Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1,18,Mod(1,1)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1, base_ring=CyclotomicField(1))
chi = DirichletCharacter(H, H._module([]))
N = Newforms(chi, 18, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1.1");
S:= CuspForms(chi, 18);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1 \) |
| Weight: | \( k \) | \(=\) | \( 18 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1.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: | \(1.83222087345\) |
| Analytic rank: | \(1\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| 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)
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 |
|
−528.000 | −4284.00 | 147712. | −1.02585e6 | 2.26195e6 | 3.22599e6 | −8.78592e6 | −1.10788e8 | 5.41649e8 | |||||||||||||||||||||
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 | 1.18.a.a | ✓ | 1 |
| 3.b | odd | 2 | 1 | 9.18.a.b | 1 | ||
| 4.b | odd | 2 | 1 | 16.18.a.b | 1 | ||
| 5.b | even | 2 | 1 | 25.18.a.a | 1 | ||
| 5.c | odd | 4 | 2 | 25.18.b.a | 2 | ||
| 7.b | odd | 2 | 1 | 49.18.a.a | 1 | ||
| 8.b | even | 2 | 1 | 64.18.a.d | 1 | ||
| 8.d | odd | 2 | 1 | 64.18.a.b | 1 | ||
| 11.b | odd | 2 | 1 | 121.18.a.b | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.18.a.a | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
| 9.18.a.b | 1 | 3.b | odd | 2 | 1 | ||
| 16.18.a.b | 1 | 4.b | odd | 2 | 1 | ||
| 25.18.a.a | 1 | 5.b | even | 2 | 1 | ||
| 25.18.b.a | 2 | 5.c | odd | 4 | 2 | ||
| 49.18.a.a | 1 | 7.b | odd | 2 | 1 | ||
| 64.18.a.b | 1 | 8.d | odd | 2 | 1 | ||
| 64.18.a.d | 1 | 8.b | even | 2 | 1 | ||
| 121.18.a.b | 1 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{18}^{\mathrm{new}}(\Gamma_0(1))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T + 528 \)
|
| $3$ |
\( T + 4284 \)
|
| $5$ |
\( T + 1025850 \)
|
| $7$ |
\( T - 3225992 \)
|
| $11$ |
\( T + 753618228 \)
|
| $13$ |
\( T - 2541064526 \)
|
| $17$ |
\( T + 5429742318 \)
|
| $19$ |
\( T - 1487499860 \)
|
| $23$ |
\( T + 317091823464 \)
|
| $29$ |
\( T - 2433410602590 \)
|
| $31$ |
\( T + 8849722053088 \)
|
| $37$ |
\( T - 12691652946662 \)
|
| $41$ |
\( T - 48864151002282 \)
|
| $43$ |
\( T + 91019974317844 \)
|
| $47$ |
\( T + 49304994276048 \)
|
| $53$ |
\( T - 22940453195766 \)
|
| $59$ |
\( T - 32695090729980 \)
|
| $61$ |
\( T + 1308285854869378 \)
|
| $67$ |
\( T - 5196143861984132 \)
|
| $71$ |
\( T + 3709489877412408 \)
|
| $73$ |
\( T - 3402372968272586 \)
|
| $79$ |
\( T - 2366533941308240 \)
|
| $83$ |
\( T + 29\!\cdots\!04 \)
|
| $89$ |
\( T - 29\!\cdots\!70 \)
|
| $97$ |
\( T + 63\!\cdots\!98 \)
|
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