Properties

Label 2007.1.d.b
Level $2007$
Weight $1$
Character orbit 2007.d
Self dual yes
Analytic conductor $1.002$
Analytic rank $0$
Dimension $3$
Projective image $D_{7}$
CM discriminant -223
Inner twists $2$

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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: \(3\)
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 223)
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.11089567.1
Artin image: $D_{14}$
Artin 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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + ( - \beta_{2} + \beta_1 - 1) q^{7} + (\beta_{2} + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + ( - \beta_{2} + \beta_1 - 1) q^{7} + (\beta_{2} + 1) q^{8} + ( - \beta_1 + 1) q^{14} + \beta_1 q^{16} - \beta_{2} q^{17} + \beta_{2} q^{19} + q^{25} - q^{28} + \beta_1 q^{29} - \beta_1 q^{31} + q^{32} + ( - \beta_{2} - 1) q^{34} + ( - \beta_{2} + \beta_1 - 1) q^{37} + (\beta_{2} + 1) q^{38} + (\beta_{2} - \beta_1 + 1) q^{41} - \beta_1 q^{43} - \beta_{2} q^{47} + ( - \beta_1 + 1) q^{49} + \beta_1 q^{50} - \beta_{2} q^{53} - q^{56} + (\beta_{2} + 2) q^{58} + ( - \beta_{2} - 2) q^{62} + ( - \beta_1 - 1) q^{68} - \beta_1 q^{73} + ( - \beta_1 + 1) q^{74} + (\beta_1 + 1) q^{76} + (\beta_1 - 1) q^{82} + \beta_1 q^{83} + ( - \beta_{2} - 2) q^{86} + (\beta_{2} - \beta_1 + 1) q^{89} + ( - \beta_{2} - 1) q^{94} + ( - \beta_{2} + \beta_1 - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 2 q^{4} - q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + 2 q^{4} - q^{7} + 2 q^{8} + 2 q^{14} + q^{16} + q^{17} - q^{19} + 3 q^{25} - 3 q^{28} + q^{29} - q^{31} + 3 q^{32} - 2 q^{34} - q^{37} + 2 q^{38} + q^{41} - q^{43} + q^{47} + 2 q^{49} + q^{50} + q^{53} - 3 q^{56} + 5 q^{58} - 5 q^{62} - 4 q^{68} - q^{73} + 2 q^{74} + 4 q^{76} - 2 q^{82} + q^{83} - 5 q^{86} + q^{89} - 2 q^{94} - 4 q^{98}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/2007\mathbb{Z}\right)^\times\).

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

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
−1.24698
0.445042
1.80194
−1.24698 0 0.554958 0 0 −1.80194 0.554958 0 0
1783.2 0.445042 0 −0.801938 0 0 1.24698 −0.801938 0 0
1783.3 1.80194 0 2.24698 0 0 −0.445042 2.24698 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
223.b odd 2 1 CM by \(\Q(\sqrt{-223}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2007.1.d.b 3
3.b odd 2 1 223.1.b.a 3
12.b even 2 1 3568.1.c.a 3
223.b odd 2 1 CM 2007.1.d.b 3
669.c even 2 1 223.1.b.a 3
2676.f odd 2 1 3568.1.c.a 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
223.1.b.a 3 3.b odd 2 1
223.1.b.a 3 669.c even 2 1
2007.1.d.b 3 1.a even 1 1 trivial
2007.1.d.b 3 223.b odd 2 1 CM
3568.1.c.a 3 12.b even 2 1
3568.1.c.a 3 2676.f odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - T_{2}^{2} - 2T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2007, [\chi])\). Copy content Toggle raw display

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
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + T^{2} - 2T - 1 \) Copy content Toggle raw display
$11$ \( T^{3} \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - T^{2} - 2T + 1 \) 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
$31$ \( T^{3} + T^{2} - 2T - 1 \) Copy content Toggle raw display
$37$ \( T^{3} + T^{2} - 2T - 1 \) Copy content Toggle raw display
$41$ \( T^{3} - T^{2} - 2T + 1 \) Copy content Toggle raw display
$43$ \( T^{3} + T^{2} - 2T - 1 \) Copy content Toggle raw display
$47$ \( T^{3} - T^{2} - 2T + 1 \) Copy content Toggle raw display
$53$ \( T^{3} - T^{2} - 2T + 1 \) 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
$71$ \( T^{3} \) Copy content Toggle raw display
$73$ \( T^{3} + T^{2} - 2T - 1 \) Copy content Toggle raw display
$79$ \( T^{3} \) 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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