Properties

Label 48.8.a.g
Level $48$
Weight $8$
Character orbit 48.a
Self dual yes
Analytic conductor $14.994$
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] = [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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
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)
 
\(f(q)\) \(=\) \( q + 27 q^{3} + 390 q^{5} + 64 q^{7} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 27 q^{3} + 390 q^{5} + 64 q^{7} + 729 q^{9} + 948 q^{11} - 5098 q^{13} + 10530 q^{15} + 28386 q^{17} + 8620 q^{19} + 1728 q^{21} + 15288 q^{23} + 73975 q^{25} + 19683 q^{27} + 36510 q^{29} + 276808 q^{31} + 25596 q^{33} + 24960 q^{35} + 268526 q^{37} - 137646 q^{39} - 629718 q^{41} - 685772 q^{43} + 284310 q^{45} - 583296 q^{47} - 819447 q^{49} + 766422 q^{51} - 428058 q^{53} + 369720 q^{55} + 232740 q^{57} - 1306380 q^{59} + 300662 q^{61} + 46656 q^{63} - 1988220 q^{65} + 507244 q^{67} + 412776 q^{69} - 5560632 q^{71} + 1369082 q^{73} + 1997325 q^{75} + 60672 q^{77} + 6913720 q^{79} + 531441 q^{81} + 4376748 q^{83} + 11070540 q^{85} + 985770 q^{87} - 8528310 q^{89} - 326272 q^{91} + 7473816 q^{93} + 3361800 q^{95} - 8826814 q^{97} + 691092 q^{99}+O(q^{100}) \) 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 0 390.000 0 64.0000 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.g 1
3.b odd 2 1 144.8.a.b 1
4.b odd 2 1 3.8.a.a 1
8.b even 2 1 192.8.a.a 1
8.d odd 2 1 192.8.a.i 1
12.b even 2 1 9.8.a.a 1
20.d odd 2 1 75.8.a.a 1
20.e even 4 2 75.8.b.c 2
24.f even 2 1 576.8.a.w 1
24.h odd 2 1 576.8.a.x 1
28.d even 2 1 147.8.a.b 1
28.f even 6 2 147.8.e.a 2
28.g odd 6 2 147.8.e.b 2
36.f odd 6 2 81.8.c.a 2
36.h even 6 2 81.8.c.c 2
44.c even 2 1 363.8.a.b 1
52.b odd 2 1 507.8.a.a 1
60.h even 2 1 225.8.a.i 1
60.l odd 4 2 225.8.b.f 2
84.h odd 2 1 441.8.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.8.a.a 1 4.b odd 2 1
9.8.a.a 1 12.b even 2 1
48.8.a.g 1 1.a even 1 1 trivial
75.8.a.a 1 20.d odd 2 1
75.8.b.c 2 20.e even 4 2
81.8.c.a 2 36.f odd 6 2
81.8.c.c 2 36.h even 6 2
144.8.a.b 1 3.b odd 2 1
147.8.a.b 1 28.d even 2 1
147.8.e.a 2 28.f even 6 2
147.8.e.b 2 28.g odd 6 2
192.8.a.a 1 8.b even 2 1
192.8.a.i 1 8.d odd 2 1
225.8.a.i 1 60.h even 2 1
225.8.b.f 2 60.l odd 4 2
363.8.a.b 1 44.c even 2 1
441.8.a.a 1 84.h odd 2 1
507.8.a.a 1 52.b odd 2 1
576.8.a.w 1 24.f even 2 1
576.8.a.x 1 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 390 \) 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
$5$ \( T - 390 \) Copy content Toggle raw display
$7$ \( T - 64 \) Copy content Toggle raw display
$11$ \( T - 948 \) Copy content Toggle raw display
$13$ \( T + 5098 \) Copy content Toggle raw display
$17$ \( T - 28386 \) Copy content Toggle raw display
$19$ \( T - 8620 \) Copy content Toggle raw display
$23$ \( T - 15288 \) Copy content Toggle raw display
$29$ \( T - 36510 \) Copy content Toggle raw display
$31$ \( T - 276808 \) Copy content Toggle raw display
$37$ \( T - 268526 \) Copy content Toggle raw display
$41$ \( T + 629718 \) Copy content Toggle raw display
$43$ \( T + 685772 \) Copy content Toggle raw display
$47$ \( T + 583296 \) Copy content Toggle raw display
$53$ \( T + 428058 \) Copy content Toggle raw display
$59$ \( T + 1306380 \) Copy content Toggle raw display
$61$ \( T - 300662 \) Copy content Toggle raw display
$67$ \( T - 507244 \) Copy content Toggle raw display
$71$ \( T + 5560632 \) Copy content Toggle raw display
$73$ \( T - 1369082 \) Copy content Toggle raw display
$79$ \( T - 6913720 \) Copy content Toggle raw display
$83$ \( T - 4376748 \) Copy content Toggle raw display
$89$ \( T + 8528310 \) Copy content Toggle raw display
$97$ \( T + 8826814 \) Copy content Toggle raw display
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