↑

Properties

Label 64.18.a.d
Level $64$
Weight $18$
Character orbit 64.a
Self dual yes
Analytic conductor $117.262$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [64,18,Mod(1,64)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(64, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 18, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("64.1"); S:= CuspForms(chi, 18); N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 64.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,4284,0,1025850] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(117.262135901\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 4284 q^{3} + 1025850 q^{5} + 3225992 q^{7} - 110787507 q^{9} + 753618228 q^{11} - 2541064526 q^{13} + 4394741400 q^{15} - 5429742318 q^{17} - 1487499860 q^{19} + 13820149728 q^{21} - 317091823464 q^{23}+ \cdots - 83\!\cdots\!96 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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 4284.00 0 1.02585e6 0 3.22599e6 0 −1.10788e8 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 64.18.a.d 1
4.b odd 2 1 64.18.a.b 1
8.b even 2 1 1.18.a.a ✓ 1
8.d odd 2 1 16.18.a.b 1
24.h odd 2 1 9.18.a.b 1
40.f even 2 1 25.18.a.a 1
40.i odd 4 2 25.18.b.a 2
56.h odd 2 1 49.18.a.a 1
88.b odd 2 1 121.18.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.18.a.a ✓ 1 8.b even 2 1
9.18.a.b 1 24.h odd 2 1
16.18.a.b 1 8.d odd 2 1
25.18.a.a 1 40.f even 2 1
25.18.b.a 2 40.i odd 4 2
49.18.a.a 1 56.h odd 2 1
64.18.a.b 1 4.b odd 2 1
64.18.a.d 1 1.a even 1 1 trivial
121.18.a.b 1 88.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 4284 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(64))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 4284 \) Copy content Toggle raw display
$5$ \( T - 1025850 \) Copy content Toggle raw display
$7$ \( T - 3225992 \) Copy content Toggle raw display
$11$ \( T - 753618228 \) Copy content Toggle raw display
$13$ \( T + 2541064526 \) Copy content Toggle raw display
$17$ \( T + 5429742318 \) Copy content Toggle raw display
$19$ \( T + 1487499860 \) Copy content Toggle raw display
$23$ \( T + 317091823464 \) Copy content Toggle raw display
$29$ \( T + 2433410602590 \) Copy content Toggle raw display
$31$ \( T + 8849722053088 \) Copy content Toggle raw display
$37$ \( T + 12691652946662 \) Copy content Toggle raw display
$41$ \( T - 48864151002282 \) Copy content Toggle raw display
$43$ \( T - 91019974317844 \) Copy content Toggle raw display
$47$ \( T + 49304994276048 \) Copy content Toggle raw display
$53$ \( T + 22940453195766 \) Copy content Toggle raw display
$59$ \( T + 32695090729980 \) Copy content Toggle raw display
$61$ \( T - 1308285854869378 \) Copy content Toggle raw display
$67$ \( T + 5196143861984132 \) Copy content Toggle raw display
$71$ \( T + 3709489877412408 \) Copy content Toggle raw display
$73$ \( T - 3402372968272586 \) Copy content Toggle raw display
$79$ \( T - 2366533941308240 \) Copy content Toggle raw display
$83$ \( T - 29\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T - 29\!\cdots\!70 \) Copy content Toggle raw display
$97$ \( T + 63\!\cdots\!98 \) Copy content Toggle raw display
show more
show less