Properties

Label 2112.2.a.b
Level $2112$
Weight $2$
Character orbit 2112.a
Self dual yes
Analytic conductor $16.864$
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] = [2112,2,Mod(1,2112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2112, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2112.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2112.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: \(16.8644049069\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 132)
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)
 
\(f(q)\) \(=\) \( q - q^{3} - 2 q^{5} - 2 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - 2 q^{5} - 2 q^{7} + q^{9} - q^{11} - 6 q^{13} + 2 q^{15} - 4 q^{17} - 2 q^{19} + 2 q^{21} + 8 q^{23} - q^{25} - q^{27} + q^{33} + 4 q^{35} + 6 q^{37} + 6 q^{39} + 10 q^{43} - 2 q^{45} - 3 q^{49} + 4 q^{51} - 14 q^{53} + 2 q^{55} + 2 q^{57} - 12 q^{59} + 14 q^{61} - 2 q^{63} + 12 q^{65} + 4 q^{67} - 8 q^{69} + 6 q^{73} + q^{75} + 2 q^{77} - 2 q^{79} + q^{81} + 16 q^{83} + 8 q^{85} - 14 q^{89} + 12 q^{91} + 4 q^{95} - 2 q^{97} - q^{99}+O(q^{100}) \) 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 −1.00000 0 −2.00000 0 −2.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(11\) \(1\)

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 2112.2.a.b 1
3.b odd 2 1 6336.2.a.bz 1
4.b odd 2 1 2112.2.a.t 1
8.b even 2 1 528.2.a.h 1
8.d odd 2 1 132.2.a.a 1
12.b even 2 1 6336.2.a.cf 1
24.f even 2 1 396.2.a.b 1
24.h odd 2 1 1584.2.a.c 1
40.e odd 2 1 3300.2.a.k 1
40.k even 4 2 3300.2.c.c 2
56.e even 2 1 6468.2.a.l 1
72.l even 6 2 3564.2.i.g 2
72.p odd 6 2 3564.2.i.b 2
88.b odd 2 1 5808.2.a.bf 1
88.g even 2 1 1452.2.a.c 1
88.k even 10 4 1452.2.i.l 4
88.l odd 10 4 1452.2.i.k 4
120.m even 2 1 9900.2.a.g 1
120.q odd 4 2 9900.2.c.k 2
264.p odd 2 1 4356.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
132.2.a.a 1 8.d odd 2 1
396.2.a.b 1 24.f even 2 1
528.2.a.h 1 8.b even 2 1
1452.2.a.c 1 88.g even 2 1
1452.2.i.k 4 88.l odd 10 4
1452.2.i.l 4 88.k even 10 4
1584.2.a.c 1 24.h odd 2 1
2112.2.a.b 1 1.a even 1 1 trivial
2112.2.a.t 1 4.b odd 2 1
3300.2.a.k 1 40.e odd 2 1
3300.2.c.c 2 40.k even 4 2
3564.2.i.b 2 72.p odd 6 2
3564.2.i.g 2 72.l even 6 2
4356.2.a.c 1 264.p odd 2 1
5808.2.a.bf 1 88.b odd 2 1
6336.2.a.bz 1 3.b odd 2 1
6336.2.a.cf 1 12.b even 2 1
6468.2.a.l 1 56.e even 2 1
9900.2.a.g 1 120.m even 2 1
9900.2.c.k 2 120.q odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2112))\):

\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{13} + 6 \) Copy content Toggle raw display
\( T_{17} + 4 \) Copy content Toggle raw display
\( T_{19} + 2 \) Copy content Toggle raw display
\( T_{23} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T + 2 \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T + 6 \) Copy content Toggle raw display
$17$ \( T + 4 \) Copy content Toggle raw display
$19$ \( T + 2 \) Copy content Toggle raw display
$23$ \( T - 8 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 6 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T - 10 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 14 \) Copy content Toggle raw display
$59$ \( T + 12 \) Copy content Toggle raw display
$61$ \( T - 14 \) Copy content Toggle raw display
$67$ \( T - 4 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 6 \) Copy content Toggle raw display
$79$ \( T + 2 \) Copy content Toggle raw display
$83$ \( T - 16 \) Copy content Toggle raw display
$89$ \( T + 14 \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
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