Properties

Label 3087.1.v.a
Level $3087$
Weight $1$
Character orbit 3087.v
Analytic conductor $1.541$
Analytic rank $0$
Dimension $6$
Projective image $D_{14}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3087,1,Mod(244,3087)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3087, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3087.244");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3087 = 3^{2} \cdot 7^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3087.v (of order \(14\), degree \(6\), 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: \(1.54061369400\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 441)
Projective image: \(D_{14}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{14} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{14} q^{4}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{14} q^{4} + ( - \zeta_{14}^{6} + \zeta_{14}^{2}) q^{13} + \zeta_{14}^{2} q^{16} + ( - \zeta_{14}^{4} - \zeta_{14}^{3}) q^{19} - \zeta_{14}^{3} q^{25} + (\zeta_{14}^{6} + \zeta_{14}) q^{31} + (\zeta_{14}^{5} - 1) q^{37} + ( - \zeta_{14}^{6} + \zeta_{14}^{5}) q^{43} + (\zeta_{14}^{3} + 1) q^{52} + (\zeta_{14}^{5} + 1) q^{61} + \zeta_{14}^{3} q^{64} + ( - \zeta_{14}^{5} + \zeta_{14}^{2}) q^{67} + ( - \zeta_{14}^{4} + \zeta_{14}^{2}) q^{73} + ( - \zeta_{14}^{5} - \zeta_{14}^{4}) q^{76} + (\zeta_{14}^{6} - \zeta_{14}) q^{79} + (\zeta_{14}^{4} + \zeta_{14}^{3}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{4} - q^{16} - q^{25} - 5 q^{37} + 2 q^{43} + 7 q^{52} + 7 q^{61} + q^{64} - 2 q^{67} - 2 q^{79}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3087\mathbb{Z}\right)^\times\).

\(n\) \(344\) \(2404\)
\(\chi(n)\) \(1\) \(\zeta_{14}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
244.1
0.900969 + 0.433884i
0.900969 0.433884i
−0.623490 + 0.781831i
0.222521 0.974928i
0.222521 + 0.974928i
−0.623490 0.781831i
0 0 0.900969 + 0.433884i 0 0 0 0 0 0
1126.1 0 0 0.900969 0.433884i 0 0 0 0 0 0
1567.1 0 0 −0.623490 + 0.781831i 0 0 0 0 0 0
2008.1 0 0 0.222521 0.974928i 0 0 0 0 0 0
2449.1 0 0 0.222521 + 0.974928i 0 0 0 0 0 0
2890.1 0 0 −0.623490 0.781831i 0 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 244.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
49.f odd 14 1 inner
147.k even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3087.1.v.a 6
3.b odd 2 1 CM 3087.1.v.a 6
7.b odd 2 1 441.1.v.a 6
7.c even 3 2 3087.1.bj.a 12
7.d odd 6 2 3087.1.bj.b 12
21.c even 2 1 441.1.v.a 6
21.g even 6 2 3087.1.bj.b 12
21.h odd 6 2 3087.1.bj.a 12
49.e even 7 1 441.1.v.a 6
49.f odd 14 1 inner 3087.1.v.a 6
49.g even 21 2 3087.1.bj.b 12
49.h odd 42 2 3087.1.bj.a 12
63.l odd 6 2 3969.1.bz.a 12
63.o even 6 2 3969.1.bz.a 12
147.k even 14 1 inner 3087.1.v.a 6
147.l odd 14 1 441.1.v.a 6
147.n odd 42 2 3087.1.bj.b 12
147.o even 42 2 3087.1.bj.a 12
441.ba even 21 2 3969.1.bz.a 12
441.be odd 42 2 3969.1.bz.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.1.v.a 6 7.b odd 2 1
441.1.v.a 6 21.c even 2 1
441.1.v.a 6 49.e even 7 1
441.1.v.a 6 147.l odd 14 1
3087.1.v.a 6 1.a even 1 1 trivial
3087.1.v.a 6 3.b odd 2 1 CM
3087.1.v.a 6 49.f odd 14 1 inner
3087.1.v.a 6 147.k even 14 1 inner
3087.1.bj.a 12 7.c even 3 2
3087.1.bj.a 12 21.h odd 6 2
3087.1.bj.a 12 49.h odd 42 2
3087.1.bj.a 12 147.o even 42 2
3087.1.bj.b 12 7.d odd 6 2
3087.1.bj.b 12 21.g even 6 2
3087.1.bj.b 12 49.g even 21 2
3087.1.bj.b 12 147.n odd 42 2
3969.1.bz.a 12 63.l odd 6 2
3969.1.bz.a 12 63.o even 6 2
3969.1.bz.a 12 441.ba even 21 2
3969.1.bz.a 12 441.be odd 42 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3087, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - 7 T^{3} + 7 T + 7 \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( T^{6} + 7 T^{4} + 14 T^{2} + 7 \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} + 7 T^{4} + 14 T^{2} + 7 \) Copy content Toggle raw display
$37$ \( T^{6} + 5 T^{5} + 11 T^{4} + 13 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{6} \) Copy content Toggle raw display
$43$ \( T^{6} - 2 T^{5} + 4 T^{4} - 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{6} \) Copy content Toggle raw display
$53$ \( T^{6} \) Copy content Toggle raw display
$59$ \( T^{6} \) Copy content Toggle raw display
$61$ \( T^{6} - 7 T^{5} + 21 T^{4} - 35 T^{3} + \cdots + 7 \) Copy content Toggle raw display
$67$ \( (T^{3} + T^{2} - 2 T - 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( T^{6} + 7 T^{2} - 14 T + 7 \) Copy content Toggle raw display
$79$ \( (T^{3} + T^{2} - 2 T - 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} \) Copy content Toggle raw display
$89$ \( T^{6} \) Copy content Toggle raw display
$97$ \( T^{6} + 7 T^{4} + 14 T^{2} + 7 \) Copy content Toggle raw display
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