Properties

Label 440.2.a.d
Level $440$
Weight $2$
Character orbit 440.a
Self dual yes
Analytic conductor $3.513$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [440,2,Mod(1,440)]
 
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, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("440.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 440.a (trivial)

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: yes
Analytic conductor: \(3.51341768894\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 3 q^{3} + q^{5} + q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + q^{5} + q^{7} + 6 q^{9} - q^{11} - 6 q^{13} + 3 q^{15} + 3 q^{17} - 5 q^{19} + 3 q^{21} - 2 q^{23} + q^{25} + 9 q^{27} - 5 q^{29} + 5 q^{31} - 3 q^{33} + q^{35} - q^{37} - 18 q^{39} - 2 q^{41} + 12 q^{43} + 6 q^{45} - 2 q^{47} - 6 q^{49} + 9 q^{51} - 13 q^{53} - q^{55} - 15 q^{57} + 2 q^{59} + q^{61} + 6 q^{63} - 6 q^{65} + 16 q^{67} - 6 q^{69} + 15 q^{71} + 10 q^{73} + 3 q^{75} - q^{77} + 2 q^{79} + 9 q^{81} - 14 q^{83} + 3 q^{85} - 15 q^{87} + 9 q^{89} - 6 q^{91} + 15 q^{93} - 5 q^{95} - 16 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 3.00000 0 1.00000 0 1.00000 0 6.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(11\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 440.2.a.d 1
3.b odd 2 1 3960.2.a.f 1
4.b odd 2 1 880.2.a.a 1
5.b even 2 1 2200.2.a.a 1
5.c odd 4 2 2200.2.b.b 2
8.b even 2 1 3520.2.a.a 1
8.d odd 2 1 3520.2.a.bh 1
11.b odd 2 1 4840.2.a.i 1
12.b even 2 1 7920.2.a.e 1
20.d odd 2 1 4400.2.a.be 1
20.e even 4 2 4400.2.b.a 2
44.c even 2 1 9680.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.a.d 1 1.a even 1 1 trivial
880.2.a.a 1 4.b odd 2 1
2200.2.a.a 1 5.b even 2 1
2200.2.b.b 2 5.c odd 4 2
3520.2.a.a 1 8.b even 2 1
3520.2.a.bh 1 8.d odd 2 1
3960.2.a.f 1 3.b odd 2 1
4400.2.a.be 1 20.d odd 2 1
4400.2.b.a 2 20.e even 4 2
4840.2.a.i 1 11.b odd 2 1
7920.2.a.e 1 12.b even 2 1
9680.2.a.a 1 44.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(440))\):

\( T_{3} - 3 \) Copy content Toggle raw display
\( T_{7} - 1 \) Copy content Toggle raw display
\( T_{13} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T + 6 \) Copy content Toggle raw display
$17$ \( T - 3 \) Copy content Toggle raw display
$19$ \( T + 5 \) Copy content Toggle raw display
$23$ \( T + 2 \) Copy content Toggle raw display
$29$ \( T + 5 \) Copy content Toggle raw display
$31$ \( T - 5 \) Copy content Toggle raw display
$37$ \( T + 1 \) Copy content Toggle raw display
$41$ \( T + 2 \) Copy content Toggle raw display
$43$ \( T - 12 \) Copy content Toggle raw display
$47$ \( T + 2 \) Copy content Toggle raw display
$53$ \( T + 13 \) Copy content Toggle raw display
$59$ \( T - 2 \) Copy content Toggle raw display
$61$ \( T - 1 \) Copy content Toggle raw display
$67$ \( T - 16 \) Copy content Toggle raw display
$71$ \( T - 15 \) Copy content Toggle raw display
$73$ \( T - 10 \) Copy content Toggle raw display
$79$ \( T - 2 \) Copy content Toggle raw display
$83$ \( T + 14 \) Copy content Toggle raw display
$89$ \( T - 9 \) Copy content Toggle raw display
$97$ \( T + 16 \) Copy content Toggle raw display
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