Properties

Label 6.26.a.b
Level $6$
Weight $26$
Character orbit 6.a
Self dual yes
Analytic conductor $23.760$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6,26,Mod(1,6)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 26, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6.1");
 
S:= CuspForms(chi, 26);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 6.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: \(23.7598067971\)
Analytic rank: \(0\)
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)
 
\(f(q)\) \(=\) \( q - 4096 q^{2} + 531441 q^{3} + 16777216 q^{4} + 590425734 q^{5} - 2176782336 q^{6} + 57857417576 q^{7} - 68719476736 q^{8} + 282429536481 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4096 q^{2} + 531441 q^{3} + 16777216 q^{4} + 590425734 q^{5} - 2176782336 q^{6} + 57857417576 q^{7} - 68719476736 q^{8} + 282429536481 q^{9} - 2418383806464 q^{10} + 9494266240140 q^{11} + 8916100448256 q^{12} - 134968021061458 q^{13} - 236983982391296 q^{14} + 313776442502694 q^{15} + 281474976710656 q^{16} - 25\!\cdots\!14 q^{17}+ \cdots + 26\!\cdots\!40 q^{99}+O(q^{100}) \) 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
−4096.00 531441. 1.67772e7 5.90426e8 −2.17678e9 5.78574e10 −6.87195e10 2.82430e11 −2.41838e12
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)

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 6.26.a.b 1
3.b odd 2 1 18.26.a.b 1
4.b odd 2 1 48.26.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.26.a.b 1 1.a even 1 1 trivial
18.26.a.b 1 3.b odd 2 1
48.26.a.a 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 590425734 \) acting on \(S_{26}^{\mathrm{new}}(\Gamma_0(6))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 4096 \) Copy content Toggle raw display
$3$ \( T - 531441 \) Copy content Toggle raw display
$5$ \( T - 590425734 \) Copy content Toggle raw display
$7$ \( T - 57857417576 \) Copy content Toggle raw display
$11$ \( T - 9494266240140 \) Copy content Toggle raw display
$13$ \( T + 134968021061458 \) Copy content Toggle raw display
$17$ \( T + 2526114016804014 \) Copy content Toggle raw display
$19$ \( T - 11\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T - 11\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T - 10\!\cdots\!74 \) Copy content Toggle raw display
$31$ \( T - 46\!\cdots\!08 \) Copy content Toggle raw display
$37$ \( T + 46\!\cdots\!58 \) Copy content Toggle raw display
$41$ \( T - 51\!\cdots\!94 \) Copy content Toggle raw display
$43$ \( T + 36\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T + 49\!\cdots\!80 \) Copy content Toggle raw display
$53$ \( T - 44\!\cdots\!34 \) Copy content Toggle raw display
$59$ \( T - 22\!\cdots\!40 \) Copy content Toggle raw display
$61$ \( T + 12\!\cdots\!90 \) Copy content Toggle raw display
$67$ \( T - 63\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T - 60\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( T + 26\!\cdots\!58 \) Copy content Toggle raw display
$79$ \( T + 26\!\cdots\!92 \) Copy content Toggle raw display
$83$ \( T - 68\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T + 25\!\cdots\!42 \) Copy content Toggle raw display
$97$ \( T - 25\!\cdots\!82 \) Copy content Toggle raw display
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