Properties

Label 3168.1.ck.a
Level $3168$
Weight $1$
Character orbit 3168.ck
Analytic conductor $1.581$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3168,1,Mod(559,3168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3168, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 5, 0, 6]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3168.559");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3168 = 2^{5} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3168.ck (of order \(10\), degree \(4\), not 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: \(1.58103796002\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 88)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.937024.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{10}^{2} q^{11} + ( - \zeta_{10}^{4} + \zeta_{10}^{3}) q^{17} + ( - \zeta_{10}^{4} - 1) q^{19} + \zeta_{10}^{2} q^{25} + (\zeta_{10}^{3} + \zeta_{10}) q^{41} + ( - \zeta_{10}^{4} + \zeta_{10}) q^{43} - \zeta_{10} q^{49} + ( - \zeta_{10} + 1) q^{59} + (\zeta_{10}^{3} - \zeta_{10}^{2}) q^{67} + (\zeta_{10}^{4} + \zeta_{10}^{2}) q^{73} + (\zeta_{10}^{2} + 1) q^{83} + ( - \zeta_{10}^{4} + \zeta_{10}) q^{89} + ( - \zeta_{10}^{3} + 1) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{11} + 2 q^{17} - 3 q^{19} - q^{25} + 2 q^{41} + 2 q^{43} - q^{49} + 3 q^{59} + 2 q^{67} - 2 q^{73} + 3 q^{83} + 2 q^{89} + 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3168\mathbb{Z}\right)^\times\).

\(n\) \(353\) \(991\) \(1189\) \(1729\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-\zeta_{10}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
−0.309017 0.951057i
0.809017 0.587785i
0.809017 + 0.587785i
−0.309017 + 0.951057i
0 0 0 0 0 0 0 0 0
1423.1 0 0 0 0 0 0 0 0 0
2863.1 0 0 0 0 0 0 0 0 0
3151.1 0 0 0 0 0 0 0 0 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
11.c even 5 1 inner
88.l odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3168.1.ck.a 4
3.b odd 2 1 352.1.t.a 4
4.b odd 2 1 792.1.bu.a 4
8.b even 2 1 792.1.bu.a 4
8.d odd 2 1 CM 3168.1.ck.a 4
11.c even 5 1 inner 3168.1.ck.a 4
12.b even 2 1 88.1.l.a 4
24.f even 2 1 352.1.t.a 4
24.h odd 2 1 88.1.l.a 4
33.d even 2 1 3872.1.t.c 4
33.f even 10 1 3872.1.f.b 2
33.f even 10 2 3872.1.t.a 4
33.f even 10 1 3872.1.t.c 4
33.h odd 10 1 352.1.t.a 4
33.h odd 10 1 3872.1.f.a 2
33.h odd 10 2 3872.1.t.b 4
44.h odd 10 1 792.1.bu.a 4
48.i odd 4 2 2816.1.v.c 8
48.k even 4 2 2816.1.v.c 8
60.h even 2 1 2200.1.cl.a 4
60.l odd 4 2 2200.1.dd.a 8
88.l odd 10 1 inner 3168.1.ck.a 4
88.o even 10 1 792.1.bu.a 4
120.i odd 2 1 2200.1.cl.a 4
120.w even 4 2 2200.1.dd.a 8
132.d odd 2 1 968.1.l.b 4
132.n odd 10 1 968.1.f.a 2
132.n odd 10 1 968.1.l.b 4
132.n odd 10 2 968.1.l.c 4
132.o even 10 1 88.1.l.a 4
132.o even 10 1 968.1.f.b 2
132.o even 10 2 968.1.l.a 4
264.m even 2 1 968.1.l.b 4
264.p odd 2 1 3872.1.t.c 4
264.r odd 10 1 3872.1.f.b 2
264.r odd 10 2 3872.1.t.a 4
264.r odd 10 1 3872.1.t.c 4
264.t odd 10 1 88.1.l.a 4
264.t odd 10 1 968.1.f.b 2
264.t odd 10 2 968.1.l.a 4
264.u even 10 1 968.1.f.a 2
264.u even 10 1 968.1.l.b 4
264.u even 10 2 968.1.l.c 4
264.w even 10 1 352.1.t.a 4
264.w even 10 1 3872.1.f.a 2
264.w even 10 2 3872.1.t.b 4
528.br even 20 2 2816.1.v.c 8
528.bu odd 20 2 2816.1.v.c 8
660.bd even 10 1 2200.1.cl.a 4
660.bp odd 20 2 2200.1.dd.a 8
1320.cp odd 10 1 2200.1.cl.a 4
1320.dh even 20 2 2200.1.dd.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.1.l.a 4 12.b even 2 1
88.1.l.a 4 24.h odd 2 1
88.1.l.a 4 132.o even 10 1
88.1.l.a 4 264.t odd 10 1
352.1.t.a 4 3.b odd 2 1
352.1.t.a 4 24.f even 2 1
352.1.t.a 4 33.h odd 10 1
352.1.t.a 4 264.w even 10 1
792.1.bu.a 4 4.b odd 2 1
792.1.bu.a 4 8.b even 2 1
792.1.bu.a 4 44.h odd 10 1
792.1.bu.a 4 88.o even 10 1
968.1.f.a 2 132.n odd 10 1
968.1.f.a 2 264.u even 10 1
968.1.f.b 2 132.o even 10 1
968.1.f.b 2 264.t odd 10 1
968.1.l.a 4 132.o even 10 2
968.1.l.a 4 264.t odd 10 2
968.1.l.b 4 132.d odd 2 1
968.1.l.b 4 132.n odd 10 1
968.1.l.b 4 264.m even 2 1
968.1.l.b 4 264.u even 10 1
968.1.l.c 4 132.n odd 10 2
968.1.l.c 4 264.u even 10 2
2200.1.cl.a 4 60.h even 2 1
2200.1.cl.a 4 120.i odd 2 1
2200.1.cl.a 4 660.bd even 10 1
2200.1.cl.a 4 1320.cp odd 10 1
2200.1.dd.a 8 60.l odd 4 2
2200.1.dd.a 8 120.w even 4 2
2200.1.dd.a 8 660.bp odd 20 2
2200.1.dd.a 8 1320.dh even 20 2
2816.1.v.c 8 48.i odd 4 2
2816.1.v.c 8 48.k even 4 2
2816.1.v.c 8 528.br even 20 2
2816.1.v.c 8 528.bu odd 20 2
3168.1.ck.a 4 1.a even 1 1 trivial
3168.1.ck.a 4 8.d odd 2 1 CM
3168.1.ck.a 4 11.c even 5 1 inner
3168.1.ck.a 4 88.l odd 10 1 inner
3872.1.f.a 2 33.h odd 10 1
3872.1.f.a 2 264.w even 10 1
3872.1.f.b 2 33.f even 10 1
3872.1.f.b 2 264.r odd 10 1
3872.1.t.a 4 33.f even 10 2
3872.1.t.a 4 264.r odd 10 2
3872.1.t.b 4 33.h odd 10 2
3872.1.t.b 4 264.w even 10 2
3872.1.t.c 4 33.d even 2 1
3872.1.t.c 4 33.f even 10 1
3872.1.t.c 4 264.p odd 2 1
3872.1.t.c 4 264.r odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3168, [\chi])\).

Hecke characteristic polynomials

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