Properties

Label 48.3.g.a
Level $48$
Weight $3$
Character orbit 48.g
Analytic conductor $1.308$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [48,3,Mod(31,48)]
 
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([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.31");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 48.g (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(1.30790526893\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{3} + 6 q^{5} - 4 \beta q^{7} - 3 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} + 6 q^{5} - 4 \beta q^{7} - 3 q^{9} + 12 \beta q^{11} - 14 q^{13} - 6 \beta q^{15} - 6 q^{17} + 4 \beta q^{19} - 12 q^{21} + 11 q^{25} + 3 \beta q^{27} + 30 q^{29} - 12 \beta q^{31} + 36 q^{33} - 24 \beta q^{35} + 26 q^{37} + 14 \beta q^{39} - 54 q^{41} + 12 \beta q^{43} - 18 q^{45} - 24 \beta q^{47} + q^{49} + 6 \beta q^{51} - 18 q^{53} + 72 \beta q^{55} + 12 q^{57} - 12 \beta q^{59} - 70 q^{61} + 12 \beta q^{63} - 84 q^{65} - 68 \beta q^{67} + 48 \beta q^{71} + 82 q^{73} - 11 \beta q^{75} + 144 q^{77} - 44 \beta q^{79} + 9 q^{81} - 12 \beta q^{83} - 36 q^{85} - 30 \beta q^{87} + 114 q^{89} + 56 \beta q^{91} - 36 q^{93} + 24 \beta q^{95} + 34 q^{97} - 36 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 12 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 12 q^{5} - 6 q^{9} - 28 q^{13} - 12 q^{17} - 24 q^{21} + 22 q^{25} + 60 q^{29} + 72 q^{33} + 52 q^{37} - 108 q^{41} - 36 q^{45} + 2 q^{49} - 36 q^{53} + 24 q^{57} - 140 q^{61} - 168 q^{65} + 164 q^{73} + 288 q^{77} + 18 q^{81} - 72 q^{85} + 228 q^{89} - 72 q^{93} + 68 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.73205i 0 6.00000 0 6.92820i 0 −3.00000 0
31.2 0 1.73205i 0 6.00000 0 6.92820i 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.3.g.a 2
3.b odd 2 1 144.3.g.b 2
4.b odd 2 1 inner 48.3.g.a 2
5.b even 2 1 1200.3.e.h 2
5.c odd 4 2 1200.3.j.a 4
7.b odd 2 1 2352.3.m.a 2
8.b even 2 1 192.3.g.a 2
8.d odd 2 1 192.3.g.a 2
9.c even 3 1 1296.3.o.a 2
9.c even 3 1 1296.3.o.c 2
9.d odd 6 1 1296.3.o.n 2
9.d odd 6 1 1296.3.o.p 2
12.b even 2 1 144.3.g.b 2
15.d odd 2 1 3600.3.e.t 2
15.e even 4 2 3600.3.j.i 4
16.e even 4 2 768.3.b.b 4
16.f odd 4 2 768.3.b.b 4
20.d odd 2 1 1200.3.e.h 2
20.e even 4 2 1200.3.j.a 4
24.f even 2 1 576.3.g.i 2
24.h odd 2 1 576.3.g.i 2
28.d even 2 1 2352.3.m.a 2
36.f odd 6 1 1296.3.o.a 2
36.f odd 6 1 1296.3.o.c 2
36.h even 6 1 1296.3.o.n 2
36.h even 6 1 1296.3.o.p 2
48.i odd 4 2 2304.3.b.n 4
48.k even 4 2 2304.3.b.n 4
60.h even 2 1 3600.3.e.t 2
60.l odd 4 2 3600.3.j.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.3.g.a 2 1.a even 1 1 trivial
48.3.g.a 2 4.b odd 2 1 inner
144.3.g.b 2 3.b odd 2 1
144.3.g.b 2 12.b even 2 1
192.3.g.a 2 8.b even 2 1
192.3.g.a 2 8.d odd 2 1
576.3.g.i 2 24.f even 2 1
576.3.g.i 2 24.h odd 2 1
768.3.b.b 4 16.e even 4 2
768.3.b.b 4 16.f odd 4 2
1200.3.e.h 2 5.b even 2 1
1200.3.e.h 2 20.d odd 2 1
1200.3.j.a 4 5.c odd 4 2
1200.3.j.a 4 20.e even 4 2
1296.3.o.a 2 9.c even 3 1
1296.3.o.a 2 36.f odd 6 1
1296.3.o.c 2 9.c even 3 1
1296.3.o.c 2 36.f odd 6 1
1296.3.o.n 2 9.d odd 6 1
1296.3.o.n 2 36.h even 6 1
1296.3.o.p 2 9.d odd 6 1
1296.3.o.p 2 36.h even 6 1
2304.3.b.n 4 48.i odd 4 2
2304.3.b.n 4 48.k even 4 2
2352.3.m.a 2 7.b odd 2 1
2352.3.m.a 2 28.d even 2 1
3600.3.e.t 2 15.d odd 2 1
3600.3.e.t 2 60.h even 2 1
3600.3.j.i 4 15.e even 4 2
3600.3.j.i 4 60.l odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(48, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3 \) Copy content Toggle raw display
$5$ \( (T - 6)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 48 \) Copy content Toggle raw display
$11$ \( T^{2} + 432 \) Copy content Toggle raw display
$13$ \( (T + 14)^{2} \) Copy content Toggle raw display
$17$ \( (T + 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 48 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 30)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 432 \) Copy content Toggle raw display
$37$ \( (T - 26)^{2} \) Copy content Toggle raw display
$41$ \( (T + 54)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 432 \) Copy content Toggle raw display
$47$ \( T^{2} + 1728 \) Copy content Toggle raw display
$53$ \( (T + 18)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 432 \) Copy content Toggle raw display
$61$ \( (T + 70)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 13872 \) Copy content Toggle raw display
$71$ \( T^{2} + 6912 \) Copy content Toggle raw display
$73$ \( (T - 82)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 5808 \) Copy content Toggle raw display
$83$ \( T^{2} + 432 \) Copy content Toggle raw display
$89$ \( (T - 114)^{2} \) Copy content Toggle raw display
$97$ \( (T - 34)^{2} \) Copy content Toggle raw display
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