Properties

Label 48.8.a.b
Level $48$
Weight $8$
Character orbit 48.a
Self dual yes
Analytic conductor $14.994$
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] = [48,8,Mod(1,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, 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("48.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 48.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: \(14.9944812232\)
Analytic rank: \(1\)
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)
 
\(f(q)\) \(=\) \( q - 27 q^{3} - 114 q^{5} + 1576 q^{7} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 27 q^{3} - 114 q^{5} + 1576 q^{7} + 729 q^{9} - 7332 q^{11} - 3802 q^{13} + 3078 q^{15} - 6606 q^{17} - 24860 q^{19} - 42552 q^{21} - 41448 q^{23} - 65129 q^{25} - 19683 q^{27} - 41610 q^{29} - 33152 q^{31} + 197964 q^{33} - 179664 q^{35} - 36466 q^{37} + 102654 q^{39} - 639078 q^{41} + 156412 q^{43} - 83106 q^{45} + 433776 q^{47} + 1660233 q^{49} + 178362 q^{51} + 786078 q^{53} + 835848 q^{55} + 671220 q^{57} - 745140 q^{59} - 1660618 q^{61} + 1148904 q^{63} + 433428 q^{65} + 3290836 q^{67} + 1119096 q^{69} - 5716152 q^{71} + 2659898 q^{73} + 1758483 q^{75} - 11555232 q^{77} - 3807440 q^{79} + 531441 q^{81} - 2229468 q^{83} + 753084 q^{85} + 1123470 q^{87} + 5991210 q^{89} - 5991952 q^{91} + 895104 q^{93} + 2834040 q^{95} - 4060126 q^{97} - 5345028 q^{99}+O(q^{100}) \) 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 −27.0000 0 −114.000 0 1576.00 0 729.000 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 48.8.a.b 1
3.b odd 2 1 144.8.a.h 1
4.b odd 2 1 6.8.a.a 1
8.b even 2 1 192.8.a.n 1
8.d odd 2 1 192.8.a.f 1
12.b even 2 1 18.8.a.a 1
20.d odd 2 1 150.8.a.e 1
20.e even 4 2 150.8.c.k 2
24.f even 2 1 576.8.a.h 1
24.h odd 2 1 576.8.a.i 1
28.d even 2 1 294.8.a.l 1
28.f even 6 2 294.8.e.d 2
28.g odd 6 2 294.8.e.c 2
36.f odd 6 2 162.8.c.d 2
36.h even 6 2 162.8.c.i 2
60.h even 2 1 450.8.a.ba 1
60.l odd 4 2 450.8.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.8.a.a 1 4.b odd 2 1
18.8.a.a 1 12.b even 2 1
48.8.a.b 1 1.a even 1 1 trivial
144.8.a.h 1 3.b odd 2 1
150.8.a.e 1 20.d odd 2 1
150.8.c.k 2 20.e even 4 2
162.8.c.d 2 36.f odd 6 2
162.8.c.i 2 36.h even 6 2
192.8.a.f 1 8.d odd 2 1
192.8.a.n 1 8.b even 2 1
294.8.a.l 1 28.d even 2 1
294.8.e.c 2 28.g odd 6 2
294.8.e.d 2 28.f even 6 2
450.8.a.ba 1 60.h even 2 1
450.8.c.a 2 60.l odd 4 2
576.8.a.h 1 24.f even 2 1
576.8.a.i 1 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 114 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(48))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 27 \) Copy content Toggle raw display
$5$ \( T + 114 \) Copy content Toggle raw display
$7$ \( T - 1576 \) Copy content Toggle raw display
$11$ \( T + 7332 \) Copy content Toggle raw display
$13$ \( T + 3802 \) Copy content Toggle raw display
$17$ \( T + 6606 \) Copy content Toggle raw display
$19$ \( T + 24860 \) Copy content Toggle raw display
$23$ \( T + 41448 \) Copy content Toggle raw display
$29$ \( T + 41610 \) Copy content Toggle raw display
$31$ \( T + 33152 \) Copy content Toggle raw display
$37$ \( T + 36466 \) Copy content Toggle raw display
$41$ \( T + 639078 \) Copy content Toggle raw display
$43$ \( T - 156412 \) Copy content Toggle raw display
$47$ \( T - 433776 \) Copy content Toggle raw display
$53$ \( T - 786078 \) Copy content Toggle raw display
$59$ \( T + 745140 \) Copy content Toggle raw display
$61$ \( T + 1660618 \) Copy content Toggle raw display
$67$ \( T - 3290836 \) Copy content Toggle raw display
$71$ \( T + 5716152 \) Copy content Toggle raw display
$73$ \( T - 2659898 \) Copy content Toggle raw display
$79$ \( T + 3807440 \) Copy content Toggle raw display
$83$ \( T + 2229468 \) Copy content Toggle raw display
$89$ \( T - 5991210 \) Copy content Toggle raw display
$97$ \( T + 4060126 \) Copy content Toggle raw display
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