Properties

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

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [220,2,Mod(89,220)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(220, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("220.89");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 220.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: no
Analytic conductor: \(1.75670884447\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{3} + \beta_{2} q^{5} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{7} + (\beta_{2} + \beta_1 - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{3} + \beta_{2} q^{5} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{7} + (\beta_{2} + \beta_1 - 3) q^{9} + q^{11} + (\beta_{3} - \beta_{2} + \beta_1) q^{13} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{15} + (\beta_{3} + \beta_{2} - \beta_1) q^{17} - 4 q^{19} + (2 \beta_{2} + 2 \beta_1 - 4) q^{21} + ( - 2 \beta_{3} - \beta_{2} + \beta_1) q^{23} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 + 2) q^{25} + (2 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{27} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{29} + ( - \beta_{2} - \beta_1 + 6) q^{31} + ( - \beta_{2} + \beta_1) q^{33} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 + 2) q^{35} + ( - 2 \beta_{3} - \beta_{2} + \beta_1) q^{37} - 8 q^{39} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{41} + (\beta_{3} + \beta_{2} - \beta_1) q^{43} + ( - \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 7) q^{45} + ( - 2 \beta_{2} + 2 \beta_1) q^{47} - 5 q^{49} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{51} + \beta_{2} q^{55} + (4 \beta_{2} - 4 \beta_1) q^{57} + ( - \beta_{2} - \beta_1 - 10) q^{59} + (2 \beta_{2} + 2 \beta_1 - 2) q^{61} + (\beta_{3} + 7 \beta_{2} - 7 \beta_1) q^{63} + (3 \beta_{3} + \beta_{2} - \beta_1 + 4) q^{65} + (2 \beta_{3} - \beta_{2} + \beta_1) q^{67} + (3 \beta_{2} + 3 \beta_1 - 2) q^{69} + ( - \beta_{2} - \beta_1 + 2) q^{71} + ( - \beta_{3} + 5 \beta_{2} - 5 \beta_1) q^{73} + (\beta_{3} + 2 \beta_{2} + 3 \beta_1 - 7) q^{75} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{77} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{79} + ( - 2 \beta_{2} - 2 \beta_1 + 5) q^{81} + (\beta_{3} + 5 \beta_{2} - 5 \beta_1) q^{83} + (\beta_{3} - \beta_{2} + 3 \beta_1 - 2) q^{85} + ( - 4 \beta_{3} - 8 \beta_{2} + 8 \beta_1) q^{87} + ( - \beta_{2} - \beta_1 - 4) q^{89} + (4 \beta_{2} + 4 \beta_1 + 4) q^{91} + ( - 2 \beta_{3} - 9 \beta_{2} + 9 \beta_1) q^{93} - 4 \beta_{2} q^{95} + ( - 3 \beta_{2} + 3 \beta_1) q^{97} + (\beta_{2} + \beta_1 - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{5} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{5} - 10 q^{9} + 4 q^{11} + 11 q^{15} - 16 q^{19} - 12 q^{21} + 9 q^{25} + 4 q^{29} + 22 q^{31} + 6 q^{35} - 32 q^{39} + 4 q^{41} + 26 q^{45} - 20 q^{49} + 12 q^{51} + q^{55} - 42 q^{59} - 4 q^{61} + 16 q^{65} - 2 q^{69} + 6 q^{71} - 23 q^{75} + 12 q^{79} + 16 q^{81} - 6 q^{85} - 18 q^{89} + 24 q^{91} - 4 q^{95} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 4\nu + 5 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} + 3\nu^{2} - 3\nu - 20 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 2\beta_{2} - \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_{2} + 3\beta _1 + 7 \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(111\) \(177\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1
−1.63746 1.52274i
2.13746 0.656712i
2.13746 + 0.656712i
−1.63746 + 1.52274i
0 3.04547i 0 −1.63746 + 1.52274i 0 3.46410i 0 −6.27492 0
89.2 0 1.31342i 0 2.13746 + 0.656712i 0 3.46410i 0 1.27492 0
89.3 0 1.31342i 0 2.13746 0.656712i 0 3.46410i 0 1.27492 0
89.4 0 3.04547i 0 −1.63746 1.52274i 0 3.46410i 0 −6.27492 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.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 220.2.b.b 4
3.b odd 2 1 1980.2.c.g 4
4.b odd 2 1 880.2.b.i 4
5.b even 2 1 inner 220.2.b.b 4
5.c odd 4 2 1100.2.a.j 4
11.b odd 2 1 2420.2.b.e 4
15.d odd 2 1 1980.2.c.g 4
15.e even 4 2 9900.2.a.cb 4
20.d odd 2 1 880.2.b.i 4
20.e even 4 2 4400.2.a.cd 4
55.d odd 2 1 2420.2.b.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.2.b.b 4 1.a even 1 1 trivial
220.2.b.b 4 5.b even 2 1 inner
880.2.b.i 4 4.b odd 2 1
880.2.b.i 4 20.d odd 2 1
1100.2.a.j 4 5.c odd 4 2
1980.2.c.g 4 3.b odd 2 1
1980.2.c.g 4 15.d odd 2 1
2420.2.b.e 4 11.b odd 2 1
2420.2.b.e 4 55.d odd 2 1
4400.2.a.cd 4 20.e even 4 2
9900.2.a.cb 4 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 11T_{3}^{2} + 16 \) 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
$3$ \( T^{4} + 11T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{4} - T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 44T^{2} + 256 \) Copy content Toggle raw display
$17$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$19$ \( (T + 4)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 83T^{2} + 1024 \) Copy content Toggle raw display
$29$ \( (T^{2} - 2 T - 56)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 11 T + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 83T^{2} + 1024 \) Copy content Toggle raw display
$41$ \( (T^{2} - 2 T - 56)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 44T^{2} + 256 \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 21 T + 96)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T - 56)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 123T^{2} + 576 \) Copy content Toggle raw display
$71$ \( (T^{2} - 3 T - 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 348 T^{2} + 28224 \) Copy content Toggle raw display
$79$ \( (T^{2} - 6 T - 48)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 248T^{2} + 784 \) Copy content Toggle raw display
$89$ \( (T^{2} + 9 T + 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 99T^{2} + 1296 \) Copy content Toggle raw display
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