Properties

Label 48.2.c.a
Level $48$
Weight $2$
Character orbit 48.c
Analytic conductor $0.383$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [48,2,Mod(47,48)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 48.c (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: \(0.383281929702\)
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: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta q^{3} + 2 \beta q^{7} - 3 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} + 2 \beta q^{7} - 3 q^{9} - 2 q^{13} - 2 \beta q^{19} + 6 q^{21} + 5 q^{25} + 3 \beta q^{27} - 6 \beta q^{31} - 10 q^{37} + 2 \beta q^{39} + 6 \beta q^{43} - 5 q^{49} - 6 q^{57} + 14 q^{61} - 6 \beta q^{63} - 2 \beta q^{67} + 10 q^{73} - 5 \beta q^{75} + 10 \beta q^{79} + 9 q^{81} - 4 \beta q^{91} - 18 q^{93} - 14 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{9} - 4 q^{13} + 12 q^{21} + 10 q^{25} - 20 q^{37} - 10 q^{49} - 12 q^{57} + 28 q^{61} + 20 q^{73} + 18 q^{81} - 36 q^{93} - 28 q^{97}+O(q^{100}) \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.73205i 0 0 0 3.46410i 0 −3.00000 0
47.2 0 1.73205i 0 0 0 3.46410i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.2.c.a 2
3.b odd 2 1 CM 48.2.c.a 2
4.b odd 2 1 inner 48.2.c.a 2
5.b even 2 1 1200.2.h.e 2
5.c odd 4 2 1200.2.o.i 4
7.b odd 2 1 2352.2.h.c 2
8.b even 2 1 192.2.c.a 2
8.d odd 2 1 192.2.c.a 2
9.c even 3 1 1296.2.s.b 2
9.c even 3 1 1296.2.s.e 2
9.d odd 6 1 1296.2.s.b 2
9.d odd 6 1 1296.2.s.e 2
12.b even 2 1 inner 48.2.c.a 2
15.d odd 2 1 1200.2.h.e 2
15.e even 4 2 1200.2.o.i 4
16.e even 4 2 768.2.f.d 4
16.f odd 4 2 768.2.f.d 4
20.d odd 2 1 1200.2.h.e 2
20.e even 4 2 1200.2.o.i 4
21.c even 2 1 2352.2.h.c 2
24.f even 2 1 192.2.c.a 2
24.h odd 2 1 192.2.c.a 2
28.d even 2 1 2352.2.h.c 2
36.f odd 6 1 1296.2.s.b 2
36.f odd 6 1 1296.2.s.e 2
36.h even 6 1 1296.2.s.b 2
36.h even 6 1 1296.2.s.e 2
48.i odd 4 2 768.2.f.d 4
48.k even 4 2 768.2.f.d 4
60.h even 2 1 1200.2.h.e 2
60.l odd 4 2 1200.2.o.i 4
84.h odd 2 1 2352.2.h.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.c.a 2 1.a even 1 1 trivial
48.2.c.a 2 3.b odd 2 1 CM
48.2.c.a 2 4.b odd 2 1 inner
48.2.c.a 2 12.b even 2 1 inner
192.2.c.a 2 8.b even 2 1
192.2.c.a 2 8.d odd 2 1
192.2.c.a 2 24.f even 2 1
192.2.c.a 2 24.h odd 2 1
768.2.f.d 4 16.e even 4 2
768.2.f.d 4 16.f odd 4 2
768.2.f.d 4 48.i odd 4 2
768.2.f.d 4 48.k even 4 2
1200.2.h.e 2 5.b even 2 1
1200.2.h.e 2 15.d odd 2 1
1200.2.h.e 2 20.d odd 2 1
1200.2.h.e 2 60.h even 2 1
1200.2.o.i 4 5.c odd 4 2
1200.2.o.i 4 15.e even 4 2
1200.2.o.i 4 20.e even 4 2
1200.2.o.i 4 60.l odd 4 2
1296.2.s.b 2 9.c even 3 1
1296.2.s.b 2 9.d odd 6 1
1296.2.s.b 2 36.f odd 6 1
1296.2.s.b 2 36.h even 6 1
1296.2.s.e 2 9.c even 3 1
1296.2.s.e 2 9.d odd 6 1
1296.2.s.e 2 36.f odd 6 1
1296.2.s.e 2 36.h even 6 1
2352.2.h.c 2 7.b odd 2 1
2352.2.h.c 2 21.c even 2 1
2352.2.h.c 2 28.d even 2 1
2352.2.h.c 2 84.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(48, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 12 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 12 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 108 \) Copy content Toggle raw display
$37$ \( (T + 10)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 108 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 14)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 12 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 10)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 300 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T + 14)^{2} \) Copy content Toggle raw display
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