Properties

Label 2.66.a.a
Level $2$
Weight $66$
Character orbit 2.a
Self dual yes
Analytic conductor $53.514$
Analytic rank $1$
Dimension $2$
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] = [2,66,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, 66, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2.1");
 
S:= CuspForms(chi, 66);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 66 \)
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: \(53.5144712945\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1961256803955162 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{4}\cdot 5\cdot 11 \)
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 \(\beta = 285120\sqrt{7845027215820649}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4294967296 q^{2} + ( - 111 \beta + 574529980102596) q^{3} + 18\!\cdots\!16 q^{4}+ \cdots + ( - 12\!\cdots\!12 \beta - 21\!\cdots\!27) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4294967296 q^{2} + ( - 111 \beta + 574529980102596) q^{3} + 18\!\cdots\!16 q^{4}+ \cdots + ( - 74\!\cdots\!65 \beta + 10\!\cdots\!56) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8589934592 q^{2} + 11\!\cdots\!92 q^{3}+ \cdots - 42\!\cdots\!54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8589934592 q^{2} + 11\!\cdots\!92 q^{3}+ \cdots + 21\!\cdots\!12 q^{99}+O(q^{100}) \) 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
4.42861e7
−4.42861e7
−4.29497e9 −2.22863e15 1.84467e19 2.44878e22 9.57189e24 8.86741e26 −7.92282e28 −5.33426e30 −1.05174e32
1.2 −4.29497e9 3.37769e15 1.84467e19 −9.92503e22 −1.45071e25 1.82190e27 −7.92282e28 1.10774e30 4.26277e32
\(n\): 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 2.66.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.66.a.a 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 1149059960205192T_{3} - 7527621327730776295070757798384 \) acting on \(S_{66}^{\mathrm{new}}(\Gamma_0(2))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 4294967296)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + \cdots - 75\!\cdots\!84 \) Copy content Toggle raw display
$5$ \( T^{2} + \cdots - 24\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots + 16\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots - 19\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 21\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots - 38\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 30\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 64\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 70\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 58\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 54\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 22\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 61\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 21\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 34\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 26\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 57\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 23\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 40\!\cdots\!76 \) Copy content Toggle raw display
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