Properties

Label 3.66.a.a
Level $3$
Weight $66$
Character orbit 3.a
Self dual yes
Analytic conductor $80.272$
Analytic rank $1$
Dimension $5$
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: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2 x^{4} + \cdots + 11\!\cdots\!50 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{37}\cdot 3^{20}\cdot 5^{2}\cdot 7^{2}\cdot 11^{2}\cdot 13^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 517306193) q^{2} - 18\!\cdots\!41 q^{3}+ \cdots + 34\!\cdots\!81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 517306193) q^{2} - 18\!\cdots\!41 q^{3}+ \cdots + ( - 17\!\cdots\!66 \beta_{4} + \cdots + 16\!\cdots\!88) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2586530964 q^{2} - 92\!\cdots\!05 q^{3}+ \cdots + 17\!\cdots\!05 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2586530964 q^{2} - 92\!\cdots\!05 q^{3}+ \cdots + 83\!\cdots\!12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2 x^{4} + \cdots + 11\!\cdots\!50 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 12\nu - 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 144\nu^{2} + 2797176876\nu - 54817211944833820613 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 8387469 \nu^{4} + \cdots - 92\!\cdots\!50 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 45198243 \nu^{4} + \cdots + 29\!\cdots\!50 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 5 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 233098073\beta _1 + 54817211943668330248 ) / 144 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 103549 \beta_{4} + 558003 \beta_{3} - 171728065 \beta_{2} + \cdots - 15\!\cdots\!22 ) / 216 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 808001857580522 \beta_{4} + 205412308022134 \beta_{3} + \cdots + 35\!\cdots\!12 ) / 1296 \) 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
8.82783e8
4.30385e8
−1.14024e8
−2.77962e8
−9.21181e8
−1.11107e10 −1.85302e15 8.65541e19 −1.70324e22 2.05883e25 4.28205e27 −5.51764e29 3.43368e30 1.89242e32
1.2 −5.68192e9 −1.85302e15 −4.60926e18 5.44876e22 1.05287e25 4.11535e26 2.35815e29 3.43368e30 −3.09594e32
1.3 8.50981e8 −1.85302e15 −3.61693e19 −8.94186e22 −1.57689e24 3.62069e27 −6.21751e28 3.43368e30 −7.60936e31
1.4 2.81824e9 −1.85302e15 −2.89510e19 1.31338e22 −5.22225e24 −3.12329e27 −1.85565e29 3.43368e30 3.70141e31
1.5 1.05369e10 −1.85302e15 7.41322e19 −4.74965e22 −1.95250e25 5.36551e26 3.92379e29 3.43368e30 −5.00464e32
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.a 5
3.b odd 2 1 9.66.a.a 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.66.a.a 5 1.a even 1 1 trivial
9.66.a.a 5 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} + 2586530964 T_{2}^{4} + \cdots - 15\!\cdots\!96 \) acting on \(S_{66}^{\mathrm{new}}(\Gamma_0(3))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + 2586530964 T^{4} + \cdots - 15\!\cdots\!96 \) Copy content Toggle raw display
$3$ \( (T + 18\!\cdots\!41)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{5} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{5} + \cdots + 21\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{5} + \cdots + 27\!\cdots\!48 \) Copy content Toggle raw display
$17$ \( T^{5} + \cdots - 21\!\cdots\!92 \) Copy content Toggle raw display
$19$ \( T^{5} + \cdots - 43\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{5} + \cdots - 37\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{5} + \cdots - 27\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{5} + \cdots - 22\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{5} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{5} + \cdots - 72\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{5} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{5} + \cdots - 67\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( T^{5} + \cdots - 38\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{5} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{5} + \cdots + 52\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( T^{5} + \cdots - 93\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{5} + \cdots + 29\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( T^{5} + \cdots - 72\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{5} + \cdots - 65\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{5} + \cdots - 18\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{5} + \cdots - 69\!\cdots\!12 \) Copy content Toggle raw display
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