Properties

Label 440.1.o.e
Level $440$
Weight $1$
Character orbit 440.o
Analytic conductor $0.220$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -55
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [440,1,Mod(109,440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(440, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("440.109");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 440.o (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: \(0.219588605559\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.38720.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} - \zeta_{8}^{2} q^{5} + (\zeta_{8}^{3} - \zeta_{8}) q^{7} - \zeta_{8}^{3} q^{8} - q^{9} + \zeta_{8}^{3} q^{10} - \zeta_{8}^{2} q^{11} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{13} + \cdots + \zeta_{8}^{2} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9} + 4 q^{14} - 4 q^{16} + 4 q^{20} - 4 q^{25} - 4 q^{26} - 4 q^{34} + 4 q^{44} + 4 q^{49} - 4 q^{55} - 4 q^{56} + 4 q^{70} + 8 q^{71} + 4 q^{81} + 4 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/440\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(177\) \(221\) \(321\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
−0.707107 0.707107i 0 1.00000i 1.00000i 0 −1.41421 0.707107 0.707107i −1.00000 −0.707107 + 0.707107i
109.2 −0.707107 + 0.707107i 0 1.00000i 1.00000i 0 −1.41421 0.707107 + 0.707107i −1.00000 −0.707107 0.707107i
109.3 0.707107 0.707107i 0 1.00000i 1.00000i 0 1.41421 −0.707107 0.707107i −1.00000 0.707107 + 0.707107i
109.4 0.707107 + 0.707107i 0 1.00000i 1.00000i 0 1.41421 −0.707107 + 0.707107i −1.00000 0.707107 0.707107i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
55.d odd 2 1 CM by \(\Q(\sqrt{-55}) \)
5.b even 2 1 inner
8.b even 2 1 inner
11.b odd 2 1 inner
40.f even 2 1 inner
88.b odd 2 1 inner
440.o odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 440.1.o.e 4
3.b odd 2 1 3960.1.x.i 4
4.b odd 2 1 1760.1.o.e 4
5.b even 2 1 inner 440.1.o.e 4
5.c odd 4 2 2200.1.d.g 4
8.b even 2 1 inner 440.1.o.e 4
8.d odd 2 1 1760.1.o.e 4
11.b odd 2 1 inner 440.1.o.e 4
15.d odd 2 1 3960.1.x.i 4
20.d odd 2 1 1760.1.o.e 4
24.h odd 2 1 3960.1.x.i 4
33.d even 2 1 3960.1.x.i 4
40.e odd 2 1 1760.1.o.e 4
40.f even 2 1 inner 440.1.o.e 4
40.i odd 4 2 2200.1.d.g 4
44.c even 2 1 1760.1.o.e 4
55.d odd 2 1 CM 440.1.o.e 4
55.e even 4 2 2200.1.d.g 4
88.b odd 2 1 inner 440.1.o.e 4
88.g even 2 1 1760.1.o.e 4
120.i odd 2 1 3960.1.x.i 4
165.d even 2 1 3960.1.x.i 4
220.g even 2 1 1760.1.o.e 4
264.m even 2 1 3960.1.x.i 4
440.c even 2 1 1760.1.o.e 4
440.o odd 2 1 inner 440.1.o.e 4
440.t even 4 2 2200.1.d.g 4
1320.u even 2 1 3960.1.x.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.1.o.e 4 1.a even 1 1 trivial
440.1.o.e 4 5.b even 2 1 inner
440.1.o.e 4 8.b even 2 1 inner
440.1.o.e 4 11.b odd 2 1 inner
440.1.o.e 4 40.f even 2 1 inner
440.1.o.e 4 55.d odd 2 1 CM
440.1.o.e 4 88.b odd 2 1 inner
440.1.o.e 4 440.o odd 2 1 inner
1760.1.o.e 4 4.b odd 2 1
1760.1.o.e 4 8.d odd 2 1
1760.1.o.e 4 20.d odd 2 1
1760.1.o.e 4 40.e odd 2 1
1760.1.o.e 4 44.c even 2 1
1760.1.o.e 4 88.g even 2 1
1760.1.o.e 4 220.g even 2 1
1760.1.o.e 4 440.c even 2 1
2200.1.d.g 4 5.c odd 4 2
2200.1.d.g 4 40.i odd 4 2
2200.1.d.g 4 55.e even 4 2
2200.1.d.g 4 440.t even 4 2
3960.1.x.i 4 3.b odd 2 1
3960.1.x.i 4 15.d odd 2 1
3960.1.x.i 4 24.h odd 2 1
3960.1.x.i 4 33.d even 2 1
3960.1.x.i 4 120.i odd 2 1
3960.1.x.i 4 165.d even 2 1
3960.1.x.i 4 264.m even 2 1
3960.1.x.i 4 1320.u even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(440, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{7}^{2} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( (T - 2)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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