Properties

Label 144.12.a.b
Level $144$
Weight $12$
Character orbit 144.a
Self dual yes
Analytic conductor $110.641$
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] = [144,12,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: \(0\)
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)
 
\(f(q)\) \(=\) \( q - 5766 q^{5} - 72464 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 5766 q^{5} - 72464 q^{7} - 408948 q^{11} + 1367558 q^{13} - 5422914 q^{17} - 15166100 q^{19} - 52194072 q^{23} - 15581369 q^{25} - 118581150 q^{29} + 57652408 q^{31} + 417827424 q^{35} - 375985186 q^{37} - 856316202 q^{41} + 1245189172 q^{43} - 1306762656 q^{47} + 3273704553 q^{49} - 409556358 q^{53} + 2357994168 q^{55} - 2882866260 q^{59} + 5731767302 q^{61} - 7885339428 q^{65} - 3893272244 q^{67} - 9075890088 q^{71} - 15571822822 q^{73} + 29634007872 q^{77} + 30196762600 q^{79} + 23135252628 q^{83} + 31268522124 q^{85} + 25614819990 q^{89} - 99098722912 q^{91} + 87447732600 q^{95} - 61937553406 q^{97}+O(q^{100}) \) 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 −5766.00 0 −72464.0 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.b 1
3.b odd 2 1 48.12.a.h 1
4.b odd 2 1 18.12.a.c 1
12.b even 2 1 6.12.a.a 1
24.f even 2 1 192.12.a.l 1
24.h odd 2 1 192.12.a.b 1
36.f odd 6 2 162.12.c.d 2
36.h even 6 2 162.12.c.g 2
60.h even 2 1 150.12.a.g 1
60.l odd 4 2 150.12.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.12.a.a 1 12.b even 2 1
18.12.a.c 1 4.b odd 2 1
48.12.a.h 1 3.b odd 2 1
144.12.a.b 1 1.a even 1 1 trivial
150.12.a.g 1 60.h even 2 1
150.12.c.f 2 60.l odd 4 2
162.12.c.d 2 36.f odd 6 2
162.12.c.g 2 36.h even 6 2
192.12.a.b 1 24.h odd 2 1
192.12.a.l 1 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 5766 \) 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
$5$ \( T + 5766 \) Copy content Toggle raw display
$7$ \( T + 72464 \) Copy content Toggle raw display
$11$ \( T + 408948 \) Copy content Toggle raw display
$13$ \( T - 1367558 \) Copy content Toggle raw display
$17$ \( T + 5422914 \) Copy content Toggle raw display
$19$ \( T + 15166100 \) Copy content Toggle raw display
$23$ \( T + 52194072 \) Copy content Toggle raw display
$29$ \( T + 118581150 \) Copy content Toggle raw display
$31$ \( T - 57652408 \) Copy content Toggle raw display
$37$ \( T + 375985186 \) Copy content Toggle raw display
$41$ \( T + 856316202 \) Copy content Toggle raw display
$43$ \( T - 1245189172 \) Copy content Toggle raw display
$47$ \( T + 1306762656 \) Copy content Toggle raw display
$53$ \( T + 409556358 \) Copy content Toggle raw display
$59$ \( T + 2882866260 \) Copy content Toggle raw display
$61$ \( T - 5731767302 \) Copy content Toggle raw display
$67$ \( T + 3893272244 \) Copy content Toggle raw display
$71$ \( T + 9075890088 \) Copy content Toggle raw display
$73$ \( T + 15571822822 \) Copy content Toggle raw display
$79$ \( T - 30196762600 \) Copy content Toggle raw display
$83$ \( T - 23135252628 \) Copy content Toggle raw display
$89$ \( T - 25614819990 \) Copy content Toggle raw display
$97$ \( T + 61937553406 \) Copy content Toggle raw display
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