Properties

Label 2.66.a.b
Level $2$
Weight $66$
Character orbit 2.a
Self dual yes
Analytic conductor $53.514$
Analytic rank $0$
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] = [2,66,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: \(0\)
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: \( x^{3} - x^{2} - 4862367805520722608042x + 130125819203569060903952569933488 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{8}\cdot 5^{3}\cdot 7\cdot 11\cdot 13 \)
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.

\(f(q)\) \(=\) \( q + 4294967296 q^{2} + ( - \beta_1 + 994852866070404) q^{3} + 18\!\cdots\!16 q^{4}+ \cdots + (1845485370 \beta_{2} + \cdots + 28\!\cdots\!73) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4294967296 q^{2} + ( - \beta_1 + 994852866070404) q^{3} + 18\!\cdots\!16 q^{4}+ \cdots + (10\!\cdots\!60 \beta_{2} + \cdots - 38\!\cdots\!44) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 12884901888 q^{2} + 29\!\cdots\!12 q^{3}+ \cdots + 85\!\cdots\!19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 12884901888 q^{2} + 29\!\cdots\!12 q^{3}+ \cdots - 11\!\cdots\!32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 4862367805520722608042x + 130125819203569060903952569933488 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 8640\nu^{2} + 341742997270080\nu - 28007238559913276554748160 ) / 439529881 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 253376640\nu^{2} + 14192563459783927680\nu - 821340278009406951018491727360 ) / 62789983 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 205282\beta _1 + 22140518400 ) / 66421555200 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -5650512521\beta_{2} + 1642657807845362\beta _1 + 30758669675914051567045988352000 ) / 9488793600 \) 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
4.20131e10
−8.04922e10
3.84791e10
4.29497e9 −2.64736e15 1.84467e19 −6.66845e22 −1.13703e25 1.54537e27 7.92282e28 −3.29254e30 −2.86408e32
1.2 4.29497e9 −5.99699e13 1.84467e19 6.16250e22 −2.57569e23 −2.55894e27 7.92282e28 −1.02975e31 2.64677e32
1.3 4.29497e9 5.69189e15 1.84467e19 8.97760e21 2.44465e25 5.30637e27 7.92282e28 2.20965e31 3.85585e31
\(n\): 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.b 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.66.a.b 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} + \cdots - 90\!\cdots\!64 \) acting on \(S_{66}^{\mathrm{new}}(\Gamma_0(2))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 4294967296)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + \cdots - 90\!\cdots\!64 \) Copy content Toggle raw display
$5$ \( T^{3} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{3} + \cdots + 20\!\cdots\!28 \) Copy content Toggle raw display
$11$ \( T^{3} + \cdots - 11\!\cdots\!48 \) Copy content Toggle raw display
$13$ \( T^{3} + \cdots - 54\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{3} + \cdots + 36\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( T^{3} + \cdots - 59\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots - 15\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots + 60\!\cdots\!12 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots - 19\!\cdots\!92 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots + 27\!\cdots\!72 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots - 68\!\cdots\!92 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots - 18\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots + 12\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots + 15\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 75\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots - 13\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 90\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 13\!\cdots\!68 \) Copy content Toggle raw display
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