Properties

Label 2007.1.d.c
Level $2007$
Weight $1$
Character orbit 2007.d
Self dual yes
Analytic conductor $1.002$
Analytic rank $0$
Dimension $6$
Projective image $D_{14}$
CM discriminant -223
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2007,1,Mod(1783,2007)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2007, 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("2007.1783");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2007 = 3^{2} \cdot 223 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2007.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: \(1.00162348035\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{28})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 7x^{4} + 14x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + ( - \beta_{4} - \beta_{2} + 1) q^{7} + ( - \beta_{3} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + ( - \beta_{4} - \beta_{2} + 1) q^{7} + ( - \beta_{3} - \beta_1) q^{8} + (\beta_{5} + \beta_{3}) q^{14} + (\beta_{4} + 2 \beta_{2}) q^{16} - \beta_{5} q^{17} - \beta_{2} q^{19} + q^{25} + ( - 2 \beta_{4} - 2 \beta_{2} + 1) q^{28} + \beta_1 q^{29} + \beta_{4} q^{31} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{32} + (2 \beta_{4} + \beta_{2} - 1) q^{34} + (\beta_{4} + \beta_{2} - 1) q^{37} + (\beta_{3} + \beta_1) q^{38} + \beta_{3} q^{41} - \beta_{4} q^{43} + \beta_{5} q^{47} + ( - \beta_{4} + 1) q^{49} - \beta_1 q^{50} + \beta_{5} q^{53} + (\beta_{5} + \beta_{3} + \beta_1) q^{56} + ( - \beta_{2} - 2) q^{58} - \beta_{5} q^{62} + (2 \beta_{4} + 2 \beta_{2}) q^{64} + ( - \beta_{5} - \beta_{3}) q^{68} + \beta_{4} q^{73} + ( - \beta_{5} - \beta_{3}) q^{74} + ( - \beta_{4} - 2 \beta_{2} - 1) q^{76} + ( - \beta_{4} - 2 \beta_{2} + 1) q^{82} + \beta_1 q^{83} + \beta_{5} q^{86} - \beta_{3} q^{89} + ( - 2 \beta_{4} - \beta_{2} + 1) q^{94} + (\beta_{5} - \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 8 q^{4} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 8 q^{4} + 2 q^{7} + 6 q^{16} - 2 q^{19} + 6 q^{25} - 2 q^{28} + 2 q^{31} - 2 q^{37} - 2 q^{43} + 4 q^{49} - 14 q^{58} + 8 q^{64} + 2 q^{73} - 12 q^{76}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{28} + \zeta_{28}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 5\nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 5\nu^{3} + 5\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 5\beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 5\beta_{3} + 10\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2007\mathbb{Z}\right)^\times\).

\(n\) \(226\) \(893\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1783.1
1.94986
1.56366
0.867767
−0.867767
−1.56366
−1.94986
−1.94986 0 2.80194 0 0 −1.24698 −3.51352 0 0
1783.2 −1.56366 0 1.44504 0 0 1.80194 −0.695895 0 0
1783.3 −0.867767 0 −0.246980 0 0 0.445042 1.08209 0 0
1783.4 0.867767 0 −0.246980 0 0 0.445042 −1.08209 0 0
1783.5 1.56366 0 1.44504 0 0 1.80194 0.695895 0 0
1783.6 1.94986 0 2.80194 0 0 −1.24698 3.51352 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1783.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
223.b odd 2 1 CM by \(\Q(\sqrt{-223}) \)
3.b odd 2 1 inner
669.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2007.1.d.c 6
3.b odd 2 1 inner 2007.1.d.c 6
223.b odd 2 1 CM 2007.1.d.c 6
669.c even 2 1 inner 2007.1.d.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2007.1.d.c 6 1.a even 1 1 trivial
2007.1.d.c 6 3.b odd 2 1 inner
2007.1.d.c 6 223.b odd 2 1 CM
2007.1.d.c 6 669.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 7T_{2}^{4} + 14T_{2}^{2} - 7 \) acting on \(S_{1}^{\mathrm{new}}(2007, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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