Properties

Label 1.86.a.a
Level $1$
Weight $86$
Character orbit 1.a
Self dual yes
Analytic conductor $45.755$
Analytic rank $1$
Dimension $6$
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] = [1,86,Mod(1,1)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1, base_ring=CyclotomicField(1))
 
chi = DirichletCharacter(H, H._module([]))
 
N = Newforms(chi, 86, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 86);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 86 \)
Character orbit: \([\chi]\) \(=\) 1.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: \(45.7549576907\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3 x^{5} + \cdots - 17\!\cdots\!50 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{65}\cdot 3^{23}\cdot 5^{6}\cdot 7^{3}\cdot 11\cdot 17^{2} \)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 599485114800) q^{2} + (\beta_{2} - 6087800 \beta_1 - 26\!\cdots\!00) q^{3}+ \cdots + ( - 10659168 \beta_{5} - 36787298418 \beta_{4} + \cdots + 95\!\cdots\!73) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 599485114800) q^{2} + (\beta_{2} - 6087800 \beta_1 - 26\!\cdots\!00) q^{3}+ \cdots + ( - 13\!\cdots\!72 \beta_{5} + \cdots + 24\!\cdots\!96) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3596910688800 q^{2} - 15\!\cdots\!00 q^{3}+ \cdots + 57\!\cdots\!38 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3596910688800 q^{2} - 15\!\cdots\!00 q^{3}+ \cdots + 14\!\cdots\!76 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3 x^{5} + \cdots - 17\!\cdots\!50 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 96\nu - 48 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 22\!\cdots\!91 \nu^{5} + \cdots - 11\!\cdots\!86 ) / 15\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 95\!\cdots\!35 \nu^{5} + \cdots - 36\!\cdots\!66 ) / 51\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\!\cdots\!35 \nu^{5} + \cdots - 24\!\cdots\!98 ) / 15\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 23\!\cdots\!25 \nu^{5} + \cdots - 33\!\cdots\!98 ) / 12\!\cdots\!32 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 48 ) / 96 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 125655\beta_{2} + 2250185547597\beta _1 + 61824811714782289642833408 ) / 9216 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 11439 \beta_{5} - 1321486 \beta_{4} + 183077915148 \beta_{3} + \cdots + 86\!\cdots\!48 ) / 55296 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 630076817944247 \beta_{5} + \cdots + 41\!\cdots\!48 ) / 55296 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 18\!\cdots\!52 \beta_{5} + \cdots + 16\!\cdots\!04 ) / 4608 \) 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
−1.03394e11
−7.73032e10
−3.27167e10
6.50618e9
8.03621e10
1.26545e11
−1.05253e13 −2.03014e20 7.20957e25 −7.53919e29 2.13678e33 −3.48209e35 −3.51650e38 5.29713e39 7.93520e42
1.2 −8.02060e12 1.02906e20 2.56443e25 8.73356e29 −8.25371e32 1.26096e35 1.04599e38 −2.53278e40 −7.00484e42
1.3 −3.74029e12 2.81970e20 −2.46958e25 −8.27734e29 −1.05465e33 1.89091e35 2.37065e38 4.35893e40 3.09596e42
1.4 2.51077e10 −1.85192e20 −3.86850e25 5.79783e28 −4.64974e30 −1.94193e31 −1.94260e36 −1.62151e39 1.45570e39
1.5 7.11528e12 1.40385e20 1.19416e25 1.39389e29 9.98881e32 −1.69824e35 −1.90292e38 −1.62095e40 9.91791e41
1.6 1.15489e13 −2.95630e20 9.46906e25 −4.26442e29 −3.41419e33 5.79630e35 6.46794e38 5.14795e40 −4.92492e42
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

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 1.86.a.a 6
3.b odd 2 1 9.86.a.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.86.a.a 6 1.a even 1 1 trivial
9.86.a.a 6 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{86}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 3596910688800 T^{5} + \cdots - 65\!\cdots\!64 \) Copy content Toggle raw display
$3$ \( T^{6} + \cdots - 45\!\cdots\!96 \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots - 15\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 40\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 70\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 45\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots - 55\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 88\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots - 22\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots - 17\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 56\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 22\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 35\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 23\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 10\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 28\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 45\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots - 21\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 34\!\cdots\!16 \) Copy content Toggle raw display
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