Properties

Label 144.4.a.c
Level $144$
Weight $4$
Character orbit 144.a
Self dual yes
Analytic conductor $8.496$
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] = [144,4,Mod(1,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, 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("144.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 144.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: \(8.49627504083\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
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)
 
\(f(q)\) \(=\) \( q - 6 q^{5} + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 6 q^{5} + 16 q^{7} + 12 q^{11} + 38 q^{13} + 126 q^{17} - 20 q^{19} + 168 q^{23} - 89 q^{25} - 30 q^{29} + 88 q^{31} - 96 q^{35} + 254 q^{37} - 42 q^{41} + 52 q^{43} - 96 q^{47} - 87 q^{49} - 198 q^{53} - 72 q^{55} - 660 q^{59} - 538 q^{61} - 228 q^{65} - 884 q^{67} + 792 q^{71} + 218 q^{73} + 192 q^{77} + 520 q^{79} - 492 q^{83} - 756 q^{85} - 810 q^{89} + 608 q^{91} + 120 q^{95} + 1154 q^{97}+O(q^{100}) \) 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 −6.00000 0 16.0000 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 144.4.a.c 1
3.b odd 2 1 48.4.a.c 1
4.b odd 2 1 18.4.a.a 1
8.b even 2 1 576.4.a.r 1
8.d odd 2 1 576.4.a.q 1
12.b even 2 1 6.4.a.a 1
15.d odd 2 1 1200.4.a.b 1
15.e even 4 2 1200.4.f.j 2
20.d odd 2 1 450.4.a.h 1
20.e even 4 2 450.4.c.e 2
21.c even 2 1 2352.4.a.e 1
24.f even 2 1 192.4.a.i 1
24.h odd 2 1 192.4.a.c 1
28.d even 2 1 882.4.a.n 1
28.f even 6 2 882.4.g.f 2
28.g odd 6 2 882.4.g.i 2
36.f odd 6 2 162.4.c.c 2
36.h even 6 2 162.4.c.f 2
44.c even 2 1 2178.4.a.e 1
48.i odd 4 2 768.4.d.c 2
48.k even 4 2 768.4.d.n 2
60.h even 2 1 150.4.a.i 1
60.l odd 4 2 150.4.c.d 2
84.h odd 2 1 294.4.a.e 1
84.j odd 6 2 294.4.e.g 2
84.n even 6 2 294.4.e.h 2
132.d odd 2 1 726.4.a.f 1
156.h even 2 1 1014.4.a.g 1
156.l odd 4 2 1014.4.b.d 2
204.h even 2 1 1734.4.a.d 1
228.b odd 2 1 2166.4.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 12.b even 2 1
18.4.a.a 1 4.b odd 2 1
48.4.a.c 1 3.b odd 2 1
144.4.a.c 1 1.a even 1 1 trivial
150.4.a.i 1 60.h even 2 1
150.4.c.d 2 60.l odd 4 2
162.4.c.c 2 36.f odd 6 2
162.4.c.f 2 36.h even 6 2
192.4.a.c 1 24.h odd 2 1
192.4.a.i 1 24.f even 2 1
294.4.a.e 1 84.h odd 2 1
294.4.e.g 2 84.j odd 6 2
294.4.e.h 2 84.n even 6 2
450.4.a.h 1 20.d odd 2 1
450.4.c.e 2 20.e even 4 2
576.4.a.q 1 8.d odd 2 1
576.4.a.r 1 8.b even 2 1
726.4.a.f 1 132.d odd 2 1
768.4.d.c 2 48.i odd 4 2
768.4.d.n 2 48.k even 4 2
882.4.a.n 1 28.d even 2 1
882.4.g.f 2 28.f even 6 2
882.4.g.i 2 28.g odd 6 2
1014.4.a.g 1 156.h even 2 1
1014.4.b.d 2 156.l odd 4 2
1200.4.a.b 1 15.d odd 2 1
1200.4.f.j 2 15.e even 4 2
1734.4.a.d 1 204.h even 2 1
2166.4.a.i 1 228.b odd 2 1
2178.4.a.e 1 44.c even 2 1
2352.4.a.e 1 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 6 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(144))\). 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
$5$ \( T + 6 \) Copy content Toggle raw display
$7$ \( T - 16 \) Copy content Toggle raw display
$11$ \( T - 12 \) Copy content Toggle raw display
$13$ \( T - 38 \) Copy content Toggle raw display
$17$ \( T - 126 \) Copy content Toggle raw display
$19$ \( T + 20 \) Copy content Toggle raw display
$23$ \( T - 168 \) Copy content Toggle raw display
$29$ \( T + 30 \) Copy content Toggle raw display
$31$ \( T - 88 \) Copy content Toggle raw display
$37$ \( T - 254 \) Copy content Toggle raw display
$41$ \( T + 42 \) Copy content Toggle raw display
$43$ \( T - 52 \) Copy content Toggle raw display
$47$ \( T + 96 \) Copy content Toggle raw display
$53$ \( T + 198 \) Copy content Toggle raw display
$59$ \( T + 660 \) Copy content Toggle raw display
$61$ \( T + 538 \) Copy content Toggle raw display
$67$ \( T + 884 \) Copy content Toggle raw display
$71$ \( T - 792 \) Copy content Toggle raw display
$73$ \( T - 218 \) Copy content Toggle raw display
$79$ \( T - 520 \) Copy content Toggle raw display
$83$ \( T + 492 \) Copy content Toggle raw display
$89$ \( T + 810 \) Copy content Toggle raw display
$97$ \( T - 1154 \) Copy content Toggle raw display
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