Properties

Label 160.2.c.b
Level $160$
Weight $2$
Character orbit 160.c
Analytic conductor $1.278$
Analytic rank $0$
Dimension $4$
CM discriminant -20
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [160,2,Mod(129,160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("160.129");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 160.c (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: no
Analytic conductor: \(1.27760643234\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{2} q^{5} + \beta_{3} q^{7} + (2 \beta_{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{2} q^{5} + \beta_{3} q^{7} + (2 \beta_{2} - 3) q^{9} + ( - \beta_{3} + 2 \beta_1) q^{15} + ( - 2 \beta_{2} - 2) q^{21} + ( - \beta_{3} - 2 \beta_1) q^{23} + 5 q^{25} + (2 \beta_{3} - 4 \beta_1) q^{27} + 6 q^{29} + ( - 2 \beta_{3} - \beta_1) q^{35} + 2 \beta_{2} q^{41} + ( - 2 \beta_{3} + 3 \beta_1) q^{43} + (3 \beta_{2} - 10) q^{45} + (\beta_{3} - 4 \beta_1) q^{47} + ( - 6 \beta_{2} - 7) q^{49} + 6 \beta_{2} q^{61} + (\beta_{3} + 2 \beta_1) q^{63} + (2 \beta_{3} + 3 \beta_1) q^{67} + ( - 2 \beta_{2} + 14) q^{69} + 5 \beta_1 q^{75} + ( - 6 \beta_{2} + 11) q^{81} + (2 \beta_{3} - 5 \beta_1) q^{83} + 6 \beta_1 q^{87} - 6 q^{89}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{9} - 8 q^{21} + 20 q^{25} + 24 q^{29} - 40 q^{45} - 28 q^{49} + 56 q^{69} + 44 q^{81} - 24 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{3} + 10\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} - 5\beta_1 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
1.61803i
0.618034i
0.618034i
1.61803i
0 3.23607i 0 2.23607 0 0.763932i 0 −7.47214 0
129.2 0 1.23607i 0 −2.23607 0 5.23607i 0 1.47214 0
129.3 0 1.23607i 0 −2.23607 0 5.23607i 0 1.47214 0
129.4 0 3.23607i 0 2.23607 0 0.763932i 0 −7.47214 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.2.c.b 4
3.b odd 2 1 1440.2.f.i 4
4.b odd 2 1 inner 160.2.c.b 4
5.b even 2 1 inner 160.2.c.b 4
5.c odd 4 1 800.2.a.j 2
5.c odd 4 1 800.2.a.n 2
8.b even 2 1 320.2.c.d 4
8.d odd 2 1 320.2.c.d 4
12.b even 2 1 1440.2.f.i 4
15.d odd 2 1 1440.2.f.i 4
15.e even 4 1 7200.2.a.cb 2
15.e even 4 1 7200.2.a.cr 2
16.e even 4 1 1280.2.f.g 4
16.e even 4 1 1280.2.f.h 4
16.f odd 4 1 1280.2.f.g 4
16.f odd 4 1 1280.2.f.h 4
20.d odd 2 1 CM 160.2.c.b 4
20.e even 4 1 800.2.a.j 2
20.e even 4 1 800.2.a.n 2
24.f even 2 1 2880.2.f.w 4
24.h odd 2 1 2880.2.f.w 4
40.e odd 2 1 320.2.c.d 4
40.f even 2 1 320.2.c.d 4
40.i odd 4 1 1600.2.a.z 2
40.i odd 4 1 1600.2.a.bd 2
40.k even 4 1 1600.2.a.z 2
40.k even 4 1 1600.2.a.bd 2
60.h even 2 1 1440.2.f.i 4
60.l odd 4 1 7200.2.a.cb 2
60.l odd 4 1 7200.2.a.cr 2
80.k odd 4 1 1280.2.f.g 4
80.k odd 4 1 1280.2.f.h 4
80.q even 4 1 1280.2.f.g 4
80.q even 4 1 1280.2.f.h 4
120.i odd 2 1 2880.2.f.w 4
120.m even 2 1 2880.2.f.w 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.c.b 4 1.a even 1 1 trivial
160.2.c.b 4 4.b odd 2 1 inner
160.2.c.b 4 5.b even 2 1 inner
160.2.c.b 4 20.d odd 2 1 CM
320.2.c.d 4 8.b even 2 1
320.2.c.d 4 8.d odd 2 1
320.2.c.d 4 40.e odd 2 1
320.2.c.d 4 40.f even 2 1
800.2.a.j 2 5.c odd 4 1
800.2.a.j 2 20.e even 4 1
800.2.a.n 2 5.c odd 4 1
800.2.a.n 2 20.e even 4 1
1280.2.f.g 4 16.e even 4 1
1280.2.f.g 4 16.f odd 4 1
1280.2.f.g 4 80.k odd 4 1
1280.2.f.g 4 80.q even 4 1
1280.2.f.h 4 16.e even 4 1
1280.2.f.h 4 16.f odd 4 1
1280.2.f.h 4 80.k odd 4 1
1280.2.f.h 4 80.q even 4 1
1440.2.f.i 4 3.b odd 2 1
1440.2.f.i 4 12.b even 2 1
1440.2.f.i 4 15.d odd 2 1
1440.2.f.i 4 60.h even 2 1
1600.2.a.z 2 40.i odd 4 1
1600.2.a.z 2 40.k even 4 1
1600.2.a.bd 2 40.i odd 4 1
1600.2.a.bd 2 40.k even 4 1
2880.2.f.w 4 24.f even 2 1
2880.2.f.w 4 24.h odd 2 1
2880.2.f.w 4 120.i odd 2 1
2880.2.f.w 4 120.m even 2 1
7200.2.a.cb 2 15.e even 4 1
7200.2.a.cb 2 60.l odd 4 1
7200.2.a.cr 2 15.e even 4 1
7200.2.a.cr 2 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 12T_{3}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(160, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 28T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 92T^{2} + 1936 \) Copy content Toggle raw display
$29$ \( (T - 6)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 172T^{2} + 5776 \) Copy content Toggle raw display
$47$ \( T^{4} + 188T^{2} + 16 \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 180)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 268 T^{2} + 13456 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 332T^{2} + 5776 \) Copy content Toggle raw display
$89$ \( (T + 6)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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