + 435600 \) Copy content Toggle raw display
$61$ \( T^{2} - 538T + 289444 \) Copy content Toggle raw display
$67$ \( T^{2} + 884T + 781456 \) Copy content Toggle raw display
$71$ \( (T - 792)^{2} \) Copy content Toggle raw display
$73$ \( (T - 218)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 520T + 270400 \) Copy content Toggle raw display
$83$ \( T^{2} - 492T + 242064 \) Copy content Toggle raw display
$89$ \( (T - 810)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 1154 T + 1331716 \) Copy content Toggle raw display
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