Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9,72,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, 72, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.1");
S:= CuspForms(chi, 72);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 72 \) |
| 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: | \(287.321544505\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{6} - 3 x^{5} + \cdots - 27\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | multiple of \( 2^{49}\cdot 3^{29}\cdot 5^{7}\cdot 7^{2} \) |
| Twist minimal: | no (minimal twist has level 3) |
| 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.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{6} - 3 x^{5} + \cdots - 27\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 6\nu - 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( 36\nu^{2} - 40711045962\nu - 3932546999134591191657 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 2647338101397 \nu^{5} + \cdots + 63\!\cdots\!80 ) / 13\!\cdots\!40 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 10\!\cdots\!51 \nu^{5} + \cdots - 54\!\cdots\!20 ) / 62\!\cdots\!80 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 25\!\cdots\!31 \nu^{5} + \cdots + 18\!\cdots\!00 ) / 12\!\cdots\!60 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 3 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 6785174327\beta _1 + 3932546999154946714638 ) / 36 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 10241 \beta_{5} + 8666 \beta_{4} - 6340812 \beta_{3} + 17570586570 \beta_{2} + \cdots + 26\!\cdots\!88 ) / 216 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 184385957659079 \beta_{5} - 147886657045898 \beta_{4} + \cdots + 12\!\cdots\!44 ) / 648 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 13\!\cdots\!97 \beta_{5} + \cdots + 54\!\cdots\!86 ) / 972 \)
|
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 |
|
−8.09473e10 | 0 | 4.19128e21 | −7.59856e24 | 0 | 1.03665e30 | −1.48142e32 | 0 | 6.15083e35 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −4.39862e10 | 0 | −4.26395e20 | −4.61068e24 | 0 | −1.72926e30 | 1.22615e32 | 0 | 2.02806e35 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −7.97054e9 | 0 | −2.29765e21 | 4.57364e24 | 0 | 9.29524e29 | 3.71335e31 | 0 | −3.64544e34 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 3.61112e10 | 0 | −1.05717e21 | −4.57686e24 | 0 | −2.20036e29 | −1.23441e32 | 0 | −1.65276e35 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 7.41748e10 | 0 | 3.14071e21 | 1.28474e25 | 0 | −1.80308e28 | 5.78216e31 | 0 | 9.52956e35 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 9.55218e10 | 0 | 6.76322e21 | −1.07436e25 | 0 | 6.00629e29 | 4.20491e32 | 0 | −1.02625e36 | |||||||||||||||||||||||||||||||||||||
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 | 9.72.a.c | 6 | |
| 3.b | odd | 2 | 1 | 3.72.a.b | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.72.a.b | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 9.72.a.c | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 72903656826 T_{2}^{5} + \cdots - 72\!\cdots\!72 \)
acting on \(S_{72}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots - 72\!\cdots\!72 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $7$ |
\( T^{6} + \cdots - 39\!\cdots\!00 \)
|
| $11$ |
\( T^{6} + \cdots - 24\!\cdots\!92 \)
|
| $13$ |
\( T^{6} + \cdots + 19\!\cdots\!88 \)
|
| $17$ |
\( T^{6} + \cdots + 42\!\cdots\!16 \)
|
| $19$ |
\( T^{6} + \cdots - 21\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots - 36\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots + 25\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots + 21\!\cdots\!00 \)
|
| $37$ |
\( T^{6} + \cdots + 44\!\cdots\!00 \)
|
| $41$ |
\( T^{6} + \cdots - 10\!\cdots\!00 \)
|
| $43$ |
\( T^{6} + \cdots - 14\!\cdots\!04 \)
|
| $47$ |
\( T^{6} + \cdots + 86\!\cdots\!04 \)
|
| $53$ |
\( T^{6} + \cdots + 33\!\cdots\!00 \)
|
| $59$ |
\( T^{6} + \cdots + 26\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots - 48\!\cdots\!04 \)
|
| $67$ |
\( T^{6} + \cdots - 12\!\cdots\!64 \)
|
| $71$ |
\( T^{6} + \cdots - 20\!\cdots\!16 \)
|
| $73$ |
\( T^{6} + \cdots - 56\!\cdots\!00 \)
|
| $79$ |
\( T^{6} + \cdots - 24\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 25\!\cdots\!32 \)
|
| $89$ |
\( T^{6} + \cdots + 12\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots - 13\!\cdots\!84 \)
|
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