Properties

Label 9.82.a.b
Level $9$
Weight $82$
Character orbit 9.a
Self dual yes
Analytic conductor $373.951$
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] = [9,82,Mod(1,9)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 82, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.1");
 
S:= CuspForms(chi, 82);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 82 \)
Character orbit: \([\chi]\) \(=\) 9.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: \(373.951156984\)
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: \( x^{6} - 3 x^{5} + \cdots - 27\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: multiple of \( 2^{58}\cdot 3^{34}\cdot 5^{7}\cdot 7^{3}\cdot 13 \)
Twist minimal: no (minimal twist has level 1)
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.

\(f(q)\) \(=\) \( q + (\beta_1 + 76812004440) q^{2} + (\beta_{3} - 2 \beta_{2} + 330453873082 \beta_1 + 74\!\cdots\!92) q^{4}+ \cdots + (29088 \beta_{5} + 73884448 \beta_{4} + \cdots + 91\!\cdots\!20) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 76812004440) q^{2} + (\beta_{3} - 2 \beta_{2} + 330453873082 \beta_1 + 74\!\cdots\!92) q^{4}+ \cdots + (54\!\cdots\!24 \beta_{5} + \cdots + 14\!\cdots\!80) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 460872026640 q^{2} + 44\!\cdots\!52 q^{4}+ \cdots + 54\!\cdots\!20 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 460872026640 q^{2} + 44\!\cdots\!52 q^{4}+ \cdots + 85\!\cdots\!80 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3 x^{5} + \cdots - 27\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 144\nu - 72 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 17\!\cdots\!09 \nu^{5} + \cdots - 37\!\cdots\!80 ) / 69\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 17\!\cdots\!09 \nu^{5} + \cdots - 11\!\cdots\!84 ) / 34\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 30\!\cdots\!91 \nu^{5} + \cdots + 34\!\cdots\!12 ) / 99\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\!\cdots\!35 \nu^{5} + \cdots + 33\!\cdots\!72 ) / 15\!\cdots\!24 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 72 ) / 144 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 2\beta_{2} + 176829864346\beta _1 + 3160943384653180063456128 ) / 20736 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 909 \beta_{5} + 2308889 \beta_{4} + 7915256204 \beta_{3} + 350150549525 \beta_{2} + \cdots + 17\!\cdots\!36 ) / 93312 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 52520409734386 \beta_{5} + \cdots + 67\!\cdots\!28 ) / 1679616 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 16\!\cdots\!47 \beta_{5} + \cdots + 40\!\cdots\!68 ) / 3779136 \) 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
−1.73117e10
−1.13941e10
−3.17952e9
2.10202e9
1.12136e10
1.85696e10
−2.41607e12 0 3.41956e24 1.95525e27 0 1.09563e34 −2.42020e36 0 −4.72402e39
1.2 −1.56393e12 0 2.80371e22 1.92800e28 0 −2.38829e34 3.73751e36 0 −3.01526e40
1.3 −3.81039e11 0 −2.27266e24 −2.26484e28 0 −1.01850e33 1.78727e36 0 8.62992e39
1.4 3.79502e11 0 −2.27383e24 3.57322e28 0 −3.75122e33 −1.78050e36 0 1.35605e40
1.5 1.69157e12 0 4.43572e23 9.34342e27 0 1.25454e34 −3.33964e36 0 1.58051e40
1.6 2.75084e12 0 5.14927e24 −2.52982e28 0 −2.62798e34 7.51371e36 0 −6.95912e40
\(n\): 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 9.82.a.b 6
3.b odd 2 1 1.82.a.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.82.a.a 6 3.b odd 2 1
9.82.a.b 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 460872026640 T_{2}^{5} + \cdots - 25\!\cdots\!84 \) acting on \(S_{82}^{\mathrm{new}}(\Gamma_0(9))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 460872026640 T^{5} + \cdots - 25\!\cdots\!84 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 32\!\cdots\!56 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots - 23\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots - 34\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots - 46\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots - 34\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 52\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots - 12\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots - 51\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots - 32\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 29\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 59\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots - 34\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots - 70\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots - 47\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 27\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots - 25\!\cdots\!24 \) Copy content Toggle raw display
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