Properties

Label 2112.2.k.f
Level $2112$
Weight $2$
Character orbit 2112.k
Analytic conductor $16.864$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2112,2,Mod(287,2112)]
 
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([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2112.287");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2112.k (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: \(16.8644049069\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{3} + 3 q^{5} - 4 \zeta_{12}^{3} q^{7} - 3 \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{3} + 3 q^{5} - 4 \zeta_{12}^{3} q^{7} - 3 \zeta_{12}^{2} q^{9} - \zeta_{12}^{3} q^{11} + (4 \zeta_{12}^{2} - 2) q^{13} + ( - 6 \zeta_{12}^{3} + 3 \zeta_{12}) q^{15} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{19} + ( - 4 \zeta_{12}^{2} - 4) q^{21} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{23} + 4 q^{25} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{27} - 5 \zeta_{12}^{3} q^{31} + ( - \zeta_{12}^{2} - 1) q^{33} - 12 \zeta_{12}^{3} q^{35} + ( - 10 \zeta_{12}^{2} + 5) q^{37} + 6 \zeta_{12} q^{39} + (12 \zeta_{12}^{2} - 6) q^{41} + ( - 6 \zeta_{12}^{3} + 12 \zeta_{12}) q^{43} - 9 \zeta_{12}^{2} q^{45} + (6 \zeta_{12}^{3} - 12 \zeta_{12}) q^{47} - 9 q^{49} - 6 q^{53} - 3 \zeta_{12}^{3} q^{55} + ( - 6 \zeta_{12}^{2} + 6) q^{57} + 9 \zeta_{12}^{3} q^{59} + ( - 8 \zeta_{12}^{2} + 4) q^{61} + (12 \zeta_{12}^{3} - 12 \zeta_{12}) q^{63} + (12 \zeta_{12}^{2} - 6) q^{65} + (5 \zeta_{12}^{3} - 10 \zeta_{12}) q^{67} + ( - 9 \zeta_{12}^{2} + 9) q^{69} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{71} + 10 q^{73} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}) q^{75} - 4 q^{77} - 14 \zeta_{12}^{3} q^{79} + (9 \zeta_{12}^{2} - 9) q^{81} - 6 \zeta_{12}^{3} q^{83} + (6 \zeta_{12}^{2} - 3) q^{89} + ( - 8 \zeta_{12}^{3} + 16 \zeta_{12}) q^{91} + ( - 5 \zeta_{12}^{2} - 5) q^{93} + ( - 6 \zeta_{12}^{3} + 12 \zeta_{12}) q^{95} - 13 q^{97} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{5} - 6 q^{9} - 24 q^{21} + 16 q^{25} - 6 q^{33} - 18 q^{45} - 36 q^{49} - 24 q^{53} + 12 q^{57} + 18 q^{69} + 40 q^{73} - 16 q^{77} - 18 q^{81} - 30 q^{93} - 52 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2112\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(1409\) \(1729\) \(2047\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
287.1
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
0 −0.866025 1.50000i 0 3.00000 0 4.00000i 0 −1.50000 + 2.59808i 0
287.2 0 −0.866025 + 1.50000i 0 3.00000 0 4.00000i 0 −1.50000 2.59808i 0
287.3 0 0.866025 1.50000i 0 3.00000 0 4.00000i 0 −1.50000 2.59808i 0
287.4 0 0.866025 + 1.50000i 0 3.00000 0 4.00000i 0 −1.50000 + 2.59808i 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
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2112.2.k.f yes 4
3.b odd 2 1 2112.2.k.c 4
4.b odd 2 1 inner 2112.2.k.f yes 4
8.b even 2 1 2112.2.k.c 4
8.d odd 2 1 2112.2.k.c 4
12.b even 2 1 2112.2.k.c 4
24.f even 2 1 inner 2112.2.k.f yes 4
24.h odd 2 1 inner 2112.2.k.f yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2112.2.k.c 4 3.b odd 2 1
2112.2.k.c 4 8.b even 2 1
2112.2.k.c 4 8.d odd 2 1
2112.2.k.c 4 12.b even 2 1
2112.2.k.f yes 4 1.a even 1 1 trivial
2112.2.k.f yes 4 4.b odd 2 1 inner
2112.2.k.f yes 4 24.f even 2 1 inner
2112.2.k.f yes 4 24.h odd 2 1 inner

Hecke kernels

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

\( T_{5} - 3 \) Copy content Toggle raw display
\( T_{19}^{2} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( (T - 3)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 75)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$53$ \( (T + 6)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 75)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$73$ \( (T - 10)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$97$ \( (T + 13)^{4} \) Copy content Toggle raw display
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