Properties

Label 48.6.a.d
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 24)
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)
 
\(f(q)\) \(=\) \( q + 9 q^{3} - 34 q^{5} + 240 q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 q^{3} - 34 q^{5} + 240 q^{7} + 81 q^{9} + 124 q^{11} + 46 q^{13} - 306 q^{15} + 1954 q^{17} + 1924 q^{19} + 2160 q^{21} - 2840 q^{23} - 1969 q^{25} + 729 q^{27} - 8922 q^{29} + 4648 q^{31} + 1116 q^{33} - 8160 q^{35} - 4362 q^{37} + 414 q^{39} - 2886 q^{41} - 11332 q^{43} - 2754 q^{45} - 7008 q^{47} + 40793 q^{49} + 17586 q^{51} - 22594 q^{53} - 4216 q^{55} + 17316 q^{57} + 28 q^{59} - 6386 q^{61} + 19440 q^{63} - 1564 q^{65} + 39076 q^{67} - 25560 q^{69} + 54872 q^{71} + 21034 q^{73} - 17721 q^{75} + 29760 q^{77} - 26632 q^{79} + 6561 q^{81} - 56188 q^{83} - 66436 q^{85} - 80298 q^{87} + 64410 q^{89} + 11040 q^{91} + 41832 q^{93} - 65416 q^{95} - 116158 q^{97} + 10044 q^{99}+O(q^{100}) \) 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 −34.0000 0 240.000 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.d 1
3.b odd 2 1 144.6.a.i 1
4.b odd 2 1 24.6.a.a 1
8.b even 2 1 192.6.a.f 1
8.d odd 2 1 192.6.a.n 1
12.b even 2 1 72.6.a.e 1
16.e even 4 2 768.6.d.a 2
16.f odd 4 2 768.6.d.r 2
20.d odd 2 1 600.6.a.i 1
20.e even 4 2 600.6.f.f 2
24.f even 2 1 576.6.a.k 1
24.h odd 2 1 576.6.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.6.a.a 1 4.b odd 2 1
48.6.a.d 1 1.a even 1 1 trivial
72.6.a.e 1 12.b even 2 1
144.6.a.i 1 3.b odd 2 1
192.6.a.f 1 8.b even 2 1
192.6.a.n 1 8.d odd 2 1
576.6.a.k 1 24.f even 2 1
576.6.a.l 1 24.h odd 2 1
600.6.a.i 1 20.d odd 2 1
600.6.f.f 2 20.e even 4 2
768.6.d.a 2 16.e even 4 2
768.6.d.r 2 16.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 34 \) 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
$5$ \( T + 34 \) Copy content Toggle raw display
$7$ \( T - 240 \) Copy content Toggle raw display
$11$ \( T - 124 \) Copy content Toggle raw display
$13$ \( T - 46 \) Copy content Toggle raw display
$17$ \( T - 1954 \) Copy content Toggle raw display
$19$ \( T - 1924 \) Copy content Toggle raw display
$23$ \( T + 2840 \) Copy content Toggle raw display
$29$ \( T + 8922 \) Copy content Toggle raw display
$31$ \( T - 4648 \) Copy content Toggle raw display
$37$ \( T + 4362 \) Copy content Toggle raw display
$41$ \( T + 2886 \) Copy content Toggle raw display
$43$ \( T + 11332 \) Copy content Toggle raw display
$47$ \( T + 7008 \) Copy content Toggle raw display
$53$ \( T + 22594 \) Copy content Toggle raw display
$59$ \( T - 28 \) Copy content Toggle raw display
$61$ \( T + 6386 \) Copy content Toggle raw display
$67$ \( T - 39076 \) Copy content Toggle raw display
$71$ \( T - 54872 \) Copy content Toggle raw display
$73$ \( T - 21034 \) Copy content Toggle raw display
$79$ \( T + 26632 \) Copy content Toggle raw display
$83$ \( T + 56188 \) Copy content Toggle raw display
$89$ \( T - 64410 \) Copy content Toggle raw display
$97$ \( T + 116158 \) Copy content Toggle raw display
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