Properties

Label 16.22.a.b
Level $16$
Weight $22$
Character orbit 16.a
Self dual yes
Analytic conductor $44.716$
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] = [16,22,Mod(1,16)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(16, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("16.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 16.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: \(44.7163750859\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2)
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)
 
\(f(q)\) \(=\) \( q - 59316 q^{3} + 4975350 q^{5} - 1427425832 q^{7} - 6941965347 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 59316 q^{3} + 4975350 q^{5} - 1427425832 q^{7} - 6941965347 q^{9} + 106767894948 q^{11} - 150150565474 q^{13} - 295117860600 q^{15} - 11203980739758 q^{17} - 11024055955460 q^{19} + 84669190650912 q^{21} - 129502845739896 q^{23} - 452083050580625 q^{25} + 10\!\cdots\!00 q^{27}+ \cdots - 74\!\cdots\!56 q^{99}+O(q^{100}) \) 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
0 −59316.0 0 4.97535e6 0 −1.42743e9 0 −6.94197e9 0
\(n\): 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 16.22.a.b 1
4.b odd 2 1 2.22.a.b 1
8.b even 2 1 64.22.a.e 1
8.d odd 2 1 64.22.a.c 1
12.b even 2 1 18.22.a.b 1
20.d odd 2 1 50.22.a.a 1
20.e even 4 2 50.22.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.22.a.b 1 4.b odd 2 1
16.22.a.b 1 1.a even 1 1 trivial
18.22.a.b 1 12.b even 2 1
50.22.a.a 1 20.d odd 2 1
50.22.b.c 2 20.e even 4 2
64.22.a.c 1 8.d odd 2 1
64.22.a.e 1 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 59316 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(16))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) 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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