Properties

Label 2.82.a.b
Level $2$
Weight $82$
Character orbit 2.a
Self dual yes
Analytic conductor $83.100$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2 x^{3} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{45}\cdot 3^{15}\cdot 5^{5}\cdot 7^{2}\cdot 11 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 1099511627776 q^{2} + ( - \beta_1 - 17\!\cdots\!76) q^{3}+ \cdots + ( - 16652 \beta_{3} + 12455607542 \beta_{2} + \cdots + 25\!\cdots\!73) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 1099511627776 q^{2} + ( - \beta_1 - 17\!\cdots\!76) q^{3}+ \cdots + ( - 25\!\cdots\!36 \beta_{3} + \cdots + 58\!\cdots\!56) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4398046511104 q^{2} - 71\!\cdots\!04 q^{3}+ \cdots + 10\!\cdots\!92 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4398046511104 q^{2} - 71\!\cdots\!04 q^{3}+ \cdots + 23\!\cdots\!24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2 x^{3} + \cdots + 15\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 1920\nu - 960 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1897922560 \nu^{3} + \cdots - 30\!\cdots\!40 ) / 35\!\cdots\!33 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 14\!\cdots\!60 \nu^{3} + \cdots - 82\!\cdots\!80 ) / 35\!\cdots\!33 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 960 ) / 1920 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 8326 \beta_{3} + 6227803771 \beta_{2} + \cdots + 34\!\cdots\!00 ) / 1843200 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 15\!\cdots\!89 \beta_{3} + \cdots + 26\!\cdots\!00 ) / 235929600 \) 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
2.03307e16
4.77338e15
−1.17482e16
−1.33560e16
1.09951e12 −4.08130e19 1.20893e24 2.59550e28 −4.48743e31 1.64669e34 1.32923e36 1.22227e39 2.85378e40
1.2 1.09951e12 −1.09428e19 1.20893e24 −1.35941e28 −1.20318e31 −2.01060e34 1.32923e36 −3.23681e38 −1.49469e40
1.3 1.09951e12 2.07785e19 1.20893e24 −2.06533e28 2.28462e31 2.86366e34 1.32923e36 −1.16794e37 −2.27085e40
1.4 1.09951e12 2.38655e19 1.20893e24 3.32056e28 2.62404e31 −6.60461e33 1.32923e36 1.26136e38 3.65099e40
\(n\): 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.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.82.a.b 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + \cdots + 22\!\cdots\!76 \) acting on \(S_{82}^{\mathrm{new}}(\Gamma_0(2))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1099511627776)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + \cdots + 22\!\cdots\!76 \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 62\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots - 92\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 25\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots - 31\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots - 66\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 60\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots - 96\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots - 23\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots - 22\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 32\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots - 16\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots - 50\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 27\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 96\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 28\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 87\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 40\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 41\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 16\!\cdots\!84 \) Copy content Toggle raw display
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