Properties

Label 144.12.a.g
Level $144$
Weight $12$
Character orbit 144.a
Self dual yes
Analytic conductor $110.641$
Analytic rank $1$
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] = [144,12,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, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 12 \)
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: \(110.641418001\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
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)
 
\(f(q)\) \(=\) \( q - 1870 q^{5} + 72312 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 1870 q^{5} + 72312 q^{7} + 147940 q^{11} - 1562858 q^{13} + 145774 q^{17} - 1096796 q^{19} - 60014264 q^{23} - 45331225 q^{25} + 19626954 q^{29} + 239950480 q^{31} - 135223440 q^{35} + 488238078 q^{37} - 47066010 q^{41} - 428866948 q^{43} + 450903216 q^{47} + 3251698601 q^{49} - 4336685950 q^{53} - 276647800 q^{55} - 8937556460 q^{59} + 4673884486 q^{61} + 2922544460 q^{65} - 7498937612 q^{67} - 27032101480 q^{71} + 11676141658 q^{73} + 10697837280 q^{77} - 2478876544 q^{79} + 42745596956 q^{83} - 272597380 q^{85} + 93270772662 q^{89} - 113013387696 q^{91} + 2051008520 q^{95} + 118032786914 q^{97}+O(q^{100}) \) 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 −1870.00 0 72312.0 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 144.12.a.g 1
3.b odd 2 1 48.12.a.b 1
4.b odd 2 1 72.12.a.a 1
12.b even 2 1 24.12.a.c 1
24.f even 2 1 192.12.a.e 1
24.h odd 2 1 192.12.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.12.a.c 1 12.b even 2 1
48.12.a.b 1 3.b odd 2 1
72.12.a.a 1 4.b odd 2 1
144.12.a.g 1 1.a even 1 1 trivial
192.12.a.e 1 24.f even 2 1
192.12.a.o 1 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 1870 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(144))\). 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
$5$ \( T + 1870 \) Copy content Toggle raw display
$7$ \( T - 72312 \) Copy content Toggle raw display
$11$ \( T - 147940 \) Copy content Toggle raw display
$13$ \( T + 1562858 \) Copy content Toggle raw display
$17$ \( T - 145774 \) Copy content Toggle raw display
$19$ \( T + 1096796 \) Copy content Toggle raw display
$23$ \( T + 60014264 \) Copy content Toggle raw display
$29$ \( T - 19626954 \) Copy content Toggle raw display
$31$ \( T - 239950480 \) Copy content Toggle raw display
$37$ \( T - 488238078 \) Copy content Toggle raw display
$41$ \( T + 47066010 \) Copy content Toggle raw display
$43$ \( T + 428866948 \) Copy content Toggle raw display
$47$ \( T - 450903216 \) Copy content Toggle raw display
$53$ \( T + 4336685950 \) Copy content Toggle raw display
$59$ \( T + 8937556460 \) Copy content Toggle raw display
$61$ \( T - 4673884486 \) Copy content Toggle raw display
$67$ \( T + 7498937612 \) Copy content Toggle raw display
$71$ \( T + 27032101480 \) Copy content Toggle raw display
$73$ \( T - 11676141658 \) Copy content Toggle raw display
$79$ \( T + 2478876544 \) Copy content Toggle raw display
$83$ \( T - 42745596956 \) Copy content Toggle raw display
$89$ \( T - 93270772662 \) Copy content Toggle raw display
$97$ \( T - 118032786914 \) Copy content Toggle raw display
show more
show less