Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9,66,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, 66, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.1");
S:= CuspForms(chi, 66);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 66 \) |
| 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: | \(240.815120825\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{5} - 2 x^{4} + \cdots + 11\!\cdots\!50 \)
|
| 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: | 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2 x^{4} + \cdots + 11\!\cdots\!50 \)
:
| \(\beta_{1}\) | \(=\) |
\( 12\nu - 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( 144\nu^{2} + 2797176876\nu - 54817211944833820613 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 8387469 \nu^{4} + \cdots - 92\!\cdots\!50 ) / 26\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 45198243 \nu^{4} + \cdots + 29\!\cdots\!50 ) / 26\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 5 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} - 233098073\beta _1 + 54817211943668330248 ) / 144 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 103549 \beta_{4} + 558003 \beta_{3} - 171728065 \beta_{2} + \cdots - 15\!\cdots\!22 ) / 216 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 808001857580522 \beta_{4} + 205412308022134 \beta_{3} + \cdots + 35\!\cdots\!12 ) / 1296 \)
|
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 |
|
−1.05369e10 | 0 | 7.41322e19 | 4.74965e22 | 0 | 5.36551e26 | −3.92379e29 | 0 | −5.00464e32 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.81824e9 | 0 | −2.89510e19 | −1.31338e22 | 0 | −3.12329e27 | 1.85565e29 | 0 | 3.70141e31 | ||||||||||||||||||||||||||||||||||
| 1.3 | −8.50981e8 | 0 | −3.61693e19 | 8.94186e22 | 0 | 3.62069e27 | 6.21751e28 | 0 | −7.60936e31 | ||||||||||||||||||||||||||||||||||
| 1.4 | 5.68192e9 | 0 | −4.60926e18 | −5.44876e22 | 0 | 4.11535e26 | −2.35815e29 | 0 | −3.09594e32 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.11107e10 | 0 | 8.65541e19 | 1.70324e22 | 0 | 4.28205e27 | 5.51764e29 | 0 | 1.89242e32 | ||||||||||||||||||||||||||||||||||
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 | 9.66.a.a | 5 | |
| 3.b | odd | 2 | 1 | 3.66.a.a | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.66.a.a | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 9.66.a.a | 5 | 1.a | even | 1 | 1 | trivial | |
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 \(S_{66}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + \cdots + 15\!\cdots\!96 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} + \cdots - 51\!\cdots\!00 \)
|
| $7$ |
\( T^{5} + \cdots + 10\!\cdots\!00 \)
|
| $11$ |
\( T^{5} + \cdots - 21\!\cdots\!36 \)
|
| $13$ |
\( T^{5} + \cdots + 27\!\cdots\!48 \)
|
| $17$ |
\( T^{5} + \cdots + 21\!\cdots\!92 \)
|
| $19$ |
\( T^{5} + \cdots - 43\!\cdots\!00 \)
|
| $23$ |
\( T^{5} + \cdots + 37\!\cdots\!00 \)
|
| $29$ |
\( T^{5} + \cdots + 27\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots - 22\!\cdots\!00 \)
|
| $37$ |
\( T^{5} + \cdots - 16\!\cdots\!00 \)
|
| $41$ |
\( T^{5} + \cdots + 72\!\cdots\!00 \)
|
| $43$ |
\( T^{5} + \cdots + 11\!\cdots\!96 \)
|
| $47$ |
\( T^{5} + \cdots + 67\!\cdots\!48 \)
|
| $53$ |
\( T^{5} + \cdots + 38\!\cdots\!00 \)
|
| $59$ |
\( T^{5} + \cdots - 90\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots + 52\!\cdots\!32 \)
|
| $67$ |
\( T^{5} + \cdots - 93\!\cdots\!32 \)
|
| $71$ |
\( T^{5} + \cdots - 29\!\cdots\!68 \)
|
| $73$ |
\( T^{5} + \cdots - 72\!\cdots\!00 \)
|
| $79$ |
\( T^{5} + \cdots - 65\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots + 18\!\cdots\!08 \)
|
| $89$ |
\( T^{5} + \cdots + 17\!\cdots\!00 \)
|
| $97$ |
\( T^{5} + \cdots - 69\!\cdots\!12 \)
|
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