↑

Properties

Label 162.9.d.e
Level $162$
Weight $9$
Character orbit 162.d
Analytic conductor $65.995$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [162,9,Mod(53,162)]
 
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([5]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.53");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 162.d (of order \(6\), 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: \(65.9953348299\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{12} \)
Twist minimal: no (minimal twist has level 54)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{5} - 2 \beta_{4}) q^{2} + ( - 128 \beta_{2} + 128) q^{4} + ( - 51 \beta_{5} + 11 \beta_1) q^{5} + (7 \beta_{6} - 77 \beta_{2}) q^{7} - 256 \beta_{4} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{5} - 2 \beta_{4}) q^{2} + ( - 128 \beta_{2} + 128) q^{4} + ( - 51 \beta_{5} + 11 \beta_1) q^{5} + (7 \beta_{6} - 77 \beta_{2}) q^{7} - 256 \beta_{4} q^{8} + ( - 22 \beta_{7} + 3264) q^{10} + ( - 870 \beta_{5} - 870 \beta_{4} + \cdots + 409 \beta_1) q^{11}+ \cdots + ( - 9231600 \beta_{4} + 68992 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 512 q^{4} - 308 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 512 q^{4} - 308 q^{7} + 26112 q^{10} + 44680 q^{13} - 65536 q^{16} - 113024 q^{19} + 222720 q^{22} - 876736 q^{25} - 78848 q^{28} + 2375428 q^{31} + 82944 q^{34} + 10394656 q^{37} + 1671168 q^{40} + 2755096 q^{43} + 13937664 q^{46} + 18463200 q^{49} - 5719040 q^{52} + 37596888 q^{55} + 21977088 q^{58} - 36842384 q^{61} - 16777216 q^{64} + 52170376 q^{67} - 15375360 q^{70} + 44881864 q^{73} - 7233536 q^{76} + 51146440 q^{79} + 28093440 q^{82} - 101486520 q^{85} - 28508160 q^{88} - 145355728 q^{91} + 4420608 q^{94} + 269355220 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 27\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 27\zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -4\zeta_{24}^{5} - 4\zeta_{24}^{3} + 4\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -4\zeta_{24}^{7} + 4\zeta_{24}^{5} + 4\zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 108\zeta_{24}^{7} + 108\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -108\zeta_{24}^{5} + 108\zeta_{24}^{3} + 108\zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{6} + 27\beta_{5} + 27\beta_{4} ) / 216 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_1 ) / 27 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} - 27\beta_{4} ) / 216 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + \beta_{6} + 27\beta_{5} ) / 216 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( \beta_{3} ) / 27 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{6} - 27\beta_{5} - 27\beta_{4} ) / 216 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

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

\(n\) \(83\)
\(\chi(n)\) \(1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
0.258819 − 0.965926i
0.965926 + 0.258819i
−0.258819 + 0.965926i
−0.965926 − 0.258819i
0.258819 + 0.965926i
0.965926 − 0.258819i
−0.258819 − 0.965926i
−0.965926 + 0.258819i
−9.79796 + 5.65685i 0 64.0000 − 110.851i −507.057 − 292.750i 0 −573.073 − 992.591i 1448.15i 0 6624.17
53.2 −9.79796 + 5.65685i 0 64.0000 − 110.851i 7.36159 + 4.25022i 0 496.073 + 859.223i 1448.15i 0 −96.1714
53.3 9.79796 − 5.65685i 0 64.0000 − 110.851i −7.36159 − 4.25022i 0 496.073 + 859.223i − 1448.15i 0 −96.1714
53.4 9.79796 − 5.65685i 0 64.0000 − 110.851i 507.057 + 292.750i 0 −573.073 − 992.591i − 1448.15i 0 6624.17
107.1 −9.79796 − 5.65685i 0 64.0000 + 110.851i −507.057 + 292.750i 0 −573.073 + 992.591i − 1448.15i 0 6624.17
107.2 −9.79796 − 5.65685i 0 64.0000 + 110.851i 7.36159 − 4.25022i 0 496.073 − 859.223i − 1448.15i 0 −96.1714
107.3 9.79796 + 5.65685i 0 64.0000 + 110.851i −7.36159 + 4.25022i 0 496.073 − 859.223i 1448.15i 0 −96.1714
107.4 9.79796 + 5.65685i 0 64.0000 + 110.851i 507.057 − 292.750i 0 −573.073 + 992.591i 1448.15i 0 6624.17
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.9.d.e 8
3.b odd 2 1 inner 162.9.d.e 8
9.c even 3 1 54.9.b.c ✓ 4
9.c even 3 1 inner 162.9.d.e 8
9.d odd 6 1 54.9.b.c ✓ 4
9.d odd 6 1 inner 162.9.d.e 8
36.f odd 6 1 432.9.e.h 4
36.h even 6 1 432.9.e.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.9.b.c ✓ 4 9.c even 3 1
54.9.b.c ✓ 4 9.d odd 6 1
162.9.d.e 8 1.a even 1 1 trivial
162.9.d.e 8 3.b odd 2 1 inner
162.9.d.e 8 9.c even 3 1 inner
162.9.d.e 8 9.d odd 6 1 inner
432.9.e.h 4 36.f odd 6 1
432.9.e.h 4 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 342882T_{5}^{6} + 117543295395T_{5}^{4} - 8493368524578T_{5}^{2} + 613579106939841 \) acting on \(S_{9}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 128 T^{2} + 16384)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 613579106939841 \) Copy content Toggle raw display
$7$ \( (T^{4} + \cdots + 1293094202449)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 91\!\cdots\!01 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 18\!\cdots\!64)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 48\!\cdots\!56)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 28256 T - 20677000064)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 26\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 79\!\cdots\!01)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + \cdots - 3296601588848)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 51\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 31\!\cdots\!36)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 34\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 10\!\cdots\!21)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 19\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 16\!\cdots\!76)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 68\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 13\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots - 465280990005839)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 81\!\cdots\!44)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 21\!\cdots\!81 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 10\!\cdots\!84)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 10\!\cdots\!69)^{2} \) Copy content Toggle raw display
show more
show less