Properties

Label 463.1.b.a
Level $463$
Weight $1$
Character orbit 463.b
Self dual yes
Analytic conductor $0.231$
Analytic rank $0$
Dimension $3$
Projective image $D_{7}$
CM discriminant -463
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [463,1,Mod(462,463)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(463, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("463.462");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 463 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 463.b (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: \(0.231067100849\)
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 7.1.99252847.1
Artin image: $D_7$
Artin field: Galois closure of 7.1.99252847.1

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + ( - \beta_{2} - 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + ( - \beta_{2} - 1) q^{8} + q^{9} + \beta_1 q^{16} + ( - \beta_{2} + \beta_1 - 1) q^{17} - \beta_1 q^{18} + q^{25} + \beta_{2} q^{29} + ( - \beta_{2} + \beta_1 - 1) q^{31} - q^{32} + (\beta_1 - 1) q^{34} + (\beta_{2} + 1) q^{36} + \beta_{2} q^{43} + \beta_{2} q^{47} + q^{49} - \beta_1 q^{50} + ( - \beta_{2} - 1) q^{58} - \beta_1 q^{59} - \beta_1 q^{61} + (\beta_1 - 1) q^{62} + ( - \beta_{2} + \beta_1 - 1) q^{67} - q^{68} + ( - \beta_{2} - 1) q^{72} - \beta_1 q^{73} - \beta_1 q^{79} + q^{81} + ( - \beta_{2} - 1) q^{86} + \beta_{2} q^{89} + ( - \beta_{2} - 1) q^{94} + \beta_{2} q^{97} - \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 2 q^{4} - 2 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 2 q^{4} - 2 q^{8} + 3 q^{9} + q^{16} - q^{17} - q^{18} + 3 q^{25} - q^{29} - q^{31} - 3 q^{32} - 2 q^{34} + 2 q^{36} - q^{43} - q^{47} + 3 q^{49} - q^{50} - 2 q^{58} - q^{59} - q^{61} - 2 q^{62} - q^{67} - 3 q^{68} - 2 q^{72} - q^{73} - q^{79} + 3 q^{81} - 2 q^{86} - q^{89} - 2 q^{94} - q^{97} - q^{98}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/463\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
462.1
1.80194
0.445042
−1.24698
−1.80194 0 2.24698 0 0 0 −2.24698 1.00000 0
462.2 −0.445042 0 −0.801938 0 0 0 0.801938 1.00000 0
462.3 1.24698 0 0.554958 0 0 0 −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
463.b odd 2 1 CM by \(\Q(\sqrt{-463}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 463.1.b.a 3
463.b odd 2 1 CM 463.1.b.a 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
463.1.b.a 3 1.a even 1 1 trivial
463.1.b.a 3 463.b odd 2 1 CM

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(463, [\chi])\).

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
$5$ \( T^{3} \) 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
$17$ \( T^{3} + T^{2} - 2T - 1 \) Copy content Toggle raw display
$19$ \( T^{3} \) 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
$37$ \( T^{3} \) 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
$47$ \( T^{3} + T^{2} - 2T - 1 \) Copy content Toggle raw display
$53$ \( T^{3} \) Copy content Toggle raw display
$59$ \( T^{3} + T^{2} - 2T - 1 \) Copy content Toggle raw display
$61$ \( T^{3} + T^{2} - 2T - 1 \) Copy content Toggle raw display
$67$ \( T^{3} + T^{2} - 2T - 1 \) 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
$79$ \( T^{3} + T^{2} - 2T - 1 \) Copy content Toggle raw display
$83$ \( T^{3} \) Copy content Toggle raw display
$89$ \( T^{3} + T^{2} - 2T - 1 \) Copy content Toggle raw display
$97$ \( T^{3} + T^{2} - 2T - 1 \) Copy content Toggle raw display
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