Properties

Label 55.2.e.b
Level $55$
Weight $2$
Character orbit 55.e
Analytic conductor $0.439$
Analytic rank $0$
Dimension $4$
CM discriminant -11
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [55,2,Mod(32,55)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(55, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("55.32");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 55.e (of order \(4\), degree \(2\), 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: no
Analytic conductor: \(0.439177211117\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

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

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1 - 1) q^{3} + 2 \beta_{2} q^{4} + ( - \beta_{2} - \beta_1) q^{5} + (2 \beta_{3} - 3 \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_1 - 1) q^{3} + 2 \beta_{2} q^{4} + ( - \beta_{2} - \beta_1) q^{5} + (2 \beta_{3} - 3 \beta_{2} + 1) q^{9} + (\beta_{2} - 2 \beta_1) q^{11} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{12} + ( - 2 \beta_{3} + 4 \beta_{2} + \cdots - 3) q^{15}+ \cdots + (6 \beta_{3} - 11 \beta_{2} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 4 q^{12} - 8 q^{15} - 16 q^{16} + 12 q^{20} + 18 q^{23} + 2 q^{25} + 22 q^{27} - 22 q^{33} + 24 q^{36} - 14 q^{37} - 18 q^{45} - 24 q^{47} + 8 q^{48} - 12 q^{53} + 22 q^{55} - 28 q^{60} + 26 q^{67} - 12 q^{71} + 32 q^{75} - 8 q^{81} + 36 q^{92} + 66 q^{93} - 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 9 \) : Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/55\mathbb{Z}\right)^\times\).

\(n\) \(12\) \(46\)
\(\chi(n)\) \(\beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1
−1.65831 + 0.500000i
1.65831 + 0.500000i
−1.65831 0.500000i
1.65831 0.500000i
0 −2.15831 + 2.15831i 2.00000i 1.65831 1.50000i 0 0 0 6.31662i 0
32.2 0 1.15831 1.15831i 2.00000i −1.65831 1.50000i 0 0 0 0.316625i 0
43.1 0 −2.15831 2.15831i 2.00000i 1.65831 + 1.50000i 0 0 0 6.31662i 0
43.2 0 1.15831 + 1.15831i 2.00000i −1.65831 + 1.50000i 0 0 0 0.316625i 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
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
5.c odd 4 1 inner
55.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 55.2.e.b 4
3.b odd 2 1 495.2.k.a 4
4.b odd 2 1 880.2.bd.d 4
5.b even 2 1 275.2.e.a 4
5.c odd 4 1 inner 55.2.e.b 4
5.c odd 4 1 275.2.e.a 4
11.b odd 2 1 CM 55.2.e.b 4
11.c even 5 4 605.2.m.a 16
11.d odd 10 4 605.2.m.a 16
15.e even 4 1 495.2.k.a 4
20.e even 4 1 880.2.bd.d 4
33.d even 2 1 495.2.k.a 4
44.c even 2 1 880.2.bd.d 4
55.d odd 2 1 275.2.e.a 4
55.e even 4 1 inner 55.2.e.b 4
55.e even 4 1 275.2.e.a 4
55.k odd 20 4 605.2.m.a 16
55.l even 20 4 605.2.m.a 16
165.l odd 4 1 495.2.k.a 4
220.i odd 4 1 880.2.bd.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.e.b 4 1.a even 1 1 trivial
55.2.e.b 4 5.c odd 4 1 inner
55.2.e.b 4 11.b odd 2 1 CM
55.2.e.b 4 55.e even 4 1 inner
275.2.e.a 4 5.b even 2 1
275.2.e.a 4 5.c odd 4 1
275.2.e.a 4 55.d odd 2 1
275.2.e.a 4 55.e even 4 1
495.2.k.a 4 3.b odd 2 1
495.2.k.a 4 15.e even 4 1
495.2.k.a 4 33.d even 2 1
495.2.k.a 4 165.l odd 4 1
605.2.m.a 16 11.c even 5 4
605.2.m.a 16 11.d odd 10 4
605.2.m.a 16 55.k odd 20 4
605.2.m.a 16 55.l even 20 4
880.2.bd.d 4 4.b odd 2 1
880.2.bd.d 4 20.e even 4 1
880.2.bd.d 4 44.c even 2 1
880.2.bd.d 4 220.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(55, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( T^{4} - T^{2} + 25 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 18 T^{3} + \cdots + 1225 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 99)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 14 T^{3} + \cdots + 625 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 24 T^{3} + \cdots + 2500 \) Copy content Toggle raw display
$53$ \( T^{4} + 12 T^{3} + \cdots + 4900 \) Copy content Toggle raw display
$59$ \( (T^{2} + 11)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} - 26 T^{3} + \cdots + 1225 \) Copy content Toggle raw display
$71$ \( (T + 3)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 34 T^{3} + \cdots + 9025 \) Copy content Toggle raw display
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