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Properties

Label 48.4.a.a
Level $48$
Weight $4$
Character orbit 48.a
Self dual yes
Analytic conductor $2.832$
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] = [48,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(2.83209168028\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
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)
 
\(f(q)\) \(=\) \( q - 3 q^{3} - 18 q^{5} - 8 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} - 18 q^{5} - 8 q^{7} + 9 q^{9} - 36 q^{11} - 10 q^{13} + 54 q^{15} + 18 q^{17} + 100 q^{19} + 24 q^{21} - 72 q^{23} + 199 q^{25} - 27 q^{27} - 234 q^{29} + 16 q^{31} + 108 q^{33} + 144 q^{35} - 226 q^{37} + 30 q^{39} + 90 q^{41} - 452 q^{43} - 162 q^{45} - 432 q^{47} - 279 q^{49} - 54 q^{51} + 414 q^{53} + 648 q^{55} - 300 q^{57} + 684 q^{59} + 422 q^{61} - 72 q^{63} + 180 q^{65} - 332 q^{67} + 216 q^{69} + 360 q^{71} + 26 q^{73} - 597 q^{75} + 288 q^{77} - 512 q^{79} + 81 q^{81} + 1188 q^{83} - 324 q^{85} + 702 q^{87} - 630 q^{89} + 80 q^{91} - 48 q^{93} - 1800 q^{95} - 1054 q^{97} - 324 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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 −3.00000 0 −18.0000 0 −8.00000 0 9.00000 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 48.4.a.a 1
3.b odd 2 1 144.4.a.g 1
4.b odd 2 1 12.4.a.a ✓ 1
5.b even 2 1 1200.4.a.be 1
5.c odd 4 2 1200.4.f.d 2
7.b odd 2 1 2352.4.a.bk 1
8.b even 2 1 192.4.a.l 1
8.d odd 2 1 192.4.a.f 1
12.b even 2 1 36.4.a.a 1
16.e even 4 2 768.4.d.j 2
16.f odd 4 2 768.4.d.g 2
20.d odd 2 1 300.4.a.b 1
20.e even 4 2 300.4.d.e 2
24.f even 2 1 576.4.a.b 1
24.h odd 2 1 576.4.a.a 1
28.d even 2 1 588.4.a.c 1
28.f even 6 2 588.4.i.e 2
28.g odd 6 2 588.4.i.d 2
36.f odd 6 2 324.4.e.h 2
36.h even 6 2 324.4.e.a 2
44.c even 2 1 1452.4.a.d 1
52.b odd 2 1 2028.4.a.c 1
52.f even 4 2 2028.4.b.c 2
60.h even 2 1 900.4.a.g 1
60.l odd 4 2 900.4.d.c 2
84.h odd 2 1 1764.4.a.b 1
84.j odd 6 2 1764.4.k.o 2
84.n even 6 2 1764.4.k.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.a.a ✓ 1 4.b odd 2 1
36.4.a.a 1 12.b even 2 1
48.4.a.a 1 1.a even 1 1 trivial
144.4.a.g 1 3.b odd 2 1
192.4.a.f 1 8.d odd 2 1
192.4.a.l 1 8.b even 2 1
300.4.a.b 1 20.d odd 2 1
300.4.d.e 2 20.e even 4 2
324.4.e.a 2 36.h even 6 2
324.4.e.h 2 36.f odd 6 2
576.4.a.a 1 24.h odd 2 1
576.4.a.b 1 24.f even 2 1
588.4.a.c 1 28.d even 2 1
588.4.i.d 2 28.g odd 6 2
588.4.i.e 2 28.f even 6 2
768.4.d.g 2 16.f odd 4 2
768.4.d.j 2 16.e even 4 2
900.4.a.g 1 60.h even 2 1
900.4.d.c 2 60.l odd 4 2
1200.4.a.be 1 5.b even 2 1
1200.4.f.d 2 5.c odd 4 2
1452.4.a.d 1 44.c even 2 1
1764.4.a.b 1 84.h odd 2 1
1764.4.k.b 2 84.n even 6 2
1764.4.k.o 2 84.j odd 6 2
2028.4.a.c 1 52.b odd 2 1
2028.4.b.c 2 52.f even 4 2
2352.4.a.bk 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 18 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(48))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T + 18 \) Copy content Toggle raw display
$7$ \( T + 8 \) Copy content Toggle raw display
$11$ \( T + 36 \) Copy content Toggle raw display
$13$ \( T + 10 \) Copy content Toggle raw display
$17$ \( T - 18 \) Copy content Toggle raw display
$19$ \( T - 100 \) Copy content Toggle raw display
$23$ \( T + 72 \) Copy content Toggle raw display
$29$ \( T + 234 \) Copy content Toggle raw display
$31$ \( T - 16 \) Copy content Toggle raw display
$37$ \( T + 226 \) Copy content Toggle raw display
$41$ \( T - 90 \) Copy content Toggle raw display
$43$ \( T + 452 \) Copy content Toggle raw display
$47$ \( T + 432 \) Copy content Toggle raw display
$53$ \( T - 414 \) Copy content Toggle raw display
$59$ \( T - 684 \) Copy content Toggle raw display
$61$ \( T - 422 \) Copy content Toggle raw display
$67$ \( T + 332 \) Copy content Toggle raw display
$71$ \( T - 360 \) Copy content Toggle raw display
$73$ \( T - 26 \) Copy content Toggle raw display
$79$ \( T + 512 \) Copy content Toggle raw display
$83$ \( T - 1188 \) Copy content Toggle raw display
$89$ \( T + 630 \) Copy content Toggle raw display
$97$ \( T + 1054 \) Copy content Toggle raw display
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