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Properties

Label 162.6.c.f
Level $162$
Weight $6$
Character orbit 162.c
Analytic conductor $25.982$
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] = [162,6,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, 6, 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, 6); N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 162.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-4,0,-16,84,0,193] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.9821788097\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 54)
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.

\(f(q)\) \(=\) \( q + (4 \zeta_{6} - 4) q^{2} - 16 \zeta_{6} q^{4} + 84 \zeta_{6} q^{5} + ( - 193 \zeta_{6} + 193) q^{7} + 64 q^{8} - 336 q^{10} + ( - 348 \zeta_{6} + 348) q^{11} - 845 \zeta_{6} q^{13} + 772 \zeta_{6} q^{14} + \cdots + 81768 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} - 16 q^{4} + 84 q^{5} + 193 q^{7} + 128 q^{8} - 672 q^{10} + 348 q^{11} - 845 q^{13} + 772 q^{14} - 256 q^{16} - 3384 q^{17} - 158 q^{19} + 1344 q^{20} + 1392 q^{22} - 564 q^{23} - 3931 q^{25}+ \cdots + 163536 q^{98}+O(q^{100}) \) 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
−2.00000 + 3.46410i 0 −8.00000 − 13.8564i 42.0000 + 72.7461i 0 96.5000 − 167.143i 64.0000 0 −336.000
109.1 −2.00000 − 3.46410i 0 −8.00000 + 13.8564i 42.0000 − 72.7461i 0 96.5000 + 167.143i 64.0000 0 −336.000
\(n\): 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 162.6.c.f 2
3.b odd 2 1 162.6.c.g 2
9.c even 3 1 54.6.a.d yes 1
9.c even 3 1 inner 162.6.c.f 2
9.d odd 6 1 54.6.a.c ✓ 1
9.d odd 6 1 162.6.c.g 2
36.f odd 6 1 432.6.a.a 1
36.h even 6 1 432.6.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.6.a.c ✓ 1 9.d odd 6 1
54.6.a.d yes 1 9.c even 3 1
162.6.c.f 2 1.a even 1 1 trivial
162.6.c.f 2 9.c even 3 1 inner
162.6.c.g 2 3.b odd 2 1
162.6.c.g 2 9.d odd 6 1
432.6.a.a 1 36.f odd 6 1
432.6.a.j 1 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 84T_{5} + 7056 \) acting on \(S_{6}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 84T + 7056 \) Copy content Toggle raw display
$7$ \( T^{2} - 193T + 37249 \) Copy content Toggle raw display
$11$ \( T^{2} - 348T + 121104 \) Copy content Toggle raw display
$13$ \( T^{2} + 845T + 714025 \) Copy content Toggle raw display
$17$ \( (T + 1692)^{2} \) Copy content Toggle raw display
$19$ \( (T + 79)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 564T + 318096 \) Copy content Toggle raw display
$29$ \( T^{2} - 6432 T + 41370624 \) Copy content Toggle raw display
$31$ \( T^{2} + 4940 T + 24403600 \) Copy content Toggle raw display
$37$ \( (T + 3805)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 12480 T + 155750400 \) Copy content Toggle raw display
$43$ \( T^{2} - 4936 T + 24364096 \) Copy content Toggle raw display
$47$ \( T^{2} - 8124 T + 65999376 \) Copy content Toggle raw display
$53$ \( (T - 33192)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 1805570064 \) Copy content Toggle raw display
$61$ \( T^{2} - 17833 T + 318015889 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 4583154601 \) Copy content Toggle raw display
$71$ \( (T + 28152)^{2} \) Copy content Toggle raw display
$73$ \( (T + 13975)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 7053144289 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 1114491456 \) Copy content Toggle raw display
$89$ \( (T + 77868)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 2083 T + 4338889 \) Copy content Toggle raw display
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