Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9,38,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, 38, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.1");
S:= CuspForms(chi, 38);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 38 \) |
| 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: | \(78.0426343121\) |
| 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: |
\( x^{6} - 3082530225x^{4} + 1655052385582963200x^{2} - 128504121697355796840448000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{30}\cdot 3^{29}\cdot 5^{4}\cdot 7 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3082530225x^{4} + 1655052385582963200x^{2} - 128504121697355796840448000 \)
:
| \(\beta_{1}\) | \(=\) |
\( 12\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -243\nu^{5} + 705620942595\nu^{3} - 284541461145485913600\nu ) / 154833817600 \)
|
| \(\beta_{3}\) | \(=\) |
\( 144\nu^{2} - 147961450800 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 96471\nu^{5} - 12578677397415\nu^{3} - 424471235179652946700800\nu ) / 154833817600 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 81\nu^{4} - 214466876865\nu^{2} + 53185905677011060800 ) / 7700 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 147961450800 ) / 144 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{4} + 397\beta_{2} + 289253237000\beta_1 ) / 1728 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 123200\beta_{5} + 23829652985\beta_{3} + 2674895536888973665200 ) / 1296 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 26134108985\beta_{4} + 465876940645\beta_{2} + 6041821160579442905800\beta_1 ) / 15552 \)
|
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 |
|
−590416. | 0 | 2.11152e11 | 7.67686e11 | 0 | 1.40013e15 | −4.35214e16 | 0 | −4.53254e17 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −286084. | 0 | −5.55949e10 | −5.54299e12 | 0 | −4.95122e15 | 5.52239e16 | 0 | 1.58576e18 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −115971. | 0 | −1.23990e11 | 1.39044e13 | 0 | 5.04868e15 | 3.03182e16 | 0 | −1.61252e18 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 115971. | 0 | −1.23990e11 | −1.39044e13 | 0 | 5.04868e15 | −3.03182e16 | 0 | −1.61252e18 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 286084. | 0 | −5.55949e10 | 5.54299e12 | 0 | −4.95122e15 | −5.52239e16 | 0 | 1.58576e18 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 590416. | 0 | 2.11152e11 | −7.67686e11 | 0 | 1.40013e15 | 4.35214e16 | 0 | −4.53254e17 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 9.38.a.d | ✓ | 6 |
| 3.b | odd | 2 | 1 | inner | 9.38.a.d | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9.38.a.d | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 9.38.a.d | ✓ | 6 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 443884352400T_{2}^{4} + 34319166267448324915200T_{2}^{2} - 383711251322357251672828280832000 \)
acting on \(S_{38}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots - 38\!\cdots\!00 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + \cdots - 35\!\cdots\!00 \)
|
| $7$ |
\( (T^{3} + \cdots + 34\!\cdots\!00)^{2} \)
|
| $11$ |
\( T^{6} + \cdots - 20\!\cdots\!00 \)
|
| $13$ |
\( (T^{3} + \cdots - 64\!\cdots\!00)^{2} \)
|
| $17$ |
\( T^{6} + \cdots - 13\!\cdots\!00 \)
|
| $19$ |
\( (T^{3} + \cdots - 29\!\cdots\!04)^{2} \)
|
| $23$ |
\( T^{6} + \cdots - 37\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots - 12\!\cdots\!00 \)
|
| $31$ |
\( (T^{3} + \cdots - 28\!\cdots\!72)^{2} \)
|
| $37$ |
\( (T^{3} + \cdots - 35\!\cdots\!00)^{2} \)
|
| $41$ |
\( T^{6} + \cdots - 20\!\cdots\!00 \)
|
| $43$ |
\( (T^{3} + \cdots - 42\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{6} + \cdots - 10\!\cdots\!00 \)
|
| $53$ |
\( T^{6} + \cdots - 80\!\cdots\!00 \)
|
| $59$ |
\( T^{6} + \cdots - 16\!\cdots\!00 \)
|
| $61$ |
\( (T^{3} + \cdots + 25\!\cdots\!52)^{2} \)
|
| $67$ |
\( (T^{3} + \cdots - 34\!\cdots\!00)^{2} \)
|
| $71$ |
\( T^{6} + \cdots - 81\!\cdots\!00 \)
|
| $73$ |
\( (T^{3} + \cdots - 23\!\cdots\!00)^{2} \)
|
| $79$ |
\( (T^{3} + \cdots + 70\!\cdots\!84)^{2} \)
|
| $83$ |
\( T^{6} + \cdots - 94\!\cdots\!00 \)
|
| $89$ |
\( T^{6} + \cdots - 44\!\cdots\!00 \)
|
| $97$ |
\( (T^{3} + \cdots + 16\!\cdots\!00)^{2} \)
|
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