Properties

Label 1183.1.d.a
Level $1183$
Weight $1$
Character orbit 1183.d
Self dual yes
Analytic conductor $0.590$
Analytic rank $0$
Dimension $3$
Projective image $D_{7}$
CM discriminant -7
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1183,1,Mod(846,1183)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1183, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1183.846");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1183.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: \(0.590393909945\)
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: yes
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.1655595487.1
Artin image: $D_7$
Artin field: Galois closure of 7.1.1655595487.1

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{7} + ( - \beta_{2} - 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{7} + ( - \beta_{2} - 1) q^{8} + q^{9} + \beta_{2} q^{11} - \beta_1 q^{14} + \beta_1 q^{16} - \beta_1 q^{18} + ( - \beta_{2} - 1) q^{22} + ( - \beta_{2} + \beta_1 - 1) q^{23} + q^{25} + (\beta_{2} + 1) q^{28} - \beta_1 q^{29} - q^{32} + (\beta_{2} + 1) q^{36} + ( - \beta_{2} + \beta_1 - 1) q^{37} - \beta_1 q^{43} + (\beta_1 + 1) q^{44} + (\beta_1 - 1) q^{46} + q^{49} - \beta_1 q^{50} + ( - \beta_{2} + \beta_1 - 1) q^{53} + ( - \beta_{2} - 1) q^{56} + (\beta_{2} + 2) q^{58} + q^{63} + ( - \beta_{2} + \beta_1 - 1) q^{67} + ( - \beta_{2} + \beta_1 - 1) q^{71} + ( - \beta_{2} - 1) q^{72} + (\beta_1 - 1) q^{74} + \beta_{2} q^{77} + \beta_{2} q^{79} + q^{81} + (\beta_{2} + 2) q^{86} + ( - \beta_1 - 1) q^{88} - q^{92} - \beta_1 q^{98} + \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 2 q^{4} + 3 q^{7} - 2 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 2 q^{4} + 3 q^{7} - 2 q^{8} + 3 q^{9} - q^{11} - q^{14} + q^{16} - q^{18} - 2 q^{22} - q^{23} + 3 q^{25} + 2 q^{28} - q^{29} - 3 q^{32} + 2 q^{36} - q^{37} - q^{43} + 4 q^{44} - 2 q^{46} + 3 q^{49} - q^{50} - q^{53} - 2 q^{56} + 5 q^{58} + 3 q^{63} - q^{67} - q^{71} - 2 q^{72} - 2 q^{74} - q^{77} - q^{79} + 3 q^{81} + 5 q^{86} - 4 q^{88} - 3 q^{92} - q^{98} - q^{99}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
846.1
1.80194
0.445042
−1.24698
−1.80194 0 2.24698 0 0 1.00000 −2.24698 1.00000 0
846.2 −0.445042 0 −0.801938 0 0 1.00000 0.801938 1.00000 0
846.3 1.24698 0 0.554958 0 0 1.00000 −0.554958 1.00000 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 \(\Q(\sqrt{-7}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.1.d.a 3
7.b odd 2 1 CM 1183.1.d.a 3
13.b even 2 1 1183.1.d.b yes 3
13.c even 3 2 1183.1.n.b 6
13.d odd 4 2 1183.1.b.a 6
13.e even 6 2 1183.1.n.a 6
13.f odd 12 4 1183.1.t.a 12
91.b odd 2 1 1183.1.d.b yes 3
91.i even 4 2 1183.1.b.a 6
91.n odd 6 2 1183.1.n.b 6
91.t odd 6 2 1183.1.n.a 6
91.bc even 12 4 1183.1.t.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1183.1.b.a 6 13.d odd 4 2
1183.1.b.a 6 91.i even 4 2
1183.1.d.a 3 1.a even 1 1 trivial
1183.1.d.a 3 7.b odd 2 1 CM
1183.1.d.b yes 3 13.b even 2 1
1183.1.d.b yes 3 91.b odd 2 1
1183.1.n.a 6 13.e even 6 2
1183.1.n.a 6 91.t odd 6 2
1183.1.n.b 6 13.c even 3 2
1183.1.n.b 6 91.n odd 6 2
1183.1.t.a 12 13.f odd 12 4
1183.1.t.a 12 91.bc even 12 4

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 \(S_{1}^{\mathrm{new}}(1183, [\chi])\). 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
$7$ \( (T - 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + T^{2} - 2T - 1 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} \) Copy content Toggle raw display
$19$ \( T^{3} \) Copy content Toggle raw display
$23$ \( T^{3} + T^{2} - 2T - 1 \) 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
$53$ \( T^{3} + T^{2} - 2T - 1 \) Copy content Toggle raw display
$59$ \( T^{3} \) Copy content Toggle raw display
$61$ \( T^{3} \) Copy content Toggle raw display
$67$ \( T^{3} + T^{2} - 2T - 1 \) Copy content Toggle raw display
$71$ \( T^{3} + T^{2} - 2T - 1 \) Copy content Toggle raw display
$73$ \( T^{3} \) Copy content Toggle raw display
$79$ \( T^{3} + T^{2} - 2T - 1 \) Copy content Toggle raw display
$83$ \( T^{3} \) Copy content Toggle raw display
$89$ \( T^{3} \) Copy content Toggle raw display
$97$ \( T^{3} \) Copy content Toggle raw display
show more
show less