Properties

Label 16.22.a.a
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)
 
\(f(q)\) \(=\) \( q - 71604 q^{3} - 28693770 q^{5} + 853202392 q^{7} - 5333220387 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 71604 q^{3} - 28693770 q^{5} + 853202392 q^{7} - 5333220387 q^{9} - 86731179612 q^{11} - 895323442786 q^{13} + 2054588707080 q^{15} + 3257566804818 q^{17} - 23032467644420 q^{19} - 61092704076768 q^{21} - 146495714575224 q^{23} + 346495278609775 q^{25} + 11\!\cdots\!60 q^{27}+ \cdots + 46\!\cdots\!44 q^{99}+O(q^{100}) \) 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 −71604.0 0 −2.86938e7 0 8.53202e8 0 −5.33322e9 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.a 1
4.b odd 2 1 2.22.a.a 1
8.b even 2 1 64.22.a.f 1
8.d odd 2 1 64.22.a.b 1
12.b even 2 1 18.22.a.e 1
20.d odd 2 1 50.22.a.c 1
20.e even 4 2 50.22.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.22.a.a 1 4.b odd 2 1
16.22.a.a 1 1.a even 1 1 trivial
18.22.a.e 1 12.b even 2 1
50.22.a.c 1 20.d odd 2 1
50.22.b.a 2 20.e even 4 2
64.22.a.b 1 8.d odd 2 1
64.22.a.f 1 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 71604 \) 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
$3$ \( T + 71604 \) Copy content Toggle raw display
$5$ \( T + 28693770 \) Copy content Toggle raw display
$7$ \( T - 853202392 \) Copy content Toggle raw display
$11$ \( T + 86731179612 \) Copy content Toggle raw display
$13$ \( T + 895323442786 \) Copy content Toggle raw display
$17$ \( T - 3257566804818 \) Copy content Toggle raw display
$19$ \( T + 23032467644420 \) Copy content Toggle raw display
$23$ \( T + 146495714575224 \) Copy content Toggle raw display
$29$ \( T + 734051633521170 \) Copy content Toggle raw display
$31$ \( T - 3146664162057568 \) Copy content Toggle raw display
$37$ \( T + 12\!\cdots\!62 \) Copy content Toggle raw display
$41$ \( T - 45\!\cdots\!42 \) Copy content Toggle raw display
$43$ \( T - 24\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T - 44\!\cdots\!52 \) Copy content Toggle raw display
$53$ \( T - 20\!\cdots\!54 \) Copy content Toggle raw display
$59$ \( T - 37\!\cdots\!40 \) Copy content Toggle raw display
$61$ \( T + 76\!\cdots\!38 \) Copy content Toggle raw display
$67$ \( T - 18\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T - 45\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T + 25\!\cdots\!26 \) Copy content Toggle raw display
$79$ \( T + 99\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T + 29\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T - 11\!\cdots\!90 \) Copy content Toggle raw display
$97$ \( T + 56\!\cdots\!02 \) Copy content Toggle raw display
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