Properties

Label 3969.1.k.e
Level $3969$
Weight $1$
Character orbit 3969.k
Analytic conductor $1.981$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -7
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3969,1,Mod(460,3969)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3969, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3969.460");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3969 = 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3969.k (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: \(1.98078903514\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 567)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.6751269.1

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} + \zeta_{12}) q^{2} + (\zeta_{12}^{4} + \zeta_{12}^{2} - 1) q^{4} + (\zeta_{12}^{5} + \zeta_{12}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} + \zeta_{12}) q^{2} + (\zeta_{12}^{4} + \zeta_{12}^{2} - 1) q^{4} + (\zeta_{12}^{5} + \zeta_{12}) q^{8} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{11} + (\zeta_{12}^{2} + 1) q^{16} + (\zeta_{12}^{4} + \zeta_{12}^{2} + 1) q^{22} + q^{25} + \zeta_{12}^{3} q^{32} + \zeta_{12}^{4} q^{37} + \zeta_{12}^{4} q^{43} + (2 \zeta_{12}^{5} + \zeta_{12}^{3} + \zeta_{12}) q^{44} + (\zeta_{12}^{3} + \zeta_{12}) q^{50} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{53} - q^{64} - \zeta_{12}^{4} q^{67} + (\zeta_{12}^{5} - \zeta_{12}) q^{71} + (\zeta_{12}^{5} - \zeta_{12}) q^{74} + \zeta_{12}^{2} q^{79} + (\zeta_{12}^{5} - \zeta_{12}) q^{86} + (\zeta_{12}^{4} - \zeta_{12}^{2} - 2) q^{88} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 2 q^{16} + 6 q^{22} + 4 q^{25} - 2 q^{37} - 2 q^{43} - 4 q^{64} + 2 q^{67} + 2 q^{79} - 12 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3969\mathbb{Z}\right)^\times\).

\(n\) \(2108\) \(3727\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(-\zeta_{12}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
460.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 1.50000i 0 −1.00000 + 1.73205i 0 0 0 1.73205 0 0
460.2 0.866025 + 1.50000i 0 −1.00000 + 1.73205i 0 0 0 −1.73205 0 0
1648.1 −0.866025 + 1.50000i 0 −1.00000 1.73205i 0 0 0 1.73205 0 0
1648.2 0.866025 1.50000i 0 −1.00000 1.73205i 0 0 0 −1.73205 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
3.b odd 2 1 inner
21.c even 2 1 inner
63.g even 3 1 inner
63.k odd 6 1 inner
63.n odd 6 1 inner
63.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3969.1.k.e 4
3.b odd 2 1 inner 3969.1.k.e 4
7.b odd 2 1 CM 3969.1.k.e 4
7.c even 3 1 567.1.l.d 4
7.c even 3 1 3969.1.t.e 4
7.d odd 6 1 567.1.l.d 4
7.d odd 6 1 3969.1.t.e 4
9.c even 3 1 3969.1.m.c 4
9.c even 3 1 3969.1.t.e 4
9.d odd 6 1 3969.1.m.c 4
9.d odd 6 1 3969.1.t.e 4
21.c even 2 1 inner 3969.1.k.e 4
21.g even 6 1 567.1.l.d 4
21.g even 6 1 3969.1.t.e 4
21.h odd 6 1 567.1.l.d 4
21.h odd 6 1 3969.1.t.e 4
63.g even 3 1 567.1.d.c 2
63.g even 3 1 inner 3969.1.k.e 4
63.h even 3 1 567.1.l.d 4
63.h even 3 1 3969.1.m.c 4
63.i even 6 1 567.1.l.d 4
63.i even 6 1 3969.1.m.c 4
63.j odd 6 1 567.1.l.d 4
63.j odd 6 1 3969.1.m.c 4
63.k odd 6 1 567.1.d.c 2
63.k odd 6 1 inner 3969.1.k.e 4
63.l odd 6 1 3969.1.m.c 4
63.l odd 6 1 3969.1.t.e 4
63.n odd 6 1 567.1.d.c 2
63.n odd 6 1 inner 3969.1.k.e 4
63.o even 6 1 3969.1.m.c 4
63.o even 6 1 3969.1.t.e 4
63.s even 6 1 567.1.d.c 2
63.s even 6 1 inner 3969.1.k.e 4
63.t odd 6 1 567.1.l.d 4
63.t odd 6 1 3969.1.m.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
567.1.d.c 2 63.g even 3 1
567.1.d.c 2 63.k odd 6 1
567.1.d.c 2 63.n odd 6 1
567.1.d.c 2 63.s even 6 1
567.1.l.d 4 7.c even 3 1
567.1.l.d 4 7.d odd 6 1
567.1.l.d 4 21.g even 6 1
567.1.l.d 4 21.h odd 6 1
567.1.l.d 4 63.h even 3 1
567.1.l.d 4 63.i even 6 1
567.1.l.d 4 63.j odd 6 1
567.1.l.d 4 63.t odd 6 1
3969.1.k.e 4 1.a even 1 1 trivial
3969.1.k.e 4 3.b odd 2 1 inner
3969.1.k.e 4 7.b odd 2 1 CM
3969.1.k.e 4 21.c even 2 1 inner
3969.1.k.e 4 63.g even 3 1 inner
3969.1.k.e 4 63.k odd 6 1 inner
3969.1.k.e 4 63.n odd 6 1 inner
3969.1.k.e 4 63.s even 6 1 inner
3969.1.m.c 4 9.c even 3 1
3969.1.m.c 4 9.d odd 6 1
3969.1.m.c 4 63.h even 3 1
3969.1.m.c 4 63.i even 6 1
3969.1.m.c 4 63.j odd 6 1
3969.1.m.c 4 63.l odd 6 1
3969.1.m.c 4 63.o even 6 1
3969.1.m.c 4 63.t odd 6 1
3969.1.t.e 4 7.c even 3 1
3969.1.t.e 4 7.d odd 6 1
3969.1.t.e 4 9.c even 3 1
3969.1.t.e 4 9.d odd 6 1
3969.1.t.e 4 21.g even 6 1
3969.1.t.e 4 21.h odd 6 1
3969.1.t.e 4 63.l odd 6 1
3969.1.t.e 4 63.o even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3969, [\chi])\):

\( T_{2}^{4} + 3T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3T^{2} + 9 \) 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
$11$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) 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
$37$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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