Properties

Label 220.2.k.b
Level $220$
Weight $2$
Character orbit 220.k
Analytic conductor $1.757$
Analytic rank $0$
Dimension $8$
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] = [220,2,Mod(153,220)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(220, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("220.153");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 220.k (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: \(1.75670884447\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.303595776.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{6} + \beta_{2}) q^{5} + ( - \beta_{7} + \beta_{5} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (\beta_{6} + \beta_{2}) q^{5} + ( - \beta_{7} + \beta_{5} + \cdots + \beta_1) q^{9}+ \cdots + ( - \beta_{7} + 6 \beta_{4} + \cdots - 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} - 16 q^{15} - 18 q^{23} - 2 q^{25} + 26 q^{27} + 22 q^{33} + 14 q^{37} + 18 q^{45} - 48 q^{47} - 24 q^{53} - 22 q^{55} - 26 q^{67} + 12 q^{71} + 64 q^{75} - 100 q^{81} + 18 q^{93} + 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -5\nu^{6} - 16\nu^{4} - 8\nu^{2} - 72\nu - 81 ) / 72 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{7} - 16\nu^{5} - 8\nu^{3} - 81\nu ) / 216 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{6} + 16\nu^{4} + 44\nu^{2} + 117 ) / 36 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8\nu^{7} - 9\nu^{6} + 40\nu^{5} - 72\nu^{4} + 128\nu^{3} - 144\nu^{2} + 360\nu - 405 ) / 216 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{7} - 3\nu^{6} + 20\nu^{5} + 12\nu^{4} + 64\nu^{3} + 60\nu^{2} + 180\nu + 81 ) / 108 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 2\nu^{5} + 10\nu^{3} + 33\nu ) / 18 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8\nu^{7} + 15\nu^{6} + 40\nu^{5} + 48\nu^{4} + 128\nu^{3} + 24\nu^{2} + 144\nu + 243 ) / 216 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{7} + \beta_{5} + \beta_{4} - 3\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -4\beta_{7} + 2\beta_{5} + 2\beta_{4} + 5\beta_{3} + 4\beta _1 - 5 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + 4\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11\beta_{7} + 7\beta_{5} - 18\beta_{4} - 15\beta_{3} - 11\beta _1 - 15 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7\beta_{7} - 40\beta_{6} + 14\beta_{5} + 14\beta_{4} - 40\beta_{2} - 7\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -8\beta_{5} + 8\beta_{4} + 8\beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2\beta_{7} + 120\beta_{6} - 61\beta_{5} - 61\beta_{4} - 120\beta_{2} + 63\beta_1 ) / 5 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/220\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(111\) \(177\)
\(\chi(n)\) \(-1\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
153.1
−1.26217 1.18614i
1.26217 1.18614i
−0.396143 + 1.68614i
0.396143 + 1.68614i
−1.26217 + 1.18614i
1.26217 + 1.18614i
−0.396143 1.68614i
0.396143 1.68614i
0 −2.44831 2.44831i 0 −0.469882 2.18614i 0 0 0 8.98844i 0
153.2 0 0.0760282 + 0.0760282i 0 0.469882 2.18614i 0 0 0 2.98844i 0
153.3 0 1.29000 + 1.29000i 0 2.12819 + 0.686141i 0 0 0 0.328185i 0
153.4 0 2.08228 + 2.08228i 0 −2.12819 + 0.686141i 0 0 0 5.67181i 0
197.1 0 −2.44831 + 2.44831i 0 −0.469882 + 2.18614i 0 0 0 8.98844i 0
197.2 0 0.0760282 0.0760282i 0 0.469882 + 2.18614i 0 0 0 2.98844i 0
197.3 0 1.29000 1.29000i 0 2.12819 0.686141i 0 0 0 0.328185i 0
197.4 0 2.08228 2.08228i 0 −2.12819 0.686141i 0 0 0 5.67181i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 153.4
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 220.2.k.b 8
3.b odd 2 1 1980.2.y.b 8
4.b odd 2 1 880.2.bd.h 8
5.b even 2 1 1100.2.k.b 8
5.c odd 4 1 inner 220.2.k.b 8
5.c odd 4 1 1100.2.k.b 8
11.b odd 2 1 CM 220.2.k.b 8
15.e even 4 1 1980.2.y.b 8
20.e even 4 1 880.2.bd.h 8
33.d even 2 1 1980.2.y.b 8
44.c even 2 1 880.2.bd.h 8
55.d odd 2 1 1100.2.k.b 8
55.e even 4 1 inner 220.2.k.b 8
55.e even 4 1 1100.2.k.b 8
165.l odd 4 1 1980.2.y.b 8
220.i odd 4 1 880.2.bd.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.2.k.b 8 1.a even 1 1 trivial
220.2.k.b 8 5.c odd 4 1 inner
220.2.k.b 8 11.b odd 2 1 CM
220.2.k.b 8 55.e even 4 1 inner
880.2.bd.h 8 4.b odd 2 1
880.2.bd.h 8 20.e even 4 1
880.2.bd.h 8 44.c even 2 1
880.2.bd.h 8 220.i odd 4 1
1100.2.k.b 8 5.b even 2 1
1100.2.k.b 8 5.c odd 4 1
1100.2.k.b 8 55.d odd 2 1
1100.2.k.b 8 55.e even 4 1
1980.2.y.b 8 3.b odd 2 1
1980.2.y.b 8 15.e even 4 1
1980.2.y.b 8 33.d even 2 1
1980.2.y.b 8 165.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 2T_{3}^{7} + 2T_{3}^{6} - 6T_{3}^{5} + 125T_{3}^{4} - 312T_{3}^{3} + 392T_{3}^{2} - 56T_{3} + 4 \) acting on \(S_{2}^{\mathrm{new}}(220, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{8} + T^{6} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{2} - 11)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} + 18 T^{7} + \cdots + 131044 \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} - 87 T^{2} + 36)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 14 T^{7} + \cdots + 12124324 \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} + 24 T^{3} + \cdots + 2500)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 12 T^{3} + \cdots + 4900)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 343 T^{2} + 27556)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} + 26 T^{7} + \cdots + 149866564 \) Copy content Toggle raw display
$71$ \( (T^{2} - 3 T - 204)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{4} + 453 T^{2} + 34596)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 34 T^{7} + \cdots + 368716804 \) Copy content Toggle raw display
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