Properties

Label 639.1.d.a
Level $639$
Weight $1$
Character orbit 639.d
Self dual yes
Analytic conductor $0.319$
Analytic rank $0$
Dimension $3$
Projective image $D_{7}$
CM discriminant -71
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [639,1,Mod(496,639)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(639, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("639.496");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 639 = 3^{2} \cdot 71 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 639.d (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: yes
Analytic conductor: \(0.318902543072\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 71)
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.357911.1
Artin image: $D_{14}$
Artin field: Galois closure of 14.0.280155320935227.1

$q$-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 basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - \beta_{2} q^{5} + (\beta_{2} + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - \beta_{2} q^{5} + (\beta_{2} + 1) q^{8} + ( - \beta_{2} - 1) q^{10} + \beta_1 q^{16} - \beta_1 q^{19} + ( - \beta_1 - 1) q^{20} + ( - \beta_{2} + \beta_1) q^{25} + (\beta_{2} - \beta_1 + 1) q^{29} + q^{32} - \beta_1 q^{37} + ( - \beta_{2} - 2) q^{38} + ( - \beta_1 - 1) q^{40} + \beta_{2} q^{43} + q^{49} + q^{50} + (\beta_1 - 1) q^{58} - q^{71} + \beta_{2} q^{73} + ( - \beta_{2} - 2) q^{74} + ( - \beta_{2} - \beta_1 - 1) q^{76} + \beta_{2} q^{79} + ( - \beta_{2} - 1) q^{80} + \beta_1 q^{83} + (\beta_{2} + 1) q^{86} + (\beta_{2} - \beta_1 + 1) q^{89} + (\beta_{2} + 1) q^{95} + \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 2 q^{4} + q^{5} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + 2 q^{4} + q^{5} + 2 q^{8} - 2 q^{10} + q^{16} - q^{19} - 4 q^{20} + 2 q^{25} + q^{29} + 3 q^{32} - q^{37} - 5 q^{38} - 4 q^{40} - q^{43} + 3 q^{49} + 3 q^{50} - 2 q^{58} - 3 q^{71} - q^{73} - 5 q^{74} - 3 q^{76} - q^{79} - 2 q^{80} + q^{83} + 2 q^{86} + q^{89} + 2 q^{95} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/639\mathbb{Z}\right)^\times\).

\(n\) \(433\) \(569\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
496.1
−1.24698
0.445042
1.80194
−1.24698 0 0.554958 0.445042 0 0 0.554958 0 −0.554958
496.2 0.445042 0 −0.801938 1.80194 0 0 −0.801938 0 0.801938
496.3 1.80194 0 2.24698 −1.24698 0 0 2.24698 0 −2.24698
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.b odd 2 1 CM by \(\Q(\sqrt{-71}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 639.1.d.a 3
3.b odd 2 1 71.1.b.a 3
12.b even 2 1 1136.1.h.a 3
15.d odd 2 1 1775.1.d.b 3
15.e even 4 2 1775.1.c.a 6
21.c even 2 1 3479.1.d.e 3
21.g even 6 2 3479.1.g.d 6
21.h odd 6 2 3479.1.g.e 6
71.b odd 2 1 CM 639.1.d.a 3
213.b even 2 1 71.1.b.a 3
852.d odd 2 1 1136.1.h.a 3
1065.h even 2 1 1775.1.d.b 3
1065.l odd 4 2 1775.1.c.a 6
1491.e odd 2 1 3479.1.d.e 3
1491.n odd 6 2 3479.1.g.d 6
1491.p even 6 2 3479.1.g.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.1.b.a 3 3.b odd 2 1
71.1.b.a 3 213.b even 2 1
639.1.d.a 3 1.a even 1 1 trivial
639.1.d.a 3 71.b odd 2 1 CM
1136.1.h.a 3 12.b even 2 1
1136.1.h.a 3 852.d odd 2 1
1775.1.c.a 6 15.e even 4 2
1775.1.c.a 6 1065.l odd 4 2
1775.1.d.b 3 15.d odd 2 1
1775.1.d.b 3 1065.h even 2 1
3479.1.d.e 3 21.c even 2 1
3479.1.d.e 3 1491.e odd 2 1
3479.1.g.d 6 21.g even 6 2
3479.1.g.d 6 1491.n odd 6 2
3479.1.g.e 6 21.h odd 6 2
3479.1.g.e 6 1491.p even 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(639, [\chi])\).

Hecke characteristic polynomials

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