Properties

Label 160.2.n.b
Level $160$
Weight $2$
Character orbit 160.n
Analytic conductor $1.278$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [160,2,Mod(63,160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(160, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("160.63");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 160.n (of order \(4\), degree \(2\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - i - 1) q^{3} + ( - 2 i + 1) q^{5} + (i - 1) q^{7} - i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - i - 1) q^{3} + ( - 2 i + 1) q^{5} + (i - 1) q^{7} - i q^{9} - 6 i q^{11} + (i - 1) q^{13} + (i - 3) q^{15} + (i + 1) q^{17} + 4 q^{19} + 2 q^{21} + (5 i + 5) q^{23} + ( - 4 i - 3) q^{25} + (4 i - 4) q^{27} + 8 i q^{29} + 2 i q^{31} + (6 i - 6) q^{33} + (3 i + 1) q^{35} + ( - 5 i - 5) q^{37} + 2 q^{39} + 6 q^{41} + (3 i + 3) q^{43} + ( - i - 2) q^{45} + ( - 7 i + 7) q^{47} + 5 i q^{49} - 2 i q^{51} + (i - 1) q^{53} + ( - 6 i - 12) q^{55} + ( - 4 i - 4) q^{57} + 4 q^{59} + 2 q^{61} + (i + 1) q^{63} + (3 i + 1) q^{65} + (7 i - 7) q^{67} - 10 i q^{69} - 6 i q^{71} + ( - 9 i + 9) q^{73} + (7 i - 1) q^{75} + (6 i + 6) q^{77} - 8 q^{79} + 5 q^{81} + ( - 5 i - 5) q^{83} + ( - i + 3) q^{85} + ( - 8 i + 8) q^{87} - 2 i q^{91} + ( - 2 i + 2) q^{93} + ( - 8 i + 4) q^{95} + ( - 3 i - 3) q^{97} - 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} - 2 q^{13} - 6 q^{15} + 2 q^{17} + 8 q^{19} + 4 q^{21} + 10 q^{23} - 6 q^{25} - 8 q^{27} - 12 q^{33} + 2 q^{35} - 10 q^{37} + 4 q^{39} + 12 q^{41} + 6 q^{43} - 4 q^{45} + 14 q^{47} - 2 q^{53} - 24 q^{55} - 8 q^{57} + 8 q^{59} + 4 q^{61} + 2 q^{63} + 2 q^{65} - 14 q^{67} + 18 q^{73} - 2 q^{75} + 12 q^{77} - 16 q^{79} + 10 q^{81} - 10 q^{83} + 6 q^{85} + 16 q^{87} + 4 q^{93} + 8 q^{95} - 6 q^{97} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1
1.00000i
1.00000i
0 −1.00000 + 1.00000i 0 1.00000 + 2.00000i 0 −1.00000 1.00000i 0 1.00000i 0
127.1 0 −1.00000 1.00000i 0 1.00000 2.00000i 0 −1.00000 + 1.00000i 0 1.00000i 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.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.2.n.b 2
3.b odd 2 1 1440.2.x.b 2
4.b odd 2 1 160.2.n.e yes 2
5.b even 2 1 800.2.n.h 2
5.c odd 4 1 160.2.n.e yes 2
5.c odd 4 1 800.2.n.c 2
8.b even 2 1 320.2.n.f 2
8.d odd 2 1 320.2.n.c 2
12.b even 2 1 1440.2.x.e 2
15.e even 4 1 1440.2.x.e 2
16.e even 4 1 1280.2.o.e 2
16.e even 4 1 1280.2.o.k 2
16.f odd 4 1 1280.2.o.d 2
16.f odd 4 1 1280.2.o.n 2
20.d odd 2 1 800.2.n.c 2
20.e even 4 1 inner 160.2.n.b 2
20.e even 4 1 800.2.n.h 2
40.e odd 2 1 1600.2.n.j 2
40.f even 2 1 1600.2.n.e 2
40.i odd 4 1 320.2.n.c 2
40.i odd 4 1 1600.2.n.j 2
40.k even 4 1 320.2.n.f 2
40.k even 4 1 1600.2.n.e 2
60.l odd 4 1 1440.2.x.b 2
80.i odd 4 1 1280.2.o.n 2
80.j even 4 1 1280.2.o.k 2
80.s even 4 1 1280.2.o.e 2
80.t odd 4 1 1280.2.o.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.n.b 2 1.a even 1 1 trivial
160.2.n.b 2 20.e even 4 1 inner
160.2.n.e yes 2 4.b odd 2 1
160.2.n.e yes 2 5.c odd 4 1
320.2.n.c 2 8.d odd 2 1
320.2.n.c 2 40.i odd 4 1
320.2.n.f 2 8.b even 2 1
320.2.n.f 2 40.k even 4 1
800.2.n.c 2 5.c odd 4 1
800.2.n.c 2 20.d odd 2 1
800.2.n.h 2 5.b even 2 1
800.2.n.h 2 20.e even 4 1
1280.2.o.d 2 16.f odd 4 1
1280.2.o.d 2 80.t odd 4 1
1280.2.o.e 2 16.e even 4 1
1280.2.o.e 2 80.s even 4 1
1280.2.o.k 2 16.e even 4 1
1280.2.o.k 2 80.j even 4 1
1280.2.o.n 2 16.f odd 4 1
1280.2.o.n 2 80.i odd 4 1
1440.2.x.b 2 3.b odd 2 1
1440.2.x.b 2 60.l odd 4 1
1440.2.x.e 2 12.b even 2 1
1440.2.x.e 2 15.e even 4 1
1600.2.n.e 2 40.f even 2 1
1600.2.n.e 2 40.k even 4 1
1600.2.n.j 2 40.e odd 2 1
1600.2.n.j 2 40.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(160, [\chi])\):

\( T_{3}^{2} + 2T_{3} + 2 \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$5$ \( T^{2} - 2T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$11$ \( T^{2} + 36 \) Copy content Toggle raw display
$13$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$17$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$29$ \( T^{2} + 64 \) Copy content Toggle raw display
$31$ \( T^{2} + 4 \) Copy content Toggle raw display
$37$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$47$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$53$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$59$ \( (T - 4)^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 14T + 98 \) Copy content Toggle raw display
$71$ \( T^{2} + 36 \) Copy content Toggle raw display
$73$ \( T^{2} - 18T + 162 \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
show more
show less