Properties

Label 4410.2.a.g
Level $4410$
Weight $2$
Character orbit 4410.a
Self dual yes
Analytic conductor $35.214$
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] = [4410,2,Mod(1,4410)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4410, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4410.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4410.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: \(35.2140272914\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
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)
 
\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{5} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} - q^{5} - q^{8} + q^{10} + q^{11} + 7 q^{13} + q^{16} + 4 q^{17} + q^{19} - q^{20} - q^{22} - q^{23} + q^{25} - 7 q^{26} + 8 q^{29} + 6 q^{31} - q^{32} - 4 q^{34} - 3 q^{37} - q^{38} + q^{40} - 9 q^{41} - 4 q^{43} + q^{44} + q^{46} + 3 q^{47} - q^{50} + 7 q^{52} + q^{53} - q^{55} - 8 q^{58} - 12 q^{59} - 4 q^{61} - 6 q^{62} + q^{64} - 7 q^{65} + 12 q^{67} + 4 q^{68} + 14 q^{71} - 14 q^{73} + 3 q^{74} + q^{76} + 4 q^{79} - q^{80} + 9 q^{82} - 12 q^{83} - 4 q^{85} + 4 q^{86} - q^{88} + 2 q^{89} - q^{92} - 3 q^{94} - q^{95} - 16 q^{97}+O(q^{100}) \) 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
−1.00000 0 1.00000 −1.00000 0 0 −1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)

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 4410.2.a.g 1
3.b odd 2 1 1470.2.a.r 1
7.b odd 2 1 4410.2.a.q 1
7.c even 3 2 630.2.k.h 2
15.d odd 2 1 7350.2.a.j 1
21.c even 2 1 1470.2.a.k 1
21.g even 6 2 1470.2.i.i 2
21.h odd 6 2 210.2.i.a 2
84.n even 6 2 1680.2.bg.k 2
105.g even 2 1 7350.2.a.ba 1
105.o odd 6 2 1050.2.i.s 2
105.x even 12 4 1050.2.o.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.i.a 2 21.h odd 6 2
630.2.k.h 2 7.c even 3 2
1050.2.i.s 2 105.o odd 6 2
1050.2.o.j 4 105.x even 12 4
1470.2.a.k 1 21.c even 2 1
1470.2.a.r 1 3.b odd 2 1
1470.2.i.i 2 21.g even 6 2
1680.2.bg.k 2 84.n even 6 2
4410.2.a.g 1 1.a even 1 1 trivial
4410.2.a.q 1 7.b odd 2 1
7350.2.a.j 1 15.d odd 2 1
7350.2.a.ba 1 105.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4410))\):

\( T_{11} - 1 \) Copy content Toggle raw display
\( T_{13} - 7 \) Copy content Toggle raw display
\( T_{17} - 4 \) Copy content Toggle raw display
\( T_{19} - 1 \) Copy content Toggle raw display
\( T_{29} - 8 \) Copy content Toggle raw display
\( T_{31} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 1 \) Copy content Toggle raw display
$13$ \( T - 7 \) Copy content Toggle raw display
$17$ \( T - 4 \) Copy content Toggle raw display
$19$ \( T - 1 \) Copy content Toggle raw display
$23$ \( T + 1 \) Copy content Toggle raw display
$29$ \( T - 8 \) Copy content Toggle raw display
$31$ \( T - 6 \) Copy content Toggle raw display
$37$ \( T + 3 \) Copy content Toggle raw display
$41$ \( T + 9 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T - 3 \) Copy content Toggle raw display
$53$ \( T - 1 \) Copy content Toggle raw display
$59$ \( T + 12 \) Copy content Toggle raw display
$61$ \( T + 4 \) Copy content Toggle raw display
$67$ \( T - 12 \) Copy content Toggle raw display
$71$ \( T - 14 \) Copy content Toggle raw display
$73$ \( T + 14 \) Copy content Toggle raw display
$79$ \( T - 4 \) Copy content Toggle raw display
$83$ \( T + 12 \) Copy content Toggle raw display
$89$ \( T - 2 \) Copy content Toggle raw display
$97$ \( T + 16 \) Copy content Toggle raw display
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