Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [27,10,Mod(1,27)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(27, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("27.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 27 = 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 27.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: | \(13.9059675764\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.203942560.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 83x^{2} + 1440 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{8} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 83x^{2} + 1440 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 35\nu ) / 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -21\nu^{3} + 1383\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( 54\nu^{2} - 2241 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 42\beta_1 ) / 324 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 2241 ) / 54 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 35\beta_{2} + 2766\beta_1 ) / 324 \)
|
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 |
|
−44.4771 | 0 | 1466.22 | 1050.62 | 0 | 6591.86 | −42440.8 | 0 | −46728.6 | ||||||||||||||||||||||||||||||
| 1.2 | −12.7978 | 0 | −348.216 | −2019.77 | 0 | −665.864 | 11008.9 | 0 | 25848.6 | |||||||||||||||||||||||||||||||
| 1.3 | 12.7978 | 0 | −348.216 | 2019.77 | 0 | −665.864 | −11008.9 | 0 | 25848.6 | |||||||||||||||||||||||||||||||
| 1.4 | 44.4771 | 0 | 1466.22 | −1050.62 | 0 | 6591.86 | 42440.8 | 0 | −46728.6 | |||||||||||||||||||||||||||||||
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 | 27.10.a.d | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 27.10.a.d | ✓ | 4 |
| 9.c | even | 3 | 2 | 81.10.c.j | 8 | ||
| 9.d | odd | 6 | 2 | 81.10.c.j | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 27.10.a.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 27.10.a.d | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 81.10.c.j | 8 | 9.c | even | 3 | 2 | ||
| 81.10.c.j | 8 | 9.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 2142T_{2}^{2} + 324000 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(27))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 2142 T^{2} + 324000 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 4502946816000 \)
|
| $7$ |
\( (T^{2} - 5926 T - 4389287)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 15\!\cdots\!00 \)
|
| $13$ |
\( (T^{2} - 89842 T + 424488865)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 38\!\cdots\!00 \)
|
| $19$ |
\( (T^{2} - 505714 T - 484090125647)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 26\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots + 19\!\cdots\!00 \)
|
| $31$ |
\( (T^{2} + \cdots - 10428415330928)^{2} \)
|
| $37$ |
\( (T^{2} + \cdots - 8057965728479)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 19\!\cdots\!00 \)
|
| $43$ |
\( (T^{2} + \cdots - 899237665076288)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 14\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 64\!\cdots\!00 \)
|
| $59$ |
\( T^{4} + \cdots + 35\!\cdots\!00 \)
|
| $61$ |
\( (T^{2} + \cdots - 24\!\cdots\!87)^{2} \)
|
| $67$ |
\( (T^{2} + \cdots - 42\!\cdots\!15)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 50\!\cdots\!00 \)
|
| $73$ |
\( (T^{2} + \cdots + 69\!\cdots\!53)^{2} \)
|
| $79$ |
\( (T^{2} + \cdots - 392927220347495)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 53\!\cdots\!00 \)
|
| $89$ |
\( T^{4} + \cdots + 52\!\cdots\!00 \)
|
| $97$ |
\( (T^{2} + \cdots + 788793992975497)^{2} \)
|
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