Properties

Label 720.2.x.c
Level $720$
Weight $2$
Character orbit 720.x
Analytic conductor $5.749$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [720,2,Mod(127,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 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("720.127");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.x (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: \(5.74922894553\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (2 i + 1) q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 i + 1) q^{5} + ( - i + 1) q^{13} + (5 i + 5) q^{17} + (4 i - 3) q^{25} + 10 i q^{29} + ( - 7 i - 7) q^{37} + 10 q^{41} + 7 i q^{49} + (5 i - 5) q^{53} + 12 q^{61} + (i + 3) q^{65} + ( - 11 i + 11) q^{73} + (15 i - 5) q^{85} - 10 i q^{89} + ( - 13 i - 13) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} + 2 q^{13} + 10 q^{17} - 6 q^{25} - 14 q^{37} + 20 q^{41} - 10 q^{53} + 24 q^{61} + 6 q^{65} + 22 q^{73} - 10 q^{85} - 26 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(-1\) \(i\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
1.00000i
1.00000i
0 0 0 1.00000 + 2.00000i 0 0 0 0 0
703.1 0 0 0 1.00000 2.00000i 0 0 0 0 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
5.c odd 4 1 inner
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 720.2.x.c yes 2
3.b odd 2 1 720.2.x.b 2
4.b odd 2 1 CM 720.2.x.c yes 2
5.b even 2 1 3600.2.x.a 2
5.c odd 4 1 inner 720.2.x.c yes 2
5.c odd 4 1 3600.2.x.a 2
12.b even 2 1 720.2.x.b 2
15.d odd 2 1 3600.2.x.b 2
15.e even 4 1 720.2.x.b 2
15.e even 4 1 3600.2.x.b 2
20.d odd 2 1 3600.2.x.a 2
20.e even 4 1 inner 720.2.x.c yes 2
20.e even 4 1 3600.2.x.a 2
60.h even 2 1 3600.2.x.b 2
60.l odd 4 1 720.2.x.b 2
60.l odd 4 1 3600.2.x.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.2.x.b 2 3.b odd 2 1
720.2.x.b 2 12.b even 2 1
720.2.x.b 2 15.e even 4 1
720.2.x.b 2 60.l odd 4 1
720.2.x.c yes 2 1.a even 1 1 trivial
720.2.x.c yes 2 4.b odd 2 1 CM
720.2.x.c yes 2 5.c odd 4 1 inner
720.2.x.c yes 2 20.e even 4 1 inner
3600.2.x.a 2 5.b even 2 1
3600.2.x.a 2 5.c odd 4 1
3600.2.x.a 2 20.d odd 2 1
3600.2.x.a 2 20.e even 4 1
3600.2.x.b 2 15.d odd 2 1
3600.2.x.b 2 15.e even 4 1
3600.2.x.b 2 60.h even 2 1
3600.2.x.b 2 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(720, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} + 2 \) Copy content Toggle raw display
\( T_{17}^{2} - 10T_{17} + 50 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 2T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$17$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 100 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 14T + 98 \) Copy content Toggle raw display
$41$ \( (T - 10)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 12)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 22T + 242 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 100 \) Copy content Toggle raw display
$97$ \( T^{2} + 26T + 338 \) Copy content Toggle raw display
show more
show less