Properties

Label 144.8.a.a
Level $144$
Weight $8$
Character orbit 144.a
Self dual yes
Analytic conductor $44.983$
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] = [144,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
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: \(44.9834436697\)
Analytic rank: \(1\)
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)
 
\(f(q)\) \(=\) \( q - 430 q^{5} + 1224 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 430 q^{5} + 1224 q^{7} - 3164 q^{11} + 6118 q^{13} + 16270 q^{17} + 5476 q^{19} + 1576 q^{23} + 106775 q^{25} - 122838 q^{29} - 251360 q^{31} - 526320 q^{35} - 52338 q^{37} + 319398 q^{41} - 710788 q^{43} + 284112 q^{47} + 674633 q^{49} - 296062 q^{53} + 1360520 q^{55} - 897548 q^{59} - 884810 q^{61} - 2630740 q^{65} - 4659692 q^{67} - 2710792 q^{71} - 5670854 q^{73} - 3872736 q^{77} + 5124176 q^{79} - 1563556 q^{83} - 6996100 q^{85} - 11605674 q^{89} + 7488432 q^{91} - 2354680 q^{95} + 10931618 q^{97}+O(q^{100}) \) 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 −430.000 0 1224.00 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 144.8.a.a 1
3.b odd 2 1 16.8.a.a 1
4.b odd 2 1 72.8.a.a 1
8.b even 2 1 576.8.a.z 1
8.d odd 2 1 576.8.a.y 1
12.b even 2 1 8.8.a.b 1
15.d odd 2 1 400.8.a.p 1
15.e even 4 2 400.8.c.f 2
24.f even 2 1 64.8.a.b 1
24.h odd 2 1 64.8.a.f 1
48.i odd 4 2 256.8.b.a 2
48.k even 4 2 256.8.b.g 2
60.h even 2 1 200.8.a.b 1
60.l odd 4 2 200.8.c.c 2
84.h odd 2 1 392.8.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.8.a.b 1 12.b even 2 1
16.8.a.a 1 3.b odd 2 1
64.8.a.b 1 24.f even 2 1
64.8.a.f 1 24.h odd 2 1
72.8.a.a 1 4.b odd 2 1
144.8.a.a 1 1.a even 1 1 trivial
200.8.a.b 1 60.h even 2 1
200.8.c.c 2 60.l odd 4 2
256.8.b.a 2 48.i odd 4 2
256.8.b.g 2 48.k even 4 2
392.8.a.b 1 84.h odd 2 1
400.8.a.p 1 15.d odd 2 1
400.8.c.f 2 15.e even 4 2
576.8.a.y 1 8.d odd 2 1
576.8.a.z 1 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 430 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(144))\). 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
$5$ \( T + 430 \) Copy content Toggle raw display
$7$ \( T - 1224 \) Copy content Toggle raw display
$11$ \( T + 3164 \) Copy content Toggle raw display
$13$ \( T - 6118 \) Copy content Toggle raw display
$17$ \( T - 16270 \) Copy content Toggle raw display
$19$ \( T - 5476 \) Copy content Toggle raw display
$23$ \( T - 1576 \) Copy content Toggle raw display
$29$ \( T + 122838 \) Copy content Toggle raw display
$31$ \( T + 251360 \) Copy content Toggle raw display
$37$ \( T + 52338 \) Copy content Toggle raw display
$41$ \( T - 319398 \) Copy content Toggle raw display
$43$ \( T + 710788 \) Copy content Toggle raw display
$47$ \( T - 284112 \) Copy content Toggle raw display
$53$ \( T + 296062 \) Copy content Toggle raw display
$59$ \( T + 897548 \) Copy content Toggle raw display
$61$ \( T + 884810 \) Copy content Toggle raw display
$67$ \( T + 4659692 \) Copy content Toggle raw display
$71$ \( T + 2710792 \) Copy content Toggle raw display
$73$ \( T + 5670854 \) Copy content Toggle raw display
$79$ \( T - 5124176 \) Copy content Toggle raw display
$83$ \( T + 1563556 \) Copy content Toggle raw display
$89$ \( T + 11605674 \) Copy content Toggle raw display
$97$ \( T - 10931618 \) Copy content Toggle raw display
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