Properties

Label 72.4.a.c
Level $72$
Weight $4$
Character orbit 72.a
Self dual yes
Analytic conductor $4.248$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [72,4,Mod(1,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 72.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: \(4.24813752041\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 8)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{5} + 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{5} + 24 q^{7} + 44 q^{11} + 22 q^{13} - 50 q^{17} + 44 q^{19} + 56 q^{23} - 121 q^{25} - 198 q^{29} - 160 q^{31} + 48 q^{35} - 162 q^{37} + 198 q^{41} + 52 q^{43} - 528 q^{47} + 233 q^{49} + 242 q^{53} + 88 q^{55} + 668 q^{59} + 550 q^{61} + 44 q^{65} + 188 q^{67} - 728 q^{71} + 154 q^{73} + 1056 q^{77} - 656 q^{79} - 236 q^{83} - 100 q^{85} - 714 q^{89} + 528 q^{91} + 88 q^{95} - 478 q^{97}+O(q^{100}) \) 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 2.00000 0 24.0000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 72.4.a.c 1
3.b odd 2 1 8.4.a.a 1
4.b odd 2 1 144.4.a.e 1
5.b even 2 1 1800.4.a.d 1
5.c odd 4 2 1800.4.f.u 2
8.b even 2 1 576.4.a.k 1
8.d odd 2 1 576.4.a.j 1
9.c even 3 2 648.4.i.e 2
9.d odd 6 2 648.4.i.h 2
12.b even 2 1 16.4.a.a 1
15.d odd 2 1 200.4.a.g 1
15.e even 4 2 200.4.c.e 2
21.c even 2 1 392.4.a.e 1
21.g even 6 2 392.4.i.b 2
21.h odd 6 2 392.4.i.g 2
24.f even 2 1 64.4.a.b 1
24.h odd 2 1 64.4.a.d 1
33.d even 2 1 968.4.a.a 1
39.d odd 2 1 1352.4.a.a 1
48.i odd 4 2 256.4.b.a 2
48.k even 4 2 256.4.b.g 2
51.c odd 2 1 2312.4.a.a 1
60.h even 2 1 400.4.a.g 1
60.l odd 4 2 400.4.c.i 2
84.h odd 2 1 784.4.a.e 1
120.i odd 2 1 1600.4.a.o 1
120.m even 2 1 1600.4.a.bm 1
132.d odd 2 1 1936.4.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.a.a 1 3.b odd 2 1
16.4.a.a 1 12.b even 2 1
64.4.a.b 1 24.f even 2 1
64.4.a.d 1 24.h odd 2 1
72.4.a.c 1 1.a even 1 1 trivial
144.4.a.e 1 4.b odd 2 1
200.4.a.g 1 15.d odd 2 1
200.4.c.e 2 15.e even 4 2
256.4.b.a 2 48.i odd 4 2
256.4.b.g 2 48.k even 4 2
392.4.a.e 1 21.c even 2 1
392.4.i.b 2 21.g even 6 2
392.4.i.g 2 21.h odd 6 2
400.4.a.g 1 60.h even 2 1
400.4.c.i 2 60.l odd 4 2
576.4.a.j 1 8.d odd 2 1
576.4.a.k 1 8.b even 2 1
648.4.i.e 2 9.c even 3 2
648.4.i.h 2 9.d odd 6 2
784.4.a.e 1 84.h odd 2 1
968.4.a.a 1 33.d even 2 1
1352.4.a.a 1 39.d odd 2 1
1600.4.a.o 1 120.i odd 2 1
1600.4.a.bm 1 120.m even 2 1
1800.4.a.d 1 5.b even 2 1
1800.4.f.u 2 5.c odd 4 2
1936.4.a.l 1 132.d odd 2 1
2312.4.a.a 1 51.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 2 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(72))\). 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
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T - 24 \) Copy content Toggle raw display
$11$ \( T - 44 \) Copy content Toggle raw display
$13$ \( T - 22 \) Copy content Toggle raw display
$17$ \( T + 50 \) Copy content Toggle raw display
$19$ \( T - 44 \) Copy content Toggle raw display
$23$ \( T - 56 \) Copy content Toggle raw display
$29$ \( T + 198 \) Copy content Toggle raw display
$31$ \( T + 160 \) Copy content Toggle raw display
$37$ \( T + 162 \) Copy content Toggle raw display
$41$ \( T - 198 \) Copy content Toggle raw display
$43$ \( T - 52 \) Copy content Toggle raw display
$47$ \( T + 528 \) Copy content Toggle raw display
$53$ \( T - 242 \) Copy content Toggle raw display
$59$ \( T - 668 \) Copy content Toggle raw display
$61$ \( T - 550 \) Copy content Toggle raw display
$67$ \( T - 188 \) Copy content Toggle raw display
$71$ \( T + 728 \) Copy content Toggle raw display
$73$ \( T - 154 \) Copy content Toggle raw display
$79$ \( T + 656 \) Copy content Toggle raw display
$83$ \( T + 236 \) Copy content Toggle raw display
$89$ \( T + 714 \) Copy content Toggle raw display
$97$ \( T + 478 \) Copy content Toggle raw display
show more
show less