Properties

Label 48.6.a.a
Level $48$
Weight $6$
Character orbit 48.a
Self dual yes
Analytic conductor $7.698$
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] = [48,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(7.69842335102\)
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)
 
\(f(q)\) \(=\) \( q - 9 q^{3} + 6 q^{5} + 40 q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 9 q^{3} + 6 q^{5} + 40 q^{7} + 81 q^{9} + 564 q^{11} + 638 q^{13} - 54 q^{15} + 882 q^{17} + 556 q^{19} - 360 q^{21} + 840 q^{23} - 3089 q^{25} - 729 q^{27} + 4638 q^{29} - 4400 q^{31} - 5076 q^{33} + 240 q^{35} - 2410 q^{37} - 5742 q^{39} - 6870 q^{41} - 9644 q^{43} + 486 q^{45} + 18672 q^{47} - 15207 q^{49} - 7938 q^{51} + 33750 q^{53} + 3384 q^{55} - 5004 q^{57} + 18084 q^{59} + 39758 q^{61} + 3240 q^{63} + 3828 q^{65} + 23068 q^{67} - 7560 q^{69} + 4248 q^{71} - 41110 q^{73} + 27801 q^{75} + 22560 q^{77} - 21920 q^{79} + 6561 q^{81} - 82452 q^{83} + 5292 q^{85} - 41742 q^{87} - 94086 q^{89} + 25520 q^{91} + 39600 q^{93} + 3336 q^{95} + 49442 q^{97} + 45684 q^{99}+O(q^{100}) \) 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 −9.00000 0 6.00000 0 40.0000 0 81.0000 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 48.6.a.a 1
3.b odd 2 1 144.6.a.f 1
4.b odd 2 1 3.6.a.a 1
8.b even 2 1 192.6.a.l 1
8.d odd 2 1 192.6.a.d 1
12.b even 2 1 9.6.a.a 1
16.e even 4 2 768.6.d.h 2
16.f odd 4 2 768.6.d.k 2
20.d odd 2 1 75.6.a.e 1
20.e even 4 2 75.6.b.b 2
24.f even 2 1 576.6.a.s 1
24.h odd 2 1 576.6.a.t 1
28.d even 2 1 147.6.a.a 1
28.f even 6 2 147.6.e.k 2
28.g odd 6 2 147.6.e.h 2
36.f odd 6 2 81.6.c.c 2
36.h even 6 2 81.6.c.a 2
44.c even 2 1 363.6.a.d 1
52.b odd 2 1 507.6.a.b 1
60.h even 2 1 225.6.a.a 1
60.l odd 4 2 225.6.b.b 2
68.d odd 2 1 867.6.a.a 1
76.d even 2 1 1083.6.a.c 1
84.h odd 2 1 441.6.a.i 1
132.d odd 2 1 1089.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.6.a.a 1 4.b odd 2 1
9.6.a.a 1 12.b even 2 1
48.6.a.a 1 1.a even 1 1 trivial
75.6.a.e 1 20.d odd 2 1
75.6.b.b 2 20.e even 4 2
81.6.c.a 2 36.h even 6 2
81.6.c.c 2 36.f odd 6 2
144.6.a.f 1 3.b odd 2 1
147.6.a.a 1 28.d even 2 1
147.6.e.h 2 28.g odd 6 2
147.6.e.k 2 28.f even 6 2
192.6.a.d 1 8.d odd 2 1
192.6.a.l 1 8.b even 2 1
225.6.a.a 1 60.h even 2 1
225.6.b.b 2 60.l odd 4 2
363.6.a.d 1 44.c even 2 1
441.6.a.i 1 84.h odd 2 1
507.6.a.b 1 52.b odd 2 1
576.6.a.s 1 24.f even 2 1
576.6.a.t 1 24.h odd 2 1
768.6.d.h 2 16.e even 4 2
768.6.d.k 2 16.f odd 4 2
867.6.a.a 1 68.d odd 2 1
1083.6.a.c 1 76.d even 2 1
1089.6.a.b 1 132.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 6 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(48))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 9 \) Copy content Toggle raw display
$5$ \( T - 6 \) Copy content Toggle raw display
$7$ \( T - 40 \) Copy content Toggle raw display
$11$ \( T - 564 \) Copy content Toggle raw display
$13$ \( T - 638 \) Copy content Toggle raw display
$17$ \( T - 882 \) Copy content Toggle raw display
$19$ \( T - 556 \) Copy content Toggle raw display
$23$ \( T - 840 \) Copy content Toggle raw display
$29$ \( T - 4638 \) Copy content Toggle raw display
$31$ \( T + 4400 \) Copy content Toggle raw display
$37$ \( T + 2410 \) Copy content Toggle raw display
$41$ \( T + 6870 \) Copy content Toggle raw display
$43$ \( T + 9644 \) Copy content Toggle raw display
$47$ \( T - 18672 \) Copy content Toggle raw display
$53$ \( T - 33750 \) Copy content Toggle raw display
$59$ \( T - 18084 \) Copy content Toggle raw display
$61$ \( T - 39758 \) Copy content Toggle raw display
$67$ \( T - 23068 \) Copy content Toggle raw display
$71$ \( T - 4248 \) Copy content Toggle raw display
$73$ \( T + 41110 \) Copy content Toggle raw display
$79$ \( T + 21920 \) Copy content Toggle raw display
$83$ \( T + 82452 \) Copy content Toggle raw display
$89$ \( T + 94086 \) Copy content Toggle raw display
$97$ \( T - 49442 \) Copy content Toggle raw display
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