Properties

Label 144.12.a.k
Level $144$
Weight $12$
Character orbit 144.a
Self dual yes
Analytic conductor $110.641$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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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 18)
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)
 
\(f(q)\) \(=\) \( q + 5280 q^{5} + 49036 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5280 q^{5} + 49036 q^{7} - 414336 q^{11} - 522982 q^{13} + 9499968 q^{17} - 13053944 q^{19} - 58755840 q^{23} - 20949725 q^{25} - 117142944 q^{29} - 142907156 q^{31} + 258910080 q^{35} + 718521806 q^{37} - 668055360 q^{41} - 141575864 q^{43} - 729235200 q^{47} + 427202553 q^{49} + 4917225312 q^{53} - 2187694080 q^{55} - 1408015104 q^{59} - 3223327018 q^{61} - 2761344960 q^{65} + 2358681328 q^{67} + 22245092352 q^{71} - 28036594330 q^{73} - 20317380096 q^{77} + 20685045676 q^{79} - 37818604416 q^{83} + 50159831040 q^{85} + 11288711808 q^{89} - 25644945352 q^{91} - 68924824320 q^{95} - 115724393266 q^{97}+O(q^{100}) \) 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 5280.00 0 49036.0 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.k 1
3.b odd 2 1 144.12.a.c 1
4.b odd 2 1 18.12.a.b 1
12.b even 2 1 18.12.a.d yes 1
36.f odd 6 2 162.12.c.h 2
36.h even 6 2 162.12.c.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.12.a.b 1 4.b odd 2 1
18.12.a.d yes 1 12.b even 2 1
144.12.a.c 1 3.b odd 2 1
144.12.a.k 1 1.a even 1 1 trivial
162.12.c.c 2 36.h even 6 2
162.12.c.h 2 36.f odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 5280 \) 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
$5$ \( T - 5280 \) Copy content Toggle raw display
$7$ \( T - 49036 \) Copy content Toggle raw display
$11$ \( T + 414336 \) Copy content Toggle raw display
$13$ \( T + 522982 \) Copy content Toggle raw display
$17$ \( T - 9499968 \) Copy content Toggle raw display
$19$ \( T + 13053944 \) Copy content Toggle raw display
$23$ \( T + 58755840 \) Copy content Toggle raw display
$29$ \( T + 117142944 \) Copy content Toggle raw display
$31$ \( T + 142907156 \) Copy content Toggle raw display
$37$ \( T - 718521806 \) Copy content Toggle raw display
$41$ \( T + 668055360 \) Copy content Toggle raw display
$43$ \( T + 141575864 \) Copy content Toggle raw display
$47$ \( T + 729235200 \) Copy content Toggle raw display
$53$ \( T - 4917225312 \) Copy content Toggle raw display
$59$ \( T + 1408015104 \) Copy content Toggle raw display
$61$ \( T + 3223327018 \) Copy content Toggle raw display
$67$ \( T - 2358681328 \) Copy content Toggle raw display
$71$ \( T - 22245092352 \) Copy content Toggle raw display
$73$ \( T + 28036594330 \) Copy content Toggle raw display
$79$ \( T - 20685045676 \) Copy content Toggle raw display
$83$ \( T + 37818604416 \) Copy content Toggle raw display
$89$ \( T - 11288711808 \) Copy content Toggle raw display
$97$ \( T + 115724393266 \) Copy content Toggle raw display
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