Properties

Label 3.66.a.b
Level $3$
Weight $66$
Character orbit 3.a
Self dual yes
Analytic conductor $80.272$
Analytic rank $0$
Dimension $6$
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] = [3,66,Mod(1,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 66, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.1");
 
S:= CuspForms(chi, 66);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 66 \)
Character orbit: \([\chi]\) \(=\) 3.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: \(80.2717069417\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3 x^{5} + \cdots - 27\!\cdots\!48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: multiple of \( 2^{43}\cdot 3^{29}\cdot 5^{6}\cdot 7^{2}\cdot 11\cdot 13 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1035163827) q^{2} + 18\!\cdots\!41 q^{3}+ \cdots + 34\!\cdots\!81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1035163827) q^{2} + 18\!\cdots\!41 q^{3}+ \cdots + (10\!\cdots\!84 \beta_{5} + \cdots - 13\!\cdots\!76) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6210982962 q^{2} + 11\!\cdots\!46 q^{3}+ \cdots + 20\!\cdots\!86 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6210982962 q^{2} + 11\!\cdots\!46 q^{3}+ \cdots - 79\!\cdots\!56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3 x^{5} + \cdots - 27\!\cdots\!48 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 6\nu - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 36\nu^{2} - 4176733326\nu - 51045408173315560047 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 34220984678163 \nu^{5} + \cdots + 39\!\cdots\!16 ) / 32\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 85\!\cdots\!07 \nu^{5} + \cdots - 19\!\cdots\!84 ) / 22\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 43\!\cdots\!99 \nu^{5} + \cdots + 47\!\cdots\!12 ) / 30\!\cdots\!60 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 696122221\beta _1 + 51045408175403926710 ) / 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 902 \beta_{5} + 957 \beta_{4} + 1535423 \beta_{3} + 2102900233 \beta_{2} + \cdots + 35\!\cdots\!36 ) / 216 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2253844874710 \beta_{5} - 3763900160107 \beta_{4} + \cdots + 22\!\cdots\!68 ) / 648 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 30\!\cdots\!62 \beta_{5} + \cdots + 22\!\cdots\!32 ) / 1944 \) 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
−1.64208e9
−1.16886e9
−2.78112e8
2.86568e8
9.77853e8
1.82463e9
−8.81729e9 1.85302e15 4.08511e19 −7.15176e22 −1.63386e25 3.06094e27 −3.48954e28 3.43368e30 6.30591e32
1.2 −5.97799e9 1.85302e15 −1.15708e18 3.17694e22 −1.10773e25 6.82141e26 2.27466e29 3.43368e30 −1.89917e32
1.3 −6.33506e8 1.85302e15 −3.64922e19 −5.06462e22 −1.17390e24 −3.41514e27 4.64902e28 3.43368e30 3.20846e31
1.4 2.75457e9 1.85302e15 −2.93058e19 9.23075e22 5.10428e24 −1.32074e27 −1.82351e29 3.43368e30 2.54268e32
1.5 6.90228e9 1.85302e15 1.07480e19 −3.03033e22 1.27901e25 3.59710e27 −1.80464e29 3.43368e30 −2.09162e32
1.6 1.19829e10 1.85302e15 1.06697e20 6.35463e22 2.22046e25 −1.28649e27 8.36449e29 3.43368e30 7.61471e32
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

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 3.66.a.b 6
3.b odd 2 1 9.66.a.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.66.a.b 6 1.a even 1 1 trivial
9.66.a.c 6 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 6210982962 T_{2}^{5} + \cdots - 76\!\cdots\!48 \) acting on \(S_{66}^{\mathrm{new}}(\Gamma_0(3))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 6210982962 T^{5} + \cdots - 76\!\cdots\!48 \) Copy content Toggle raw display
$3$ \( (T - 18\!\cdots\!41)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots - 20\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots - 43\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots - 84\!\cdots\!32 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots - 63\!\cdots\!08 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots - 67\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots - 20\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots - 39\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 91\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots - 56\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots - 64\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots - 27\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots - 79\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 47\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 85\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 15\!\cdots\!84 \) Copy content Toggle raw display
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