Properties

Label 10.12.a.a
Level $10$
Weight $12$
Character orbit 10.a
Self dual yes
Analytic conductor $7.683$
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] = [10,12,Mod(1,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 10.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: \(7.68343180560\)
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)
 
\(f(q)\) \(=\) \( q - 32 q^{2} - 12 q^{3} + 1024 q^{4} + 3125 q^{5} + 384 q^{6} - 14176 q^{7} - 32768 q^{8} - 177003 q^{9} - 100000 q^{10} - 756348 q^{11} - 12288 q^{12} - 905482 q^{13} + 453632 q^{14} - 37500 q^{15} + 1048576 q^{16}+ \cdots + 133875865044 q^{99}+O(q^{100}) \) 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
−32.0000 −12.0000 1024.00 3125.00 384.000 −14176.0 −32768.0 −177003. −100000.
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( -1 \)

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 10.12.a.a 1
3.b odd 2 1 90.12.a.g 1
4.b odd 2 1 80.12.a.d 1
5.b even 2 1 50.12.a.d 1
5.c odd 4 2 50.12.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.12.a.a 1 1.a even 1 1 trivial
50.12.a.d 1 5.b even 2 1
50.12.b.c 2 5.c odd 4 2
80.12.a.d 1 4.b odd 2 1
90.12.a.g 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 12 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(10))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 32 \) Copy content Toggle raw display
$3$ \( T + 12 \) Copy content Toggle raw display
$5$ \( T - 3125 \) Copy content Toggle raw display
$7$ \( T + 14176 \) Copy content Toggle raw display
$11$ \( T + 756348 \) Copy content Toggle raw display
$13$ \( T + 905482 \) Copy content Toggle raw display
$17$ \( T - 2803794 \) Copy content Toggle raw display
$19$ \( T + 5428660 \) Copy content Toggle raw display
$23$ \( T + 10236672 \) Copy content Toggle raw display
$29$ \( T + 197498010 \) Copy content Toggle raw display
$31$ \( T + 44362288 \) Copy content Toggle raw display
$37$ \( T - 576737054 \) Copy content Toggle raw display
$41$ \( T - 930058362 \) Copy content Toggle raw display
$43$ \( T - 1605598988 \) Copy content Toggle raw display
$47$ \( T + 1803684456 \) Copy content Toggle raw display
$53$ \( T - 1558674798 \) Copy content Toggle raw display
$59$ \( T + 9501997020 \) Copy content Toggle raw display
$61$ \( T - 6736320422 \) Copy content Toggle raw display
$67$ \( T - 8402906564 \) Copy content Toggle raw display
$71$ \( T + 4806306168 \) Copy content Toggle raw display
$73$ \( T - 7462713338 \) Copy content Toggle raw display
$79$ \( T + 20644540720 \) Copy content Toggle raw display
$83$ \( T + 68013349212 \) Copy content Toggle raw display
$89$ \( T - 69871323210 \) Copy content Toggle raw display
$97$ \( T - 39960952514 \) Copy content Toggle raw display
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