Properties

Label 220.2.k.a
Level $220$
Weight $2$
Character orbit 220.k
Analytic conductor $1.757$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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: \(4\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{3} + (2 \beta_1 - 1) q^{5} + ( - \beta_{3} - \beta_{2}) q^{7} - \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{3} + (2 \beta_1 - 1) q^{5} + ( - \beta_{3} - \beta_{2}) q^{7} - \beta_1 q^{9} - \beta_{3} q^{11} + (\beta_{3} - \beta_{2}) q^{13} + ( - \beta_1 + 3) q^{15} + ( - \beta_{3} - \beta_{2}) q^{17} + 2 \beta_{3} q^{21} + (3 \beta_1 + 3) q^{23} + ( - 4 \beta_1 - 3) q^{25} + (4 \beta_1 - 4) q^{27} + 2 \beta_{2} q^{29} - 4 q^{31} + (\beta_{3} - \beta_{2}) q^{33} + ( - \beta_{3} + 3 \beta_{2}) q^{35} + ( - 5 \beta_1 + 5) q^{37} + 2 \beta_{2} q^{39} + ( - \beta_{3} + \beta_{2}) q^{43} + (\beta_1 + 2) q^{45} + ( - 5 \beta_1 + 5) q^{47} + 15 \beta_1 q^{49} + 2 \beta_{3} q^{51} + ( - 3 \beta_1 - 3) q^{53} + (\beta_{3} + 2 \beta_{2}) q^{55} - 10 \beta_1 q^{59} + 4 \beta_{3} q^{61} + (\beta_{3} - \beta_{2}) q^{63} + ( - 3 \beta_{3} - \beta_{2}) q^{65} + (3 \beta_1 - 3) q^{67} - 6 \beta_1 q^{69} - 4 q^{71} + (\beta_{3} - \beta_{2}) q^{73} + (7 \beta_1 - 1) q^{75} + (11 \beta_1 - 11) q^{77} - 4 \beta_{2} q^{79} + 5 q^{81} + ( - \beta_{3} + \beta_{2}) q^{83} + ( - \beta_{3} + 3 \beta_{2}) q^{85} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{87} - 12 \beta_1 q^{89} + 22 q^{91} + (4 \beta_1 + 4) q^{93} + ( - 5 \beta_1 + 5) q^{97} - \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} - 4 q^{5} + 12 q^{15} + 12 q^{23} - 12 q^{25} - 16 q^{27} - 16 q^{31} + 20 q^{37} + 8 q^{45} + 20 q^{47} - 12 q^{53} - 12 q^{67} - 16 q^{71} - 4 q^{75} - 44 q^{77} + 20 q^{81} + 88 q^{91} + 16 q^{93} + 20 q^{97}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 2\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 8\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} - 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + 4\beta_1 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
153.1
1.65831 + 0.500000i
−1.65831 + 0.500000i
1.65831 0.500000i
−1.65831 0.500000i
0 −1.00000 1.00000i 0 −1.00000 + 2.00000i 0 −3.31662 3.31662i 0 1.00000i 0
153.2 0 −1.00000 1.00000i 0 −1.00000 + 2.00000i 0 3.31662 + 3.31662i 0 1.00000i 0
197.1 0 −1.00000 + 1.00000i 0 −1.00000 2.00000i 0 −3.31662 + 3.31662i 0 1.00000i 0
197.2 0 −1.00000 + 1.00000i 0 −1.00000 2.00000i 0 3.31662 3.31662i 0 1.00000i 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
5.c odd 4 1 inner
11.b odd 2 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.a 4
3.b odd 2 1 1980.2.y.a 4
4.b odd 2 1 880.2.bd.f 4
5.b even 2 1 1100.2.k.a 4
5.c odd 4 1 inner 220.2.k.a 4
5.c odd 4 1 1100.2.k.a 4
11.b odd 2 1 inner 220.2.k.a 4
15.e even 4 1 1980.2.y.a 4
20.e even 4 1 880.2.bd.f 4
33.d even 2 1 1980.2.y.a 4
44.c even 2 1 880.2.bd.f 4
55.d odd 2 1 1100.2.k.a 4
55.e even 4 1 inner 220.2.k.a 4
55.e even 4 1 1100.2.k.a 4
165.l odd 4 1 1980.2.y.a 4
220.i odd 4 1 880.2.bd.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.2.k.a 4 1.a even 1 1 trivial
220.2.k.a 4 5.c odd 4 1 inner
220.2.k.a 4 11.b odd 2 1 inner
220.2.k.a 4 55.e even 4 1 inner
880.2.bd.f 4 4.b odd 2 1
880.2.bd.f 4 20.e even 4 1
880.2.bd.f 4 44.c even 2 1
880.2.bd.f 4 220.i odd 4 1
1100.2.k.a 4 5.b even 2 1
1100.2.k.a 4 5.c odd 4 1
1100.2.k.a 4 55.d odd 2 1
1100.2.k.a 4 55.e even 4 1
1980.2.y.a 4 3.b odd 2 1
1980.2.y.a 4 15.e even 4 1
1980.2.y.a 4 33.d even 2 1
1980.2.y.a 4 165.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 2T_{3} + 2 \) acting on \(S_{2}^{\mathrm{new}}(220, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 2 T + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 484 \) Copy content Toggle raw display
$11$ \( (T^{2} + 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 484 \) Copy content Toggle raw display
$17$ \( T^{4} + 484 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 6 T + 18)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 44)^{2} \) Copy content Toggle raw display
$31$ \( (T + 4)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 10 T + 50)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 484 \) Copy content Toggle raw display
$47$ \( (T^{2} - 10 T + 50)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 6 T + 18)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 176)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 6 T + 18)^{2} \) Copy content Toggle raw display
$71$ \( (T + 4)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 484 \) Copy content Toggle raw display
$79$ \( (T^{2} - 176)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 484 \) Copy content Toggle raw display
$89$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 10 T + 50)^{2} \) Copy content Toggle raw display
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