Properties

Label 1459.1.b.a
Level $1459$
Weight $1$
Character orbit 1459.b
Self dual yes
Analytic conductor $0.728$
Analytic rank $0$
Dimension $5$
Projective image $D_{11}$
CM discriminant -1459
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1459,1,Mod(1458,1459)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1459, 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("1459.1458");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1459 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1459.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: \(0.728135853432\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\Q(\zeta_{22})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{11}\)
Projective field: Galois closure of 11.1.6611141604851299.1

$q$-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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{4} + \beta_{4} q^{5} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{4} + \beta_{4} q^{5} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{7} + q^{9} + \beta_{2} q^{11} - \beta_1 q^{13} + q^{16} - \beta_{3} q^{19} + \beta_{4} q^{20} - \beta_{3} q^{23} + ( - \beta_{3} + 1) q^{25} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{28} - \beta_1 q^{29} + (\beta_{2} - \beta_1) q^{35} + q^{36} + \beta_{2} q^{37} + \beta_{2} q^{44} + \beta_{4} q^{45} + ( - \beta_1 + 1) q^{49} - \beta_1 q^{52} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{53} + ( - \beta_{4} + \beta_{3} + \beta_1 - 1) q^{55} - \beta_{3} q^{59} + \beta_{2} q^{61} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{63} + q^{64} + ( - \beta_{4} - \beta_{2} + \beta_1 - 1) q^{65} - \beta_1 q^{67} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{71} + \beta_{4} q^{73} - \beta_{3} q^{76} + (\beta_{4} - \beta_{3}) q^{77} + \beta_{4} q^{79} + \beta_{4} q^{80} + q^{81} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{91} - \beta_{3} q^{92} + (\beta_{4} - \beta_1) q^{95} + \beta_{2} q^{97} + \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{4} - q^{5} - q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 5 q^{4} - q^{5} - q^{7} + 5 q^{9} - q^{11} - q^{13} + 5 q^{16} - q^{19} - q^{20} - q^{23} + 4 q^{25} - q^{28} - q^{29} - 2 q^{35} + 5 q^{36} - q^{37} - q^{44} - q^{45} + 4 q^{49} - q^{52} - q^{53} - 2 q^{55} - q^{59} - q^{61} - q^{63} + 5 q^{64} - 2 q^{65} - q^{67} - q^{71} - q^{73} - q^{76} - 2 q^{77} - q^{79} - q^{80} + 5 q^{81} - 2 q^{91} - q^{92} - 2 q^{95} - q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{22} + \zeta_{22}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) 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
\(\beta_{4}\)\(=\) \( \nu^{4} - 4\nu^{2} + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 4\beta_{2} + 6 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1459\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1458.1
1.30972
−1.68251
−0.830830
1.91899
0.284630
0 0 1.00000 −1.91899 0 0.830830 0 1.00000 0
1458.2 0 0 1.00000 −1.30972 0 −1.91899 0 1.00000 0
1458.3 0 0 1.00000 −0.284630 0 1.68251 0 1.00000 0
1458.4 0 0 1.00000 0.830830 0 −0.284630 0 1.00000 0
1458.5 0 0 1.00000 1.68251 0 −1.30972 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1458.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
1459.b odd 2 1 CM by \(\Q(\sqrt{-1459}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1459.1.b.a 5
1459.b odd 2 1 CM 1459.1.b.a 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1459.1.b.a 5 1.a even 1 1 trivial
1459.1.b.a 5 1459.b odd 2 1 CM

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1459, [\chi])\).

Hecke characteristic polynomials

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