Properties

Label 164.1.j.a
Level $164$
Weight $1$
Character orbit 164.j
Analytic conductor $0.082$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
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] = [164,1,Mod(51,164)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(164, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("164.51");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 164.j (of order \(10\), degree \(4\), 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: \(0.0818466620718\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.45212176.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{10} q^{2} + \zeta_{10}^{2} q^{4} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{5} - \zeta_{10}^{3} q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{10} q^{2} + \zeta_{10}^{2} q^{4} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{5} - \zeta_{10}^{3} q^{8} + q^{9} + (\zeta_{10}^{4} + \zeta_{10}^{2}) q^{10} + (\zeta_{10}^{4} - \zeta_{10}^{3}) q^{13} + \zeta_{10}^{4} q^{16} + (\zeta_{10}^{4} + \zeta_{10}^{2}) q^{17} - \zeta_{10} q^{18} + ( - \zeta_{10}^{3} + 1) q^{20} + (\zeta_{10}^{4} + \zeta_{10}^{2} - \zeta_{10}) q^{25} + (\zeta_{10}^{4} + 1) q^{26} + \zeta_{10}^{2} q^{29} + q^{32} + ( - \zeta_{10}^{3} + 1) q^{34} + \zeta_{10}^{2} q^{36} + (\zeta_{10}^{4} + 1) q^{37} + (\zeta_{10}^{4} - \zeta_{10}) q^{40} - \zeta_{10} q^{41} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{45} - \zeta_{10}^{3} q^{49} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + 1) q^{50} + ( - \zeta_{10} + 1) q^{52} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{53} - 2 \zeta_{10}^{3} q^{58} + (\zeta_{10}^{2} - \zeta_{10}) q^{61} - \zeta_{10} q^{64} + (\zeta_{10}^{4} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{65} + (\zeta_{10}^{4} - \zeta_{10}) q^{68} - \zeta_{10}^{3} q^{72} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{73} + ( - \zeta_{10} + 1) q^{74} + (\zeta_{10}^{2} + 1) q^{80} + q^{81} + \zeta_{10}^{2} q^{82} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + 2) q^{85} + \zeta_{10}^{4} q^{89} + (\zeta_{10}^{4} + \zeta_{10}^{2}) q^{90} + (\zeta_{10}^{4} + 1) q^{97} + \zeta_{10}^{4} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - q^{4} - 2 q^{5} - q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} - q^{4} - 2 q^{5} - q^{8} + 4 q^{9} - 2 q^{10} - 2 q^{13} - q^{16} - 2 q^{17} - q^{18} + 3 q^{20} - 3 q^{25} + 3 q^{26} - 2 q^{29} + 4 q^{32} + 3 q^{34} - q^{36} + 3 q^{37} - 2 q^{40} - q^{41} - 2 q^{45} - q^{49} + 2 q^{50} + 3 q^{52} - 2 q^{53} - 2 q^{58} - 2 q^{61} - q^{64} + q^{65} - 2 q^{68} - q^{72} - 2 q^{73} + 3 q^{74} + 3 q^{80} + 4 q^{81} - q^{82} + 6 q^{85} - 2 q^{89} - 2 q^{90} + 3 q^{97} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(83\) \(129\)
\(\chi(n)\) \(-1\) \(\zeta_{10}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
0.809017 + 0.587785i
−0.309017 0.951057i
0.809017 0.587785i
−0.309017 + 0.951057i
−0.809017 0.587785i 0 0.309017 + 0.951057i −0.500000 1.53884i 0 0 0.309017 0.951057i 1.00000 −0.500000 + 1.53884i
59.1 0.309017 + 0.951057i 0 −0.809017 + 0.587785i −0.500000 + 0.363271i 0 0 −0.809017 0.587785i 1.00000 −0.500000 0.363271i
119.1 −0.809017 + 0.587785i 0 0.309017 0.951057i −0.500000 + 1.53884i 0 0 0.309017 + 0.951057i 1.00000 −0.500000 1.53884i
139.1 0.309017 0.951057i 0 −0.809017 0.587785i −0.500000 0.363271i 0 0 −0.809017 + 0.587785i 1.00000 −0.500000 + 0.363271i
\(n\): e.g. 2-40 or 990-1000
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.d even 5 1 inner
164.j odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 164.1.j.a 4
3.b odd 2 1 1476.1.ba.a 4
4.b odd 2 1 CM 164.1.j.a 4
8.b even 2 1 2624.1.bt.a 4
8.d odd 2 1 2624.1.bt.a 4
12.b even 2 1 1476.1.ba.a 4
41.d even 5 1 inner 164.1.j.a 4
123.k odd 10 1 1476.1.ba.a 4
164.j odd 10 1 inner 164.1.j.a 4
328.u even 10 1 2624.1.bt.a 4
328.x odd 10 1 2624.1.bt.a 4
492.w even 10 1 1476.1.ba.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.1.j.a 4 1.a even 1 1 trivial
164.1.j.a 4 4.b odd 2 1 CM
164.1.j.a 4 41.d even 5 1 inner
164.1.j.a 4 164.j odd 10 1 inner
1476.1.ba.a 4 3.b odd 2 1
1476.1.ba.a 4 12.b even 2 1
1476.1.ba.a 4 123.k odd 10 1
1476.1.ba.a 4 492.w even 10 1
2624.1.bt.a 4 8.b even 2 1
2624.1.bt.a 4 8.d odd 2 1
2624.1.bt.a 4 328.u even 10 1
2624.1.bt.a 4 328.x odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(164, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1 \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$41$ \( T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1 \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1 \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + T - 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$97$ \( T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1 \) Copy content Toggle raw display
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