Properties

Label 164.3.d.a
Level $164$
Weight $3$
Character orbit 164.d
Self dual yes
Analytic conductor $4.469$
Analytic rank $0$
Dimension $4$
CM discriminant -164
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [164,3,Mod(163,164)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(164, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("164.163");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 164.d (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(4.46867633551\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.3442688.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 20x^{2} + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} - \beta_{2} q^{3} + 4 q^{4} + \beta_{3} q^{5} + 2 \beta_{2} q^{6} + ( - 2 \beta_{2} - \beta_1) q^{7} - 8 q^{8} + ( - \beta_{3} + 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} - \beta_{2} q^{3} + 4 q^{4} + \beta_{3} q^{5} + 2 \beta_{2} q^{6} + ( - 2 \beta_{2} - \beta_1) q^{7} - 8 q^{8} + ( - \beta_{3} + 9) q^{9} - 2 \beta_{3} q^{10} + ( - 2 \beta_{2} + 5 \beta_1) q^{11} - 4 \beta_{2} q^{12} + (4 \beta_{2} + 2 \beta_1) q^{14} + (7 \beta_{2} - 11 \beta_1) q^{15} + 16 q^{16} + (2 \beta_{3} - 18) q^{18} + (\beta_{2} + 8 \beta_1) q^{19} + 4 \beta_{3} q^{20} + ( - \beta_{3} + 40) q^{21} + (4 \beta_{2} - 10 \beta_1) q^{22} + 8 \beta_{2} q^{24} + 57 q^{25} + ( - 7 \beta_{2} + 11 \beta_1) q^{27} + ( - 8 \beta_{2} - 4 \beta_1) q^{28} + ( - 14 \beta_{2} + 22 \beta_1) q^{30} - 32 q^{32} + ( - 7 \beta_{3} + 16) q^{33} + (11 \beta_{2} - 29 \beta_1) q^{35} + ( - 4 \beta_{3} + 36) q^{36} + 7 \beta_{3} q^{37} + ( - 2 \beta_{2} - 16 \beta_1) q^{38} - 8 \beta_{3} q^{40} - 41 q^{41} + (2 \beta_{3} - 80) q^{42} + ( - 8 \beta_{2} + 20 \beta_1) q^{44} + (9 \beta_{3} - 82) q^{45} + ( - 11 \beta_{2} + 20 \beta_1) q^{47} - 16 \beta_{2} q^{48} + (\beta_{3} + 49) q^{49} - 114 q^{50} + (14 \beta_{2} - 22 \beta_1) q^{54} + (29 \beta_{2} + 13 \beta_1) q^{55} + (16 \beta_{2} + 8 \beta_1) q^{56} + ( - 7 \beta_{3} - 50) q^{57} + (28 \beta_{2} - 44 \beta_1) q^{60} - 40 q^{61} + ( - 29 \beta_{2} + 20 \beta_1) q^{63} + 64 q^{64} + (14 \beta_{3} - 32) q^{66} + (22 \beta_{2} - 19 \beta_1) q^{67} + ( - 22 \beta_{2} + 58 \beta_1) q^{70} + (19 \beta_{2} - 28 \beta_1) q^{71} + (8 \beta_{3} - 72) q^{72} + 7 \beta_{3} q^{73} - 14 \beta_{3} q^{74} - 57 \beta_{2} q^{75} + (4 \beta_{2} + 32 \beta_1) q^{76} + ( - 17 \beta_{3} - 10) q^{77} + ( - 11 \beta_{2} - 28 \beta_1) q^{79} + 16 \beta_{3} q^{80} + ( - 9 \beta_{3} + 1) q^{81} + 82 q^{82} + ( - 4 \beta_{3} + 160) q^{84} + (16 \beta_{2} - 40 \beta_1) q^{88} + ( - 18 \beta_{3} + 164) q^{90} + (22 \beta_{2} - 40 \beta_1) q^{94} + (17 \beta_{2} + 67 \beta_1) q^{95} + 32 \beta_{2} q^{96} + ( - 2 \beta_{3} - 98) q^{98} + ( - 47 \beta_{2} + 32 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{2} + 16 q^{4} - 32 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{2} + 16 q^{4} - 32 q^{8} + 36 q^{9} + 64 q^{16} - 72 q^{18} + 160 q^{21} + 228 q^{25} - 128 q^{32} + 64 q^{33} + 144 q^{36} - 164 q^{41} - 320 q^{42} - 328 q^{45} + 196 q^{49} - 456 q^{50} - 200 q^{57} - 160 q^{61} + 256 q^{64} - 128 q^{66} - 288 q^{72} - 40 q^{77} + 4 q^{81} + 328 q^{82} + 640 q^{84} + 656 q^{90} - 392 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 20x^{2} + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 17\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{2} + 17\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(83\) \(129\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
−0.971913
4.36525
−4.36525
0.971913
−2.00000 −5.20148 4.00000 −9.05539 10.4030 −9.43105 −8.00000 18.0554 18.1108
163.2 −2.00000 −2.99075 4.00000 9.05539 5.98151 −10.3468 −8.00000 −0.0553851 −18.1108
163.3 −2.00000 2.99075 4.00000 9.05539 −5.98151 10.3468 −8.00000 −0.0553851 −18.1108
163.4 −2.00000 5.20148 4.00000 −9.05539 −10.4030 9.43105 −8.00000 18.0554 18.1108
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 164.3.d.a 4
4.b odd 2 1 inner 164.3.d.a 4
41.b even 2 1 inner 164.3.d.a 4
164.d odd 2 1 CM 164.3.d.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.3.d.a 4 1.a even 1 1 trivial
164.3.d.a 4 4.b odd 2 1 inner
164.3.d.a 4 41.b even 2 1 inner
164.3.d.a 4 164.d odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 36T_{3}^{2} + 242 \) acting on \(S_{3}^{\mathrm{new}}(164, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 36T^{2} + 242 \) Copy content Toggle raw display
$5$ \( (T^{2} - 82)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 196T^{2} + 9522 \) Copy content Toggle raw display
$11$ \( T^{4} - 484 T^{2} + 58482 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 1444 T^{2} + 9522 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 4018)^{2} \) Copy content Toggle raw display
$41$ \( (T + 41)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 8836 T^{2} + \cdots + 17393202 \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T + 40)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} - 17956 T^{2} + \cdots + 5190642 \) Copy content Toggle raw display
$71$ \( T^{4} - 20164 T^{2} + \cdots + 67954482 \) Copy content Toggle raw display
$73$ \( (T^{2} - 4018)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 24964 T^{2} + \cdots + 21661362 \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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