Properties

Label 720.2.a.h
Level $720$
Weight $2$
Character orbit 720.a
Self dual yes
Analytic conductor $5.749$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [720,2,Mod(1,720)]
 
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([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.a (trivial)

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: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} - 2 q^{7} + 2 q^{13} + 6 q^{17} + 4 q^{19} + 6 q^{23} + q^{25} - 6 q^{29} + 4 q^{31} - 2 q^{35} + 2 q^{37} - 6 q^{41} + 10 q^{43} - 6 q^{47} - 3 q^{49} + 6 q^{53} + 12 q^{59} + 2 q^{61} + 2 q^{65} - 2 q^{67} - 12 q^{71} + 2 q^{73} - 8 q^{79} + 6 q^{83} + 6 q^{85} + 6 q^{89} - 4 q^{91} + 4 q^{95} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 0 1.00000 0 −2.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(5\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.a.h 1
3.b odd 2 1 80.2.a.b 1
4.b odd 2 1 180.2.a.a 1
5.b even 2 1 3600.2.a.be 1
5.c odd 4 2 3600.2.f.j 2
8.b even 2 1 2880.2.a.f 1
8.d odd 2 1 2880.2.a.m 1
12.b even 2 1 20.2.a.a 1
15.d odd 2 1 400.2.a.c 1
15.e even 4 2 400.2.c.b 2
20.d odd 2 1 900.2.a.b 1
20.e even 4 2 900.2.d.c 2
21.c even 2 1 3920.2.a.h 1
24.f even 2 1 320.2.a.f 1
24.h odd 2 1 320.2.a.a 1
28.d even 2 1 8820.2.a.g 1
33.d even 2 1 9680.2.a.ba 1
36.f odd 6 2 1620.2.i.b 2
36.h even 6 2 1620.2.i.h 2
48.i odd 4 2 1280.2.d.g 2
48.k even 4 2 1280.2.d.c 2
60.h even 2 1 100.2.a.a 1
60.l odd 4 2 100.2.c.a 2
84.h odd 2 1 980.2.a.h 1
84.j odd 6 2 980.2.i.c 2
84.n even 6 2 980.2.i.i 2
120.i odd 2 1 1600.2.a.w 1
120.m even 2 1 1600.2.a.c 1
120.q odd 4 2 1600.2.c.d 2
120.w even 4 2 1600.2.c.e 2
132.d odd 2 1 2420.2.a.a 1
156.h even 2 1 3380.2.a.c 1
156.l odd 4 2 3380.2.f.b 2
204.h even 2 1 5780.2.a.f 1
204.l even 4 2 5780.2.c.a 2
228.b odd 2 1 7220.2.a.f 1
420.o odd 2 1 4900.2.a.e 1
420.w even 4 2 4900.2.e.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.a.a 1 12.b even 2 1
80.2.a.b 1 3.b odd 2 1
100.2.a.a 1 60.h even 2 1
100.2.c.a 2 60.l odd 4 2
180.2.a.a 1 4.b odd 2 1
320.2.a.a 1 24.h odd 2 1
320.2.a.f 1 24.f even 2 1
400.2.a.c 1 15.d odd 2 1
400.2.c.b 2 15.e even 4 2
720.2.a.h 1 1.a even 1 1 trivial
900.2.a.b 1 20.d odd 2 1
900.2.d.c 2 20.e even 4 2
980.2.a.h 1 84.h odd 2 1
980.2.i.c 2 84.j odd 6 2
980.2.i.i 2 84.n even 6 2
1280.2.d.c 2 48.k even 4 2
1280.2.d.g 2 48.i odd 4 2
1600.2.a.c 1 120.m even 2 1
1600.2.a.w 1 120.i odd 2 1
1600.2.c.d 2 120.q odd 4 2
1600.2.c.e 2 120.w even 4 2
1620.2.i.b 2 36.f odd 6 2
1620.2.i.h 2 36.h even 6 2
2420.2.a.a 1 132.d odd 2 1
2880.2.a.f 1 8.b even 2 1
2880.2.a.m 1 8.d odd 2 1
3380.2.a.c 1 156.h even 2 1
3380.2.f.b 2 156.l odd 4 2
3600.2.a.be 1 5.b even 2 1
3600.2.f.j 2 5.c odd 4 2
3920.2.a.h 1 21.c even 2 1
4900.2.a.e 1 420.o odd 2 1
4900.2.e.f 2 420.w even 4 2
5780.2.a.f 1 204.h even 2 1
5780.2.c.a 2 204.l even 4 2
7220.2.a.f 1 228.b odd 2 1
8820.2.a.g 1 28.d even 2 1
9680.2.a.ba 1 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(720))\):

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

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
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T - 6 \) Copy content Toggle raw display
$19$ \( T - 4 \) Copy content Toggle raw display
$23$ \( T - 6 \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T - 4 \) Copy content Toggle raw display
$37$ \( T - 2 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T - 10 \) Copy content Toggle raw display
$47$ \( T + 6 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T - 12 \) Copy content Toggle raw display
$61$ \( T - 2 \) Copy content Toggle raw display
$67$ \( T + 2 \) Copy content Toggle raw display
$71$ \( T + 12 \) Copy content Toggle raw display
$73$ \( T - 2 \) Copy content Toggle raw display
$79$ \( T + 8 \) Copy content Toggle raw display
$83$ \( T - 6 \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T - 2 \) Copy content Toggle raw display
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