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Properties

Label 162.8.c.m
Level $162$
Weight $8$
Character orbit 162.c
Analytic conductor $50.606$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [162,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.55");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 8 \)
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: \(50.6063741284\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{329})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 83x^{2} + 82x + 6724 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{6} \)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (8 \beta_1 - 8) q^{2} - 64 \beta_1 q^{4} + (\beta_{2} + 24 \beta_1) q^{5} + (\beta_{3} + \beta_{2} + 440 \beta_1 - 440) q^{7} + 512 q^{8} + (8 \beta_{3} - 192) q^{10} + (8 \beta_{3} + 8 \beta_{2} + \cdots - 3615) q^{11}+ \cdots + ( - 7040 \beta_{3} - 3120816) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{2} - 128 q^{4} + 48 q^{5} - 880 q^{7} + 2048 q^{8} - 768 q^{10} - 7230 q^{11} - 8560 q^{13} - 7040 q^{14} - 8192 q^{16} + 51408 q^{17} + 74096 q^{19} + 3072 q^{20} - 57840 q^{22} + 59628 q^{23}+ \cdots - 12483264 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 83x^{2} + 82x + 6724 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 83\nu^{2} - 83\nu + 6724 ) / 6806 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 27\nu^{3} - 2241\nu^{2} + 369765\nu - 181548 ) / 6806 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 54\nu^{3} + 6669 ) / 83 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{2} + 27\beta_1 ) / 54 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 4455\beta _1 - 4455 ) / 54 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 83\beta_{3} - 6669 ) / 54 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

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

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
−4.28459 − 7.42113i
4.78459 + 8.28715i
−4.28459 + 7.42113i
4.78459 − 8.28715i
−4.00000 + 6.92820i 0 −32.0000 − 55.4256i −232.868 − 403.339i 0 24.8678 − 43.0723i 512.000 0 3725.89
55.2 −4.00000 + 6.92820i 0 −32.0000 − 55.4256i 256.868 + 444.908i 0 −464.868 + 805.175i 512.000 0 −4109.89
109.1 −4.00000 − 6.92820i 0 −32.0000 + 55.4256i −232.868 + 403.339i 0 24.8678 + 43.0723i 512.000 0 3725.89
109.2 −4.00000 − 6.92820i 0 −32.0000 + 55.4256i 256.868 − 444.908i 0 −464.868 − 805.175i 512.000 0 −4109.89
\(n\): 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 162.8.c.m 4
3.b odd 2 1 162.8.c.p 4
9.c even 3 1 54.8.a.h yes 2
9.c even 3 1 inner 162.8.c.m 4
9.d odd 6 1 54.8.a.g ✓ 2
9.d odd 6 1 162.8.c.p 4
36.f odd 6 1 432.8.a.k 2
36.h even 6 1 432.8.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.8.a.g ✓ 2 9.d odd 6 1
54.8.a.h yes 2 9.c even 3 1
162.8.c.m 4 1.a even 1 1 trivial
162.8.c.m 4 9.c even 3 1 inner
162.8.c.p 4 3.b odd 2 1
162.8.c.p 4 9.d odd 6 1
432.8.a.k 2 36.f odd 6 1
432.8.a.p 2 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 48T_{5}^{3} + 241569T_{5}^{2} + 11484720T_{5} + 57247740225 \) acting on \(S_{8}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots + 57247740225 \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 2138230081 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 5205693996801 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{2} - 25704 T + 69237504)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 37048 T - 832082324)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 62\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 19\!\cdots\!21 \) Copy content Toggle raw display
$37$ \( (T^{2} + 71060 T - 179439584684)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 37\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( (T^{2} + 1382112 T + 463161939111)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 51\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 51\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{2} + \cdots - 5355488275200)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots - 5381037627791)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 37\!\cdots\!25 \) Copy content Toggle raw display
$89$ \( (T^{2} + \cdots - 20597975704836)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 26\!\cdots\!81 \) Copy content Toggle raw display
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