Properties

Label 2.82.a.a
Level $2$
Weight $82$
Character orbit 2.a
Self dual yes
Analytic conductor $83.100$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2,82,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, 82, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2.1");
 
S:= CuspForms(chi, 82);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 82 \)
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: \(83.1002571076\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 215736607049989005852988854472x + 10253113298782277175624314846636353390456960 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{23}\cdot 3^{10}\cdot 5^{3}\cdot 7 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 1099511627776 q^{2} + (\beta_1 + 42\!\cdots\!96) q^{3}+ \cdots + (9842345463582 \beta_{2} + \cdots - 39\!\cdots\!87) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 1099511627776 q^{2} + (\beta_1 + 42\!\cdots\!96) q^{3}+ \cdots + (92\!\cdots\!96 \beta_{2} + \cdots - 18\!\cdots\!44) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3298534883328 q^{2} + 12\!\cdots\!88 q^{3}+ \cdots - 11\!\cdots\!61 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3298534883328 q^{2} + 12\!\cdots\!88 q^{3}+ \cdots - 54\!\cdots\!32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 215736607049989005852988854472x + 10253113298782277175624314846636353390456960 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 51840\nu - 17280 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 204800\nu^{2} + 14599998377556263040\nu - 29455238082558503799127537449399680 ) / 750064431 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 17280 ) / 51840 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 20251739637\beta_{2} - 7604165821643887\beta _1 + 795291428229079471176458113127424000 ) / 5529600 \) 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
−4.86628e14
4.80400e13
4.38588e14
−1.09951e12 −2.10253e19 1.20893e24 −2.29740e27 2.31176e31 2.01848e34 −1.32923e36 −1.36253e36 2.52602e39
1.2 −1.09951e12 6.69185e18 1.20893e24 −3.52449e28 −7.35776e30 −2.28912e34 −1.32923e36 −3.98646e38 3.87522e40
1.3 −1.09951e12 2.69378e19 1.20893e24 1.65593e28 −2.96185e31 2.15367e33 −1.32923e36 2.82220e38 −1.82071e40
\(n\): 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 2.82.a.a 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.82.a.a 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} + \cdots + 37\!\cdots\!64 \) acting on \(S_{82}^{\mathrm{new}}(\Gamma_0(2))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1099511627776)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + \cdots + 37\!\cdots\!64 \) Copy content Toggle raw display
$5$ \( T^{3} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{3} + \cdots + 99\!\cdots\!12 \) Copy content Toggle raw display
$11$ \( T^{3} + \cdots - 69\!\cdots\!28 \) Copy content Toggle raw display
$13$ \( T^{3} + \cdots - 15\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{3} + \cdots - 22\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( T^{3} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots + 23\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots - 64\!\cdots\!68 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots + 86\!\cdots\!72 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots + 68\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots - 16\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots - 10\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots - 34\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots - 96\!\cdots\!28 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots - 40\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 84\!\cdots\!52 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 97\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 72\!\cdots\!92 \) Copy content Toggle raw display
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