Properties

Label 720.1.j.a
Level $720$
Weight $1$
Character orbit 720.j
Self dual yes
Analytic conductor $0.359$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -4, -20, 5
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [720,1,Mod(559,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.559");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 720.j (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: \(0.359326809096\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 80)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(i, \sqrt{5})\)
Artin image: $D_4$
Artin field: Galois closure of 4.0.2880.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} + q^{25} - 2 q^{29} + 2 q^{41} - q^{49} - 2 q^{61} - 2 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
0
0 0 0 1.00000 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.b even 2 1 RM by \(\Q(\sqrt{5}) \)
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.1.j.a 1
3.b odd 2 1 80.1.h.a 1
4.b odd 2 1 CM 720.1.j.a 1
5.b even 2 1 RM 720.1.j.a 1
5.c odd 4 2 3600.1.e.a 1
8.b even 2 1 2880.1.j.a 1
8.d odd 2 1 2880.1.j.a 1
12.b even 2 1 80.1.h.a 1
15.d odd 2 1 80.1.h.a 1
15.e even 4 2 400.1.b.a 1
20.d odd 2 1 CM 720.1.j.a 1
20.e even 4 2 3600.1.e.a 1
21.c even 2 1 3920.1.j.a 1
21.g even 6 2 3920.1.bt.a 2
21.h odd 6 2 3920.1.bt.b 2
24.f even 2 1 320.1.h.a 1
24.h odd 2 1 320.1.h.a 1
40.e odd 2 1 2880.1.j.a 1
40.f even 2 1 2880.1.j.a 1
48.i odd 4 2 1280.1.e.a 2
48.k even 4 2 1280.1.e.a 2
60.h even 2 1 80.1.h.a 1
60.l odd 4 2 400.1.b.a 1
84.h odd 2 1 3920.1.j.a 1
84.j odd 6 2 3920.1.bt.a 2
84.n even 6 2 3920.1.bt.b 2
105.g even 2 1 3920.1.j.a 1
105.o odd 6 2 3920.1.bt.b 2
105.p even 6 2 3920.1.bt.a 2
120.i odd 2 1 320.1.h.a 1
120.m even 2 1 320.1.h.a 1
120.q odd 4 2 1600.1.b.a 1
120.w even 4 2 1600.1.b.a 1
240.t even 4 2 1280.1.e.a 2
240.bm odd 4 2 1280.1.e.a 2
420.o odd 2 1 3920.1.j.a 1
420.ba even 6 2 3920.1.bt.b 2
420.be odd 6 2 3920.1.bt.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.1.h.a 1 3.b odd 2 1
80.1.h.a 1 12.b even 2 1
80.1.h.a 1 15.d odd 2 1
80.1.h.a 1 60.h even 2 1
320.1.h.a 1 24.f even 2 1
320.1.h.a 1 24.h odd 2 1
320.1.h.a 1 120.i odd 2 1
320.1.h.a 1 120.m even 2 1
400.1.b.a 1 15.e even 4 2
400.1.b.a 1 60.l odd 4 2
720.1.j.a 1 1.a even 1 1 trivial
720.1.j.a 1 4.b odd 2 1 CM
720.1.j.a 1 5.b even 2 1 RM
720.1.j.a 1 20.d odd 2 1 CM
1280.1.e.a 2 48.i odd 4 2
1280.1.e.a 2 48.k even 4 2
1280.1.e.a 2 240.t even 4 2
1280.1.e.a 2 240.bm odd 4 2
1600.1.b.a 1 120.q odd 4 2
1600.1.b.a 1 120.w even 4 2
2880.1.j.a 1 8.b even 2 1
2880.1.j.a 1 8.d odd 2 1
2880.1.j.a 1 40.e odd 2 1
2880.1.j.a 1 40.f even 2 1
3600.1.e.a 1 5.c odd 4 2
3600.1.e.a 1 20.e even 4 2
3920.1.j.a 1 21.c even 2 1
3920.1.j.a 1 84.h odd 2 1
3920.1.j.a 1 105.g even 2 1
3920.1.j.a 1 420.o odd 2 1
3920.1.bt.a 2 21.g even 6 2
3920.1.bt.a 2 84.j odd 6 2
3920.1.bt.a 2 105.p even 6 2
3920.1.bt.a 2 420.be odd 6 2
3920.1.bt.b 2 21.h odd 6 2
3920.1.bt.b 2 84.n even 6 2
3920.1.bt.b 2 105.o odd 6 2
3920.1.bt.b 2 420.ba even 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(720, [\chi])\).

Hecke characteristic polynomials

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