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Properties

Label 164.4.o.a
Level $164$
Weight $4$
Character orbit 164.o
Analytic conductor $9.676$
Analytic rank $0$
Dimension $16$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [164,4,Mod(7,164)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("164.7"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(164, base_ring=CyclotomicField(40)) chi = DirichletCharacter(H, H._module([20, 39])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 164.o (of order \(40\), degree \(16\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.67631324094\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{40})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{40}]$

$q$-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 \(\zeta_{40}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \zeta_{40}^{14} + \cdots - 2 \zeta_{40}^{2}) q^{2} + 8 \zeta_{40}^{6} q^{4} + ( - 11 \zeta_{40}^{15} + \cdots + 22 \zeta_{40}^{3}) q^{5} + ( - 16 \zeta_{40}^{14} - 16 \zeta_{40}^{4}) q^{8} + 27 \zeta_{40}^{5} q^{9}+ \cdots + (686 \zeta_{40}^{13} + \cdots - 686 \zeta_{40}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{2} - 64 q^{8} + 184 q^{13} + 256 q^{16} + 188 q^{17} - 440 q^{26} + 260 q^{29} + 2048 q^{32} + 3168 q^{34} + 460 q^{41} - 5872 q^{50} - 1152 q^{52} + 1036 q^{53} - 1656 q^{58} + 2596 q^{61}+ \cdots + 1188 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/164\mathbb{Z}\right)^\times\).

\(n\) \(83\) \(129\)
\(\chi(n)\) \(-1\) \(\zeta_{40}^{11}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−0.156434 − 0.987688i
0.453990 − 0.891007i
0.453990 + 0.891007i
−0.987688 + 0.156434i
0.156434 − 0.987688i
−0.156434 + 0.987688i
0.987688 − 0.156434i
−0.453990 − 0.891007i
−0.453990 + 0.891007i
0.156434 + 0.987688i
−0.987688 − 0.156434i
0.891007 + 0.453990i
0.891007 − 0.453990i
−0.891007 + 0.453990i
−0.891007 − 0.453990i
0.987688 + 0.156434i
1.28408 + 2.52015i 0 −4.70228 + 6.47214i 3.07975 + 19.4448i 0 0 −22.3488 − 3.53971i −19.0919 − 19.0919i −45.0491 + 32.7301i
11.1 2.79360 − 0.442463i 0 7.60845 − 2.47214i −7.33982 + 14.4052i 0 0 20.1612 − 10.2726i 19.0919 + 19.0919i −14.1308 + 43.4901i
15.1 2.79360 + 0.442463i 0 7.60845 + 2.47214i −7.33982 − 14.4052i 0 0 20.1612 + 10.2726i 19.0919 − 19.0919i −14.1308 − 43.4901i
19.1 −2.52015 + 1.28408i 0 4.70228 − 6.47214i −21.8865 + 3.46648i 0 0 −3.53971 + 22.3488i −19.0919 + 19.0919i 50.7059 − 36.8400i
35.1 1.28408 − 2.52015i 0 −4.70228 − 6.47214i −3.07975 + 19.4448i 0 0 −22.3488 + 3.53971i 19.0919 − 19.0919i 45.0491 + 32.7301i
47.1 1.28408 − 2.52015i 0 −4.70228 − 6.47214i 3.07975 − 19.4448i 0 0 −22.3488 + 3.53971i −19.0919 + 19.0919i −45.0491 − 32.7301i
63.1 −2.52015 + 1.28408i 0 4.70228 − 6.47214i 21.8865 − 3.46648i 0 0 −3.53971 + 22.3488i 19.0919 − 19.0919i −50.7059 + 36.8400i
67.1 2.79360 + 0.442463i 0 7.60845 + 2.47214i 7.33982 + 14.4052i 0 0 20.1612 + 10.2726i −19.0919 + 19.0919i 14.1308 + 43.4901i
71.1 2.79360 − 0.442463i 0 7.60845 − 2.47214i 7.33982 − 14.4052i 0 0 20.1612 − 10.2726i −19.0919 − 19.0919i 14.1308 − 43.4901i
75.1 1.28408 + 2.52015i 0 −4.70228 + 6.47214i −3.07975 − 19.4448i 0 0 −22.3488 − 3.53971i 19.0919 + 19.0919i 45.0491 − 32.7301i
95.1 −2.52015 − 1.28408i 0 4.70228 + 6.47214i −21.8865 − 3.46648i 0 0 −3.53971 − 22.3488i −19.0919 − 19.0919i 50.7059 + 36.8400i
99.1 0.442463 + 2.79360i 0 −7.60845 + 2.47214i 8.63849 + 4.40153i 0 0 −10.2726 − 20.1612i −19.0919 + 19.0919i −8.47392 + 26.0801i
111.1 0.442463 − 2.79360i 0 −7.60845 − 2.47214i 8.63849 − 4.40153i 0 0 −10.2726 + 20.1612i −19.0919 − 19.0919i −8.47392 − 26.0801i
135.1 0.442463 − 2.79360i 0 −7.60845 − 2.47214i −8.63849 + 4.40153i 0 0 −10.2726 + 20.1612i 19.0919 + 19.0919i 8.47392 + 26.0801i
147.1 0.442463 + 2.79360i 0 −7.60845 + 2.47214i −8.63849 − 4.40153i 0 0 −10.2726 − 20.1612i 19.0919 − 19.0919i 8.47392 − 26.0801i
151.1 −2.52015 − 1.28408i 0 4.70228 + 6.47214i 21.8865 + 3.46648i 0 0 −3.53971 − 22.3488i 19.0919 + 19.0919i −50.7059 − 36.8400i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 7.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
41.h odd 40 1 inner
164.o even 40 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 164.4.o.a ✓ 16
4.b odd 2 1 CM 164.4.o.a ✓ 16
41.h odd 40 1 inner 164.4.o.a ✓ 16
164.o even 40 1 inner 164.4.o.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.4.o.a ✓ 16 1.a even 1 1 trivial
164.4.o.a ✓ 16 4.b odd 2 1 CM
164.4.o.a ✓ 16 41.h odd 40 1 inner
164.4.o.a ✓ 16 164.o even 40 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{4}^{\mathrm{new}}(164, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 4 T^{7} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 21\!\cdots\!21 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 29\!\cdots\!01 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 65\!\cdots\!01 \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 31\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 22\!\cdots\!81)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 37\!\cdots\!21 \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 12\!\cdots\!01)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 15\!\cdots\!01 \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 38\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 99\!\cdots\!61 \) Copy content Toggle raw display
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