Properties

Label 2128.1.cl.c
Level $2128$
Weight $1$
Character orbit 2128.cl
Analytic conductor $1.062$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -19
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2128,1,Mod(417,2128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2128, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2128.417");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2128 = 2^{4} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2128.cl (of order \(6\), degree \(2\), 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: \(1.06201034688\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 133)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.931.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{6}^{2} q^{5} - \zeta_{6}^{2} q^{7} + \zeta_{6}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6}^{2} q^{5} - \zeta_{6}^{2} q^{7} + \zeta_{6}^{2} q^{9} - \zeta_{6} q^{11} - \zeta_{6} q^{17} - \zeta_{6}^{2} q^{19} + \zeta_{6}^{2} q^{23} - \zeta_{6} q^{35} + q^{43} + \zeta_{6} q^{45} + \zeta_{6}^{2} q^{47} - \zeta_{6} q^{49} - q^{55} - \zeta_{6}^{2} q^{61} + \zeta_{6} q^{63} + \zeta_{6} q^{73} - q^{77} - \zeta_{6} q^{81} + q^{83} - 2 q^{85} - \zeta_{6} q^{95} + q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{5} + q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{5} + q^{7} - q^{9} - q^{11} - 2 q^{17} + q^{19} - q^{23} - q^{35} + 2 q^{43} + q^{45} - q^{47} - q^{49} - 2 q^{55} + q^{61} + q^{63} + q^{73} - 2 q^{77} - q^{81} + 2 q^{83} - 4 q^{85} - q^{95} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2128\mathbb{Z}\right)^\times\).

\(n\) \(533\) \(799\) \(913\) \(1009\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{6}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
417.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 −0.500000 0.866025i 0
1633.1 0 0 0 0.500000 0.866025i 0 0.500000 0.866025i 0 −0.500000 + 0.866025i 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
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
7.c even 3 1 inner
133.r odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2128.1.cl.c 2
4.b odd 2 1 133.1.r.a 2
7.c even 3 1 inner 2128.1.cl.c 2
12.b even 2 1 1197.1.cz.a 2
19.b odd 2 1 CM 2128.1.cl.c 2
20.d odd 2 1 3325.1.bm.a 2
20.e even 4 2 3325.1.y.a 4
28.d even 2 1 931.1.r.a 2
28.f even 6 1 931.1.b.b 1
28.f even 6 1 931.1.r.a 2
28.g odd 6 1 133.1.r.a 2
28.g odd 6 1 931.1.b.a 1
76.d even 2 1 133.1.r.a 2
76.f even 6 1 2527.1.j.a 2
76.f even 6 1 2527.1.n.a 2
76.g odd 6 1 2527.1.j.a 2
76.g odd 6 1 2527.1.n.a 2
76.k even 18 3 2527.1.bd.a 6
76.k even 18 3 2527.1.be.a 6
76.l odd 18 3 2527.1.bd.a 6
76.l odd 18 3 2527.1.be.a 6
84.n even 6 1 1197.1.cz.a 2
133.r odd 6 1 inner 2128.1.cl.c 2
140.p odd 6 1 3325.1.bm.a 2
140.w even 12 2 3325.1.y.a 4
228.b odd 2 1 1197.1.cz.a 2
380.d even 2 1 3325.1.bm.a 2
380.j odd 4 2 3325.1.y.a 4
532.b odd 2 1 931.1.r.a 2
532.n odd 6 1 2527.1.j.a 2
532.t even 6 1 133.1.r.a 2
532.t even 6 1 931.1.b.a 1
532.y even 6 1 2527.1.n.a 2
532.bh odd 6 1 931.1.b.b 1
532.bh odd 6 1 931.1.r.a 2
532.bk odd 6 1 2527.1.n.a 2
532.bl even 6 1 2527.1.j.a 2
532.bs even 18 3 2527.1.bd.a 6
532.bt odd 18 3 2527.1.bd.a 6
532.ce even 18 3 2527.1.be.a 6
532.cf odd 18 3 2527.1.be.a 6
1596.cb odd 6 1 1197.1.cz.a 2
2660.cx even 6 1 3325.1.bm.a 2
2660.en odd 12 2 3325.1.y.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.1.r.a 2 4.b odd 2 1
133.1.r.a 2 28.g odd 6 1
133.1.r.a 2 76.d even 2 1
133.1.r.a 2 532.t even 6 1
931.1.b.a 1 28.g odd 6 1
931.1.b.a 1 532.t even 6 1
931.1.b.b 1 28.f even 6 1
931.1.b.b 1 532.bh odd 6 1
931.1.r.a 2 28.d even 2 1
931.1.r.a 2 28.f even 6 1
931.1.r.a 2 532.b odd 2 1
931.1.r.a 2 532.bh odd 6 1
1197.1.cz.a 2 12.b even 2 1
1197.1.cz.a 2 84.n even 6 1
1197.1.cz.a 2 228.b odd 2 1
1197.1.cz.a 2 1596.cb odd 6 1
2128.1.cl.c 2 1.a even 1 1 trivial
2128.1.cl.c 2 7.c even 3 1 inner
2128.1.cl.c 2 19.b odd 2 1 CM
2128.1.cl.c 2 133.r odd 6 1 inner
2527.1.j.a 2 76.f even 6 1
2527.1.j.a 2 76.g odd 6 1
2527.1.j.a 2 532.n odd 6 1
2527.1.j.a 2 532.bl even 6 1
2527.1.n.a 2 76.f even 6 1
2527.1.n.a 2 76.g odd 6 1
2527.1.n.a 2 532.y even 6 1
2527.1.n.a 2 532.bk odd 6 1
2527.1.bd.a 6 76.k even 18 3
2527.1.bd.a 6 76.l odd 18 3
2527.1.bd.a 6 532.bs even 18 3
2527.1.bd.a 6 532.bt odd 18 3
2527.1.be.a 6 76.k even 18 3
2527.1.be.a 6 76.l odd 18 3
2527.1.be.a 6 532.ce even 18 3
2527.1.be.a 6 532.cf odd 18 3
3325.1.y.a 4 20.e even 4 2
3325.1.y.a 4 140.w even 12 2
3325.1.y.a 4 380.j odd 4 2
3325.1.y.a 4 2660.en odd 12 2
3325.1.bm.a 2 20.d odd 2 1
3325.1.bm.a 2 140.p odd 6 1
3325.1.bm.a 2 380.d even 2 1
3325.1.bm.a 2 2660.cx even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2128, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{2} - T_{5} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} + T_{11} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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