Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9,76,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, 76, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.1");
S:= CuspForms(chi, 76);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 76 \) |
| 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: | \(320.605553540\) |
| 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: |
\( x^{6} + \cdots - 27\!\cdots\!96 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | multiple of \( 2^{60}\cdot 3^{33}\cdot 5^{7}\cdot 7^{3}\cdot 11\cdot 19 \) |
| 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
\( x^{6} + \cdots - 27\!\cdots\!96 \)
:
| \(\beta_{1}\) | \(=\) |
\( 288\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 82944\nu^{2} - 10245541690080\nu - 53875980220374115605504 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 10\!\cdots\!01 \nu^{5} + \cdots - 26\!\cdots\!76 ) / 21\!\cdots\!80 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 64\!\cdots\!63 \nu^{5} + \cdots + 34\!\cdots\!88 ) / 24\!\cdots\!80 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 85\!\cdots\!13 \nu^{5} + \cdots + 62\!\cdots\!00 ) / 24\!\cdots\!68 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 288 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 35574797535\beta _1 + 53875980220374115605504 ) / 82944 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 781 \beta_{5} + 847 \beta_{4} - 5288979 \beta_{3} + 5533906039 \beta_{2} + \cdots + 47\!\cdots\!00 ) / 5971968 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 571314150045 \beta_{5} + 1429650255233 \beta_{4} + \cdots + 10\!\cdots\!68 ) / 13436928 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 11\!\cdots\!77 \beta_{5} + \cdots + 86\!\cdots\!00 ) / 483729408 \)
|
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 |
|
−3.25116e11 | 0 | 6.79214e22 | 8.28346e25 | 0 | −3.35443e31 | −9.79978e33 | 0 | −2.69308e37 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.80537e11 | 0 | −5.18547e21 | −2.29711e26 | 0 | 5.15128e31 | 7.75665e33 | 0 | 4.14713e37 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −3.14392e10 | 0 | −3.67905e22 | 2.86859e26 | 0 | 4.24926e31 | 2.34440e33 | 0 | −9.01862e36 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 9.98963e9 | 0 | −3.76791e22 | −1.02795e26 | 0 | −7.89296e31 | −7.53798e32 | 0 | −1.02689e36 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.14419e11 | 0 | 8.19676e21 | −3.21494e24 | 0 | 4.19507e31 | −6.34299e33 | 0 | −6.89345e35 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 3.72139e11 | 0 | 1.00708e23 | −1.92935e26 | 0 | −2.31137e31 | 2.34185e34 | 0 | −7.17986e37 | |||||||||||||||||||||||||||||||||||||
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 | 9.76.a.d | 6 | |
| 3.b | odd | 2 | 1 | 3.76.a.a | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.76.a.a | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 9.76.a.d | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 59456439936 T_{2}^{5} + \cdots - 14\!\cdots\!32 \)
acting on \(S_{76}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots - 14\!\cdots\!32 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 34\!\cdots\!00 \)
|
| $7$ |
\( T^{6} + \cdots - 56\!\cdots\!00 \)
|
| $11$ |
\( T^{6} + \cdots + 47\!\cdots\!08 \)
|
| $13$ |
\( T^{6} + \cdots - 16\!\cdots\!92 \)
|
| $17$ |
\( T^{6} + \cdots - 28\!\cdots\!84 \)
|
| $19$ |
\( T^{6} + \cdots + 36\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots + 15\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots - 81\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots - 11\!\cdots\!00 \)
|
| $37$ |
\( T^{6} + \cdots - 45\!\cdots\!00 \)
|
| $41$ |
\( T^{6} + \cdots + 11\!\cdots\!00 \)
|
| $43$ |
\( T^{6} + \cdots + 19\!\cdots\!76 \)
|
| $47$ |
\( T^{6} + \cdots - 14\!\cdots\!36 \)
|
| $53$ |
\( T^{6} + \cdots - 76\!\cdots\!00 \)
|
| $59$ |
\( T^{6} + \cdots - 29\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots - 27\!\cdots\!44 \)
|
| $67$ |
\( T^{6} + \cdots + 21\!\cdots\!76 \)
|
| $71$ |
\( T^{6} + \cdots - 19\!\cdots\!96 \)
|
| $73$ |
\( T^{6} + \cdots - 23\!\cdots\!00 \)
|
| $79$ |
\( T^{6} + \cdots + 25\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 12\!\cdots\!72 \)
|
| $89$ |
\( T^{6} + \cdots - 36\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots - 18\!\cdots\!64 \)
|
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