Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1080,6,Mod(1,1080)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1080, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1080.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1080.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: | \(173.214525398\) |
| 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: |
\( x^{5} - x^{4} - 762x^{3} - 4517x^{2} + 60219x + 91480 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{6} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 762x^{3} - 4517x^{2} + 60219x + 91480 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 13\nu^{4} - 250\nu^{3} - 5423\nu^{2} + 50452\nu - 12345 ) / 971 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -39\nu^{4} + 750\nu^{3} + 16269\nu^{2} - 116400\nu + 29267 ) / 971 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 37\nu^{4} - 114\nu^{3} - 25070\nu^{2} - 131647\nu + 1067248 ) / 1942 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -169\nu^{4} + 3250\nu^{3} + 79238\nu^{2} - 682093\nu - 2502968 ) / 1942 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 3\beta _1 + 8 ) / 36 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 8\beta_{4} + 3\beta_{2} + 61\beta _1 + 10996 ) / 36 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 129\beta_{4} + 117\beta_{3} + 509\beta_{2} + 2199\beta _1 + 114580 ) / 36 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 5818\beta_{4} + 2250\beta_{3} + 7159\beta_{2} + 58781\beta _1 + 6793624 ) / 36 \)
|
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 |
|
0 | 0 | 0 | 25.0000 | 0 | −175.181 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 25.0000 | 0 | −95.0320 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 25.0000 | 0 | 2.69456 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 25.0000 | 0 | 69.3582 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0 | 0 | 25.0000 | 0 | 202.160 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
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 | 1080.6.a.p | yes | 5 |
| 3.b | odd | 2 | 1 | 1080.6.a.k | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1080.6.a.k | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 1080.6.a.p | yes | 5 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(1080))\):
|
\( T_{7}^{5} - 4T_{7}^{4} - 42695T_{7}^{3} - 616346T_{7}^{2} + 235397228T_{7} - 628981784 \)
|
|
\( T_{11}^{5} - 197T_{11}^{4} - 424697T_{11}^{3} + 49735673T_{11}^{2} + 18359581328T_{11} + 815319053492 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( (T - 25)^{5} \)
|
| $7$ |
\( T^{5} - 4 T^{4} + \cdots - 628981784 \)
|
| $11$ |
\( T^{5} + \cdots + 815319053492 \)
|
| $13$ |
\( T^{5} + \cdots - 11461112282887 \)
|
| $17$ |
\( T^{5} + \cdots - 15\!\cdots\!32 \)
|
| $19$ |
\( T^{5} + \cdots - 19\!\cdots\!24 \)
|
| $23$ |
\( T^{5} + \cdots - 15\!\cdots\!44 \)
|
| $29$ |
\( T^{5} + \cdots - 861746289819940 \)
|
| $31$ |
\( T^{5} + \cdots + 26\!\cdots\!44 \)
|
| $37$ |
\( T^{5} + \cdots + 52\!\cdots\!92 \)
|
| $41$ |
\( T^{5} + \cdots - 41\!\cdots\!00 \)
|
| $43$ |
\( T^{5} + \cdots - 33\!\cdots\!76 \)
|
| $47$ |
\( T^{5} + \cdots - 69\!\cdots\!12 \)
|
| $53$ |
\( T^{5} + \cdots + 28\!\cdots\!16 \)
|
| $59$ |
\( T^{5} + \cdots + 22\!\cdots\!08 \)
|
| $61$ |
\( T^{5} + \cdots - 14\!\cdots\!00 \)
|
| $67$ |
\( T^{5} + \cdots - 21\!\cdots\!84 \)
|
| $71$ |
\( T^{5} + \cdots - 23\!\cdots\!80 \)
|
| $73$ |
\( T^{5} + \cdots + 12\!\cdots\!12 \)
|
| $79$ |
\( T^{5} + \cdots + 93\!\cdots\!35 \)
|
| $83$ |
\( T^{5} + \cdots - 58\!\cdots\!04 \)
|
| $89$ |
\( T^{5} + \cdots - 29\!\cdots\!92 \)
|
| $97$ |
\( T^{5} + \cdots + 10\!\cdots\!20 \)
|
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