Properties

Label 2.32.a.a
Level $2$
Weight $32$
Character orbit 2.a
Self dual yes
Analytic conductor $12.175$
Analytic rank $1$
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] = [2,32,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, 32, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2.1");
 
S:= CuspForms(chi, 32);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 32 \)
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: \(12.1754265638\)
Analytic rank: \(1\)
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)
 
\(f(q)\) \(=\) \( q + 32768 q^{2} - 19984212 q^{3} + 1073741824 q^{4} + 42951708750 q^{5} - 654842658816 q^{6} - 16835358997576 q^{7} + 35184372088832 q^{8} - 218304667023003 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 32768 q^{2} - 19984212 q^{3} + 1073741824 q^{4} + 42951708750 q^{5} - 654842658816 q^{6} - 16835358997576 q^{7} + 35184372088832 q^{8} - 218304667023003 q^{9} + 14\!\cdots\!00 q^{10}+ \cdots + 15\!\cdots\!44 q^{99}+O(q^{100}) \) 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
32768.0 −1.99842e7 1.07374e9 4.29517e10 −6.54843e11 −1.68354e13 3.51844e13 −2.18305e14 1.40744e15
\(n\): 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 2.32.a.a 1
3.b odd 2 1 18.32.a.a 1
4.b odd 2 1 16.32.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.32.a.a 1 1.a even 1 1 trivial
16.32.a.a 1 4.b odd 2 1
18.32.a.a 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 19984212 \) acting on \(S_{32}^{\mathrm{new}}(\Gamma_0(2))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 32768 \) Copy content Toggle raw display
$3$ \( T + 19984212 \) Copy content Toggle raw display
$5$ \( T - 42951708750 \) Copy content Toggle raw display
$7$ \( T + 16835358997576 \) Copy content Toggle raw display
$11$ \( T + 7207832704992348 \) Copy content Toggle raw display
$13$ \( T + 27\!\cdots\!82 \) Copy content Toggle raw display
$17$ \( T + 16\!\cdots\!06 \) Copy content Toggle raw display
$19$ \( T - 10\!\cdots\!40 \) Copy content Toggle raw display
$23$ \( T + 30\!\cdots\!72 \) Copy content Toggle raw display
$29$ \( T - 34\!\cdots\!90 \) Copy content Toggle raw display
$31$ \( T - 16\!\cdots\!12 \) Copy content Toggle raw display
$37$ \( T + 25\!\cdots\!46 \) Copy content Toggle raw display
$41$ \( T + 92\!\cdots\!38 \) Copy content Toggle raw display
$43$ \( T - 58\!\cdots\!88 \) Copy content Toggle raw display
$47$ \( T + 91\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T - 58\!\cdots\!98 \) Copy content Toggle raw display
$59$ \( T + 32\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T - 40\!\cdots\!22 \) Copy content Toggle raw display
$67$ \( T - 16\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T + 68\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( T + 23\!\cdots\!62 \) Copy content Toggle raw display
$79$ \( T + 63\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T + 28\!\cdots\!12 \) Copy content Toggle raw display
$89$ \( T - 19\!\cdots\!10 \) Copy content Toggle raw display
$97$ \( T + 77\!\cdots\!86 \) Copy content Toggle raw display
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