Properties

Label 2112.4.a.f
Level $2112$
Weight $4$
Character orbit 2112.a
Self dual yes
Analytic conductor $124.612$
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] = [2112,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2112.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(124.612033932\)
Analytic rank: \(1\)
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)
 
\(f(q)\) \(=\) \( q - 3 q^{3} - 2 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} - 2 q^{7} + 9 q^{9} - 11 q^{11} + 88 q^{13} - 66 q^{17} - 40 q^{19} + 6 q^{21} - 6 q^{23} - 125 q^{25} - 27 q^{27} + 54 q^{29} - 8 q^{31} + 33 q^{33} + 106 q^{37} - 264 q^{39} + 354 q^{41} - 124 q^{43} - 546 q^{47} - 339 q^{49} + 198 q^{51} + 408 q^{53} + 120 q^{57} + 552 q^{59} - 404 q^{61} - 18 q^{63} - 4 q^{67} + 18 q^{69} - 126 q^{71} - 166 q^{73} + 375 q^{75} + 22 q^{77} + 874 q^{79} + 81 q^{81} + 444 q^{83} - 162 q^{87} + 1002 q^{89} - 176 q^{91} + 24 q^{93} - 802 q^{97} - 99 q^{99}+O(q^{100}) \) 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 −3.00000 0 0 0 −2.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 2112.4.a.f 1
4.b odd 2 1 2112.4.a.t 1
8.b even 2 1 528.4.a.i 1
8.d odd 2 1 132.4.a.b 1
24.f even 2 1 396.4.a.d 1
24.h odd 2 1 1584.4.a.j 1
88.g even 2 1 1452.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
132.4.a.b 1 8.d odd 2 1
396.4.a.d 1 24.f even 2 1
528.4.a.i 1 8.b even 2 1
1452.4.a.b 1 88.g even 2 1
1584.4.a.j 1 24.h odd 2 1
2112.4.a.f 1 1.a even 1 1 trivial
2112.4.a.t 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2112))\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7} + 2 \) 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
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T + 11 \) Copy content Toggle raw display
$13$ \( T - 88 \) Copy content Toggle raw display
$17$ \( T + 66 \) Copy content Toggle raw display
$19$ \( T + 40 \) Copy content Toggle raw display
$23$ \( T + 6 \) Copy content Toggle raw display
$29$ \( T - 54 \) Copy content Toggle raw display
$31$ \( T + 8 \) Copy content Toggle raw display
$37$ \( T - 106 \) Copy content Toggle raw display
$41$ \( T - 354 \) Copy content Toggle raw display
$43$ \( T + 124 \) Copy content Toggle raw display
$47$ \( T + 546 \) Copy content Toggle raw display
$53$ \( T - 408 \) Copy content Toggle raw display
$59$ \( T - 552 \) Copy content Toggle raw display
$61$ \( T + 404 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T + 126 \) Copy content Toggle raw display
$73$ \( T + 166 \) Copy content Toggle raw display
$79$ \( T - 874 \) Copy content Toggle raw display
$83$ \( T - 444 \) Copy content Toggle raw display
$89$ \( T - 1002 \) Copy content Toggle raw display
$97$ \( T + 802 \) Copy content Toggle raw display
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