Properties

Label 2.22.a.b
Level $2$
Weight $22$
Character orbit 2.a
Self dual yes
Analytic conductor $5.590$
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] = [2,22,Mod(1,2)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2, base_ring=CyclotomicField(1))
 
chi = DirichletCharacter(H, H._module([]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 2.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: \(5.58954688574\)
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)
 
\(f(q)\) \(=\) \( q + 1024 q^{2} + 59316 q^{3} + 1048576 q^{4} + 4975350 q^{5} + 60739584 q^{6} + 1427425832 q^{7} + 1073741824 q^{8} - 6941965347 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 1024 q^{2} + 59316 q^{3} + 1048576 q^{4} + 4975350 q^{5} + 60739584 q^{6} + 1427425832 q^{7} + 1073741824 q^{8} - 6941965347 q^{9} + 5094758400 q^{10} - 106767894948 q^{11} + 62197334016 q^{12} - 150150565474 q^{13} + 1461684051968 q^{14} + 295117860600 q^{15} + 1099511627776 q^{16} - 11203980739758 q^{17} - 7108572515328 q^{18} + 11024055955460 q^{19} + 5217032601600 q^{20} + 84669190650912 q^{21} - 109330324426752 q^{22} + 129502845739896 q^{23} + 63690070032384 q^{24} - 452083050580625 q^{25} - 153754179045376 q^{26} - 10\!\cdots\!00 q^{27}+ \cdots + 74\!\cdots\!56 q^{99}+O(q^{100}) \) 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
1024.00 59316.0 1.04858e6 4.97535e6 6.07396e7 1.42743e9 1.07374e9 −6.94197e9 5.09476e9
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 2.22.a.b 1
3.b odd 2 1 18.22.a.b 1
4.b odd 2 1 16.22.a.b 1
5.b even 2 1 50.22.a.a 1
5.c odd 4 2 50.22.b.c 2
8.b even 2 1 64.22.a.c 1
8.d odd 2 1 64.22.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.22.a.b 1 1.a even 1 1 trivial
16.22.a.b 1 4.b odd 2 1
18.22.a.b 1 3.b odd 2 1
50.22.a.a 1 5.b even 2 1
50.22.b.c 2 5.c odd 4 2
64.22.a.c 1 8.b even 2 1
64.22.a.e 1 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 59316 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(2))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1024 \) Copy content Toggle raw display
$3$ \( T - 59316 \) Copy content Toggle raw display
$5$ \( T - 4975350 \) Copy content Toggle raw display
$7$ \( T - 1427425832 \) Copy content Toggle raw display
$11$ \( T + 106767894948 \) Copy content Toggle raw display
$13$ \( T + 150150565474 \) Copy content Toggle raw display
$17$ \( T + 11203980739758 \) Copy content Toggle raw display
$19$ \( T - 11024055955460 \) Copy content Toggle raw display
$23$ \( T - 129502845739896 \) Copy content Toggle raw display
$29$ \( T - 2382370826608110 \) Copy content Toggle raw display
$31$ \( T + 878552957377888 \) Copy content Toggle raw display
$37$ \( T - 31\!\cdots\!22 \) Copy content Toggle raw display
$41$ \( T + 24\!\cdots\!38 \) Copy content Toggle raw display
$43$ \( T + 13\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T + 19\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( T + 59\!\cdots\!14 \) Copy content Toggle raw display
$59$ \( T + 29\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T - 79\!\cdots\!22 \) Copy content Toggle raw display
$67$ \( T - 48\!\cdots\!52 \) Copy content Toggle raw display
$71$ \( T - 88\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T - 36\!\cdots\!66 \) Copy content Toggle raw display
$79$ \( T - 33\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T - 20\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T + 41\!\cdots\!10 \) Copy content Toggle raw display
$97$ \( T + 72\!\cdots\!98 \) Copy content Toggle raw display
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