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Properties

Label 164.2.o.a
Level $164$
Weight $2$
Character orbit 164.o
Analytic conductor $1.310$
Analytic rank $0$
Dimension $16$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [164,2,Mod(7,164)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(164, base_ring=CyclotomicField(40))
 
chi = DirichletCharacter(H, H._module([20, 39]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("164.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 164.o (of order \(40\), degree \(16\), 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: \(1.30954659315\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{40})\)
comment: defining polynomial
 
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

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

\(f(q)\) \(=\) \( q + ( - \zeta_{40}^{14} + \cdots + \zeta_{40}^{2}) q^{2}+ \cdots + 3 \zeta_{40}^{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{40}^{14} + \cdots + \zeta_{40}^{2}) q^{2}+ \cdots + ( - 7 \zeta_{40}^{13} + \cdots + 7 \zeta_{40}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{2} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{2} + 8 q^{8} - 8 q^{13} + 16 q^{16} - 4 q^{17} - 20 q^{26} + 20 q^{29} - 64 q^{32} - 48 q^{34} - 20 q^{41} + 56 q^{50} + 96 q^{52} + 28 q^{53} + 12 q^{58} - 44 q^{61} - 112 q^{65} + 32 q^{68} - 36 q^{82} + 56 q^{85} + 32 q^{89} + 48 q^{90} + 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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
−0.642040 − 1.26007i 0 −1.17557 + 1.61803i −0.490920 − 3.09954i 0 0 2.79360 + 0.442463i −2.12132 − 2.12132i −3.59046 + 2.60863i
11.1 −1.39680 + 0.221232i 0 1.90211 − 0.618034i −0.935446 + 1.83592i 0 0 −2.52015 + 1.28408i 2.12132 + 2.12132i 0.900470 − 2.77136i
15.1 −1.39680 − 0.221232i 0 1.90211 + 0.618034i −0.935446 − 1.83592i 0 0 −2.52015 − 1.28408i 2.12132 − 2.12132i 0.900470 + 2.77136i
19.1 1.26007 − 0.642040i 0 1.17557 − 1.61803i 0.657845 − 0.104192i 0 0 0.442463 − 2.79360i −2.12132 + 2.12132i 0.762037 − 0.553653i
35.1 −0.642040 + 1.26007i 0 −1.17557 − 1.61803i 0.490920 − 3.09954i 0 0 2.79360 − 0.442463i 2.12132 − 2.12132i 3.59046 + 2.60863i
47.1 −0.642040 + 1.26007i 0 −1.17557 − 1.61803i −0.490920 + 3.09954i 0 0 2.79360 − 0.442463i −2.12132 + 2.12132i −3.59046 − 2.60863i
63.1 1.26007 − 0.642040i 0 1.17557 − 1.61803i −0.657845 + 0.104192i 0 0 0.442463 − 2.79360i 2.12132 − 2.12132i −0.762037 + 0.553653i
67.1 −1.39680 − 0.221232i 0 1.90211 + 0.618034i 0.935446 + 1.83592i 0 0 −2.52015 − 1.28408i −2.12132 + 2.12132i −0.900470 − 2.77136i
71.1 −1.39680 + 0.221232i 0 1.90211 − 0.618034i 0.935446 − 1.83592i 0 0 −2.52015 + 1.28408i −2.12132 − 2.12132i −0.900470 + 2.77136i
75.1 −0.642040 − 1.26007i 0 −1.17557 + 1.61803i 0.490920 + 3.09954i 0 0 2.79360 + 0.442463i 2.12132 + 2.12132i 3.59046 − 2.60863i
95.1 1.26007 + 0.642040i 0 1.17557 + 1.61803i 0.657845 + 0.104192i 0 0 0.442463 + 2.79360i −2.12132 − 2.12132i 0.762037 + 0.553653i
99.1 −0.221232 − 1.39680i 0 −1.90211 + 0.618034i −3.93080 − 2.00284i 0 0 1.28408 + 2.52015i −2.12132 + 2.12132i −1.92796 + 5.93364i
111.1 −0.221232 + 1.39680i 0 −1.90211 − 0.618034i −3.93080 + 2.00284i 0 0 1.28408 − 2.52015i −2.12132 − 2.12132i −1.92796 − 5.93364i
135.1 −0.221232 + 1.39680i 0 −1.90211 − 0.618034i 3.93080 − 2.00284i 0 0 1.28408 − 2.52015i 2.12132 + 2.12132i 1.92796 + 5.93364i
147.1 −0.221232 − 1.39680i 0 −1.90211 + 0.618034i 3.93080 + 2.00284i 0 0 1.28408 + 2.52015i 2.12132 − 2.12132i 1.92796 − 5.93364i
151.1 1.26007 + 0.642040i 0 1.17557 + 1.61803i −0.657845 − 0.104192i 0 0 0.442463 + 2.79360i 2.12132 + 2.12132i −0.762037 − 0.553653i
\(n\): e.g. 2-40 or 990-1000
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 164.2.o.a ✓ 16
4.b odd 2 1 CM 164.2.o.a ✓ 16
41.h odd 40 1 inner 164.2.o.a ✓ 16
164.o even 40 1 inner 164.2.o.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.2.o.a ✓ 16 1.a even 1 1 trivial
164.2.o.a ✓ 16 4.b odd 2 1 CM
164.2.o.a ✓ 16 41.h odd 40 1 inner
164.2.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 \(S_{2}^{\mathrm{new}}(164, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 2 T^{7} + 2 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 44 T^{12} + \cdots + 130321 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 132733441 \) Copy content Toggle raw display
$17$ \( T^{16} + 4 T^{15} + \cdots + 66569281 \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 127758803665936 \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} + 220 T^{14} + \cdots + 81450625 \) Copy content Toggle raw display
$41$ \( (T^{8} + 10 T^{7} + \cdots + 2825761)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 11247699430081 \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( (T^{8} + 22 T^{7} + \cdots + 11282881)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 2512481776561 \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 127758803665936 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
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