Properties

Label 6.6.a.a
Level $6$
Weight $6$
Character orbit 6.a
Self dual yes
Analytic conductor $0.962$
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] = [6,6,Mod(1,6)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 6.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: \(0.962302918878\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 
\(f(q)\) \(=\) \( q + 4 q^{2} - 9 q^{3} + 16 q^{4} - 66 q^{5} - 36 q^{6} + 176 q^{7} + 64 q^{8} + 81 q^{9} - 264 q^{10} - 60 q^{11} - 144 q^{12} - 658 q^{13} + 704 q^{14} + 594 q^{15} + 256 q^{16} - 414 q^{17} + 324 q^{18}+ \cdots - 4860 q^{99}+O(q^{100}) \) 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
4.00000 −9.00000 16.0000 −66.0000 −36.0000 176.000 64.0000 81.0000 −264.000
\(n\): 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 6.6.a.a 1
3.b odd 2 1 18.6.a.b 1
4.b odd 2 1 48.6.a.c 1
5.b even 2 1 150.6.a.d 1
5.c odd 4 2 150.6.c.b 2
7.b odd 2 1 294.6.a.m 1
7.c even 3 2 294.6.e.g 2
7.d odd 6 2 294.6.e.a 2
8.b even 2 1 192.6.a.o 1
8.d odd 2 1 192.6.a.g 1
9.c even 3 2 162.6.c.e 2
9.d odd 6 2 162.6.c.h 2
11.b odd 2 1 726.6.a.a 1
12.b even 2 1 144.6.a.j 1
13.b even 2 1 1014.6.a.c 1
15.d odd 2 1 450.6.a.m 1
15.e even 4 2 450.6.c.j 2
16.e even 4 2 768.6.d.c 2
16.f odd 4 2 768.6.d.p 2
21.c even 2 1 882.6.a.a 1
24.f even 2 1 576.6.a.i 1
24.h odd 2 1 576.6.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.6.a.a 1 1.a even 1 1 trivial
18.6.a.b 1 3.b odd 2 1
48.6.a.c 1 4.b odd 2 1
144.6.a.j 1 12.b even 2 1
150.6.a.d 1 5.b even 2 1
150.6.c.b 2 5.c odd 4 2
162.6.c.e 2 9.c even 3 2
162.6.c.h 2 9.d odd 6 2
192.6.a.g 1 8.d odd 2 1
192.6.a.o 1 8.b even 2 1
294.6.a.m 1 7.b odd 2 1
294.6.e.a 2 7.d odd 6 2
294.6.e.g 2 7.c even 3 2
450.6.a.m 1 15.d odd 2 1
450.6.c.j 2 15.e even 4 2
576.6.a.i 1 24.f even 2 1
576.6.a.j 1 24.h odd 2 1
726.6.a.a 1 11.b odd 2 1
768.6.d.c 2 16.e even 4 2
768.6.d.p 2 16.f odd 4 2
882.6.a.a 1 21.c even 2 1
1014.6.a.c 1 13.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(\Gamma_0(6))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 4 \) Copy content Toggle raw display
$3$ \( T + 9 \) Copy content Toggle raw display
$5$ \( T + 66 \) Copy content Toggle raw display
$7$ \( T - 176 \) Copy content Toggle raw display
$11$ \( T + 60 \) Copy content Toggle raw display
$13$ \( T + 658 \) Copy content Toggle raw display
$17$ \( T + 414 \) Copy content Toggle raw display
$19$ \( T - 956 \) Copy content Toggle raw display
$23$ \( T - 600 \) Copy content Toggle raw display
$29$ \( T - 5574 \) Copy content Toggle raw display
$31$ \( T + 3592 \) Copy content Toggle raw display
$37$ \( T + 8458 \) Copy content Toggle raw display
$41$ \( T - 19194 \) Copy content Toggle raw display
$43$ \( T - 13316 \) Copy content Toggle raw display
$47$ \( T + 19680 \) Copy content Toggle raw display
$53$ \( T + 31266 \) Copy content Toggle raw display
$59$ \( T - 26340 \) Copy content Toggle raw display
$61$ \( T + 31090 \) Copy content Toggle raw display
$67$ \( T + 16804 \) Copy content Toggle raw display
$71$ \( T - 6120 \) Copy content Toggle raw display
$73$ \( T + 25558 \) Copy content Toggle raw display
$79$ \( T - 74408 \) Copy content Toggle raw display
$83$ \( T + 6468 \) Copy content Toggle raw display
$89$ \( T + 32742 \) Copy content Toggle raw display
$97$ \( T - 166082 \) Copy content Toggle raw display
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