Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9,60,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, 60, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.1");
S:= CuspForms(chi, 60);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 60 \) |
| 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: | \(198.412204959\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{5} - x^{4} + \cdots - 23\!\cdots\!36 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{39}\cdot 3^{19}\cdot 5^{3}\cdot 7^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} + \cdots - 23\!\cdots\!36 \)
:
| \(\beta_{1}\) | \(=\) |
\( 24\nu - 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 567 \nu^{4} - 37846914417 \nu^{3} + \cdots - 12\!\cdots\!92 ) / 12\!\cdots\!68 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 567 \nu^{4} - 37846914417 \nu^{3} + \cdots - 12\!\cdots\!24 ) / 12\!\cdots\!68 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 56341737 \nu^{4} + 450652208893551 \nu^{3} + \cdots + 85\!\cdots\!68 ) / 63\!\cdots\!40 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 5 ) / 24 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - \beta_{2} - 82073154\beta _1 + 916135717213006029 ) / 576 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 33320 \beta_{4} + 11352147 \beta_{3} - 673541451 \beta_{2} + \cdots - 93\!\cdots\!29 ) / 1728 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 6672270837960 \beta_{4} + \cdots + 21\!\cdots\!53 ) / 5184 \)
|
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 |
|
−1.25878e9 | 0 | 1.00806e18 | 4.05126e20 | 0 | 2.66317e24 | −5.43293e26 | 0 | −5.09964e29 | |||||||||||||||||||||||||||||||||
| 1.2 | −6.92842e8 | 0 | −9.64308e16 | −3.87033e20 | 0 | −7.47157e23 | 4.66207e26 | 0 | 2.68153e29 | ||||||||||||||||||||||||||||||||||
| 1.3 | 3.23738e8 | 0 | −4.71654e17 | 7.71676e20 | 0 | −4.97804e24 | −3.39315e26 | 0 | 2.49821e29 | ||||||||||||||||||||||||||||||||||
| 1.4 | 6.55640e8 | 0 | −1.46597e17 | −7.23713e20 | 0 | 1.08717e25 | −4.74066e26 | 0 | −4.74495e29 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.42193e9 | 0 | 1.44544e18 | −2.46018e20 | 0 | −6.31094e24 | 1.23563e27 | 0 | −3.49822e29 | ||||||||||||||||||||||||||||||||||
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 | 9.60.a.c | 5 | |
| 3.b | odd | 2 | 1 | 1.60.a.a | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.60.a.a | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 9.60.a.c | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 449691864 T_{2}^{4} + \cdots - 26\!\cdots\!24 \)
acting on \(S_{60}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + \cdots - 26\!\cdots\!24 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} + \cdots + 21\!\cdots\!00 \)
|
| $7$ |
\( T^{5} + \cdots + 67\!\cdots\!24 \)
|
| $11$ |
\( T^{5} + \cdots - 94\!\cdots\!68 \)
|
| $13$ |
\( T^{5} + \cdots + 37\!\cdots\!68 \)
|
| $17$ |
\( T^{5} + \cdots - 20\!\cdots\!24 \)
|
| $19$ |
\( T^{5} + \cdots - 10\!\cdots\!00 \)
|
| $23$ |
\( T^{5} + \cdots - 81\!\cdots\!68 \)
|
| $29$ |
\( T^{5} + \cdots - 37\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots - 29\!\cdots\!32 \)
|
| $37$ |
\( T^{5} + \cdots + 31\!\cdots\!24 \)
|
| $41$ |
\( T^{5} + \cdots - 48\!\cdots\!68 \)
|
| $43$ |
\( T^{5} + \cdots - 16\!\cdots\!32 \)
|
| $47$ |
\( T^{5} + \cdots - 63\!\cdots\!24 \)
|
| $53$ |
\( T^{5} + \cdots - 47\!\cdots\!68 \)
|
| $59$ |
\( T^{5} + \cdots + 34\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots - 67\!\cdots\!32 \)
|
| $67$ |
\( T^{5} + \cdots - 18\!\cdots\!76 \)
|
| $71$ |
\( T^{5} + \cdots + 22\!\cdots\!32 \)
|
| $73$ |
\( T^{5} + \cdots + 43\!\cdots\!68 \)
|
| $79$ |
\( T^{5} + \cdots + 81\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots - 37\!\cdots\!68 \)
|
| $89$ |
\( T^{5} + \cdots + 32\!\cdots\!00 \)
|
| $97$ |
\( T^{5} + \cdots - 16\!\cdots\!76 \)
|
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