Properties

Label 556.1.c.a
Level $556$
Weight $1$
Character orbit 556.c
Self dual yes
Analytic conductor $0.277$
Analytic rank $0$
Dimension $3$
Projective image $D_{9}$
CM discriminant -139
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [556,1,Mod(277,556)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(556, 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("556.277");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 556 = 2^{2} \cdot 139 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 556.c (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: \(0.277480147024\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.23891266624.1
Artin image: $D_9$
Artin field: Galois closure of 9.1.23891266624.1

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} + \beta_{2} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{5} + \beta_{2} q^{7} + q^{9} + ( - \beta_{2} + \beta_1) q^{11} + ( - \beta_{2} + \beta_1) q^{13} + (\beta_{2} + 1) q^{25} + \beta_{2} q^{29} - \beta_1 q^{31} + ( - \beta_1 - 1) q^{35} - q^{37} - q^{41} - \beta_1 q^{45} - q^{47} + ( - \beta_{2} + \beta_1 + 1) q^{49} + ( - \beta_{2} + \beta_1 - 1) q^{55} + \beta_{2} q^{63} + ( - \beta_{2} + \beta_1 - 1) q^{65} - \beta_1 q^{67} + ( - \beta_{2} + \beta_1) q^{71} + (\beta_{2} - 1) q^{77} - \beta_1 q^{79} + q^{81} + \beta_{2} q^{83} + \beta_{2} q^{89} + (\beta_{2} - 1) q^{91} + ( - \beta_{2} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{9} + 3 q^{25} - 3 q^{35} - 3 q^{37} - 3 q^{41} - 3 q^{47} + 3 q^{49} - 3 q^{55} - 3 q^{65} - 3 q^{77} + 3 q^{81} - 3 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) 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 \(\left(\mathbb{Z}/556\mathbb{Z}\right)^\times\).

\(n\) \(141\) \(279\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
277.1
1.87939
−0.347296
−1.53209
0 0 0 −1.87939 0 1.53209 0 1.00000 0
277.2 0 0 0 0.347296 0 −1.87939 0 1.00000 0
277.3 0 0 0 1.53209 0 0.347296 0 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
139.b odd 2 1 CM by \(\Q(\sqrt{-139}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 556.1.c.a 3
4.b odd 2 1 2224.1.e.b 3
139.b odd 2 1 CM 556.1.c.a 3
556.d even 2 1 2224.1.e.b 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
556.1.c.a 3 1.a even 1 1 trivial
556.1.c.a 3 139.b odd 2 1 CM
2224.1.e.b 3 4.b odd 2 1
2224.1.e.b 3 556.d even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(556, [\chi])\).

Hecke characteristic polynomials

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