Properties

Label 2.88.a.b
Level $2$
Weight $88$
Character orbit 2.a
Self dual yes
Analytic conductor $95.867$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2,88,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, 88, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2.1");
 
S:= CuspForms(chi, 88);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 88 \)
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: \(95.8667262922\)
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: \( x^{4} - 2 x^{3} + \cdots + 13\!\cdots\!68 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{45}\cdot 3^{15}\cdot 5^{4}\cdot 7^{2}\cdot 11\cdot 17\cdot 29 \)
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.

\(f(q)\) \(=\) \( q - 8796093022208 q^{2} + ( - \beta_1 + 10\!\cdots\!32) q^{3}+ \cdots + (197927972 \beta_{3} - 19291700786 \beta_{2} + \cdots + 20\!\cdots\!37) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 8796093022208 q^{2} + ( - \beta_1 + 10\!\cdots\!32) q^{3}+ \cdots + (28\!\cdots\!76 \beta_{3} + \cdots + 55\!\cdots\!44) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 35184372088832 q^{2} + 40\!\cdots\!28 q^{3}+ \cdots + 82\!\cdots\!48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 35184372088832 q^{2} + 40\!\cdots\!28 q^{3}+ \cdots + 22\!\cdots\!76 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2 x^{3} + \cdots + 13\!\cdots\!68 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 1920\nu - 960 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 9433750962176 \nu^{3} + \cdots - 15\!\cdots\!88 ) / 52\!\cdots\!57 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10\!\cdots\!68 \nu^{3} + \cdots - 16\!\cdots\!24 ) / 57\!\cdots\!27 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 960 ) / 1920 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 98963986 \beta_{3} - 9645850393 \beta_{2} + \cdots + 25\!\cdots\!00 ) / 1843200 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 23\!\cdots\!49 \beta_{3} + \cdots + 22\!\cdots\!00 ) / 235929600 \) 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
5.03541e17
2.13103e17
−3.01536e17
−4.15109e17
−8.79609e12 −8.65990e20 7.73713e25 −4.55634e30 7.61733e33 8.09122e36 −6.80565e38 4.26681e41 4.00780e43
1.2 −8.79609e12 −3.08348e20 7.73713e25 1.64822e30 2.71226e33 −1.87890e36 −6.80565e38 −2.28179e41 −1.44979e43
1.3 −8.79609e12 6.79758e20 7.73713e25 −4.34651e30 −5.97921e33 −6.14091e36 −6.80565e38 1.38813e41 3.82323e43
1.4 −8.79609e12 8.97819e20 7.73713e25 3.21672e30 −7.89730e33 8.51076e36 −6.80565e38 4.82821e41 −2.82946e43
\(n\): 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 2.88.a.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.88.a.b 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + \cdots + 16\!\cdots\!76 \) acting on \(S_{88}^{\mathrm{new}}(\Gamma_0(2))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 8796093022208)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 79\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 87\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 40\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 29\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots - 83\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots - 29\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots - 73\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots - 82\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 28\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots - 11\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots - 53\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots - 71\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots - 42\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots - 22\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots - 71\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots - 60\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 39\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 87\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 32\!\cdots\!44 \) Copy content Toggle raw display
show more
show less