Properties

Label 2.62.a.a
Level $2$
Weight $62$
Character orbit 2.a
Self dual yes
Analytic conductor $47.131$
Analytic rank $1$
Dimension $3$
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] = [2,62,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, 62, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2.1");
 
S:= CuspForms(chi, 62);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 62 \)
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: \(47.1312366529\)
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: \( x^{3} - x^{2} - 194555203040264843222x + 827041374696033876051589154700 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{8}\cdot 5^{2} \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 1073741824 q^{2} + ( - \beta_1 + 25004357142804) q^{3} + 11\!\cdots\!76 q^{4}+ \cdots + (171307710 \beta_{2} - 270377102688588 \beta_1 + 28\!\cdots\!13) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 1073741824 q^{2} + ( - \beta_1 + 25004357142804) q^{3} + 11\!\cdots\!76 q^{4}+ \cdots + ( - 11\!\cdots\!20 \beta_{2} + \cdots - 21\!\cdots\!44) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3221225472 q^{2} + 75013071428412 q^{3} + 34\!\cdots\!28 q^{4}+ \cdots + 85\!\cdots\!39 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3221225472 q^{2} + 75013071428412 q^{3} + 34\!\cdots\!28 q^{4}+ \cdots - 64\!\cdots\!32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 194555203040264843222x + 827041374696033876051589154700 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 34560\nu - 11520 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 163840\nu^{2} + 1044709396764160\nu - 21250616311092897741304320 ) / 23499 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 11520 ) / 34560 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 634473\beta_{2} - 816179216222\beta _1 + 573766640390105854444339200 ) / 4423680 \) 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
1.08912e10
4.83015e9
−1.57214e10
−1.07374e9 −3.51396e14 1.15292e18 −2.66736e21 3.77309e23 −4.09515e24 −1.23794e27 −3.69417e27 2.86405e30
1.2 −1.07374e9 −1.41926e14 1.15292e18 3.81119e21 1.52391e23 −3.88187e24 −1.23794e27 −1.07031e29 −4.09224e30
1.3 −1.07374e9 5.68335e14 1.15292e18 −9.14860e20 −6.10245e23 4.48127e25 −1.23794e27 1.95831e29 9.82324e29
\(n\): 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 2.62.a.a 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.62.a.a 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 75013071428412 T_{3}^{2} + \cdots - 28\!\cdots\!64 \) acting on \(S_{62}^{\mathrm{new}}(\Gamma_0(2))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1073741824)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 75013071428412 T^{2} + \cdots - 28\!\cdots\!64 \) Copy content Toggle raw display
$5$ \( T^{3} + \cdots - 93\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{3} + \cdots - 71\!\cdots\!12 \) Copy content Toggle raw display
$11$ \( T^{3} + \cdots - 27\!\cdots\!28 \) Copy content Toggle raw display
$13$ \( T^{3} + \cdots - 72\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{3} + \cdots + 10\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( T^{3} + \cdots - 66\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots + 37\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots - 64\!\cdots\!68 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots - 73\!\cdots\!72 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots - 26\!\cdots\!88 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots + 69\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots - 82\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots - 14\!\cdots\!28 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots + 80\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 80\!\cdots\!52 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots - 12\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 12\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 45\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 62\!\cdots\!08 \) Copy content Toggle raw display
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