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Properties

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

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [162,8,Mod(55,162)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage: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")
 
Copy content magma://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

Copy content comment:select newform
 
Copy content sage:traces = [2,-8,0,-64,165] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(50.6063741284\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content 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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + (8 \zeta_{6} - 8) q^{2} - 64 \zeta_{6} q^{4} + 165 \zeta_{6} q^{5} + ( - 508 \zeta_{6} + 508) q^{7} + 512 q^{8} - 1320 q^{10} + (3024 \zeta_{6} - 3024) q^{11} - 5039 \zeta_{6} q^{13} + 4064 \zeta_{6} q^{14} + \cdots - 4523832 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} - 64 q^{4} + 165 q^{5} + 508 q^{7} + 1024 q^{8} - 2640 q^{10} - 3024 q^{11} - 5039 q^{13} + 4064 q^{14} - 4096 q^{16} - 6378 q^{17} + 3016 q^{19} + 10560 q^{20} - 24192 q^{22} + 75600 q^{23}+ \cdots - 9047664 q^{98}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−4.00000 + 6.92820i 0 −32.0000 − 55.4256i 82.5000 + 142.894i 0 254.000 − 439.941i 512.000 0 −1320.00
109.1 −4.00000 − 6.92820i 0 −32.0000 + 55.4256i 82.5000 − 142.894i 0 254.000 + 439.941i 512.000 0 −1320.00
\(n\): 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.e 2
3.b odd 2 1 162.8.c.h 2
9.c even 3 1 162.8.a.b yes 1
9.c even 3 1 inner 162.8.c.e 2
9.d odd 6 1 162.8.a.a ✓ 1
9.d odd 6 1 162.8.c.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.8.a.a ✓ 1 9.d odd 6 1
162.8.a.b yes 1 9.c even 3 1
162.8.c.e 2 1.a even 1 1 trivial
162.8.c.e 2 9.c even 3 1 inner
162.8.c.h 2 3.b odd 2 1
162.8.c.h 2 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 165T_{5} + 27225 \) acting on \(S_{8}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 165T + 27225 \) Copy content Toggle raw display
$7$ \( T^{2} - 508T + 258064 \) Copy content Toggle raw display
$11$ \( T^{2} + 3024 T + 9144576 \) Copy content Toggle raw display
$13$ \( T^{2} + 5039 T + 25391521 \) Copy content Toggle raw display
$17$ \( (T + 3189)^{2} \) Copy content Toggle raw display
$19$ \( (T - 1508)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 5715360000 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 6833502225 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 30587211664 \) Copy content Toggle raw display
$37$ \( (T + 323569)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 94936701924 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 113353422400 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 146839174416 \) Copy content Toggle raw display
$53$ \( (T - 760206)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 4953580240896 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 5039194384225 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 2170282883344 \) Copy content Toggle raw display
$71$ \( (T + 5006892)^{2} \) Copy content Toggle raw display
$73$ \( (T + 5898301)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 49403579597824 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 7028840230416 \) Copy content Toggle raw display
$89$ \( (T + 6770901)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 261675464020996 \) Copy content Toggle raw display
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