Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [18,42,Mod(1,18)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(18, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 42, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("18.1");
S:= CuspForms(chi, 42);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 18 = 2 \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 42 \) |
| Character orbit: | \([\chi]\) | \(=\) | 18.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: | \(191.649006822\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + \cdots - 77\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{27}\cdot 3^{26}\cdot 5\cdot 7 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + \cdots - 77\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 14776720812 \nu^{3} + \cdots - 14\!\cdots\!32 ) / 13\!\cdots\!53 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 23636004760224 \nu^{3} + \cdots + 12\!\cdots\!80 ) / 66\!\cdots\!65 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 98\!\cdots\!04 \nu^{3} + \cdots + 14\!\cdots\!20 ) / 66\!\cdots\!65 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} - 48139\beta_{2} - 28751039\beta_1 ) / 914248581120 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 1086887249 \beta_{3} - 16857535330811 \beta_{2} + \cdots + 57\!\cdots\!80 ) / 32651735040 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 27\!\cdots\!37 \beta_{3} + \cdots + 11\!\cdots\!00 ) / 457124290560 \)
|
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 |
|
−1.04858e6 | 0 | 1.09951e12 | −3.43762e14 | 0 | 3.27280e17 | −1.15292e18 | 0 | 3.60460e20 | ||||||||||||||||||||||||||||||
| 1.2 | −1.04858e6 | 0 | 1.09951e12 | −2.18342e14 | 0 | −2.43749e17 | −1.15292e18 | 0 | 2.28948e20 | |||||||||||||||||||||||||||||||
| 1.3 | −1.04858e6 | 0 | 1.09951e12 | 6.18862e12 | 0 | 8.94216e16 | −1.15292e18 | 0 | −6.48924e18 | |||||||||||||||||||||||||||||||
| 1.4 | −1.04858e6 | 0 | 1.09951e12 | 2.92378e14 | 0 | 4.85219e16 | −1.15292e18 | 0 | −3.06581e20 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
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 | 18.42.a.g | ✓ | 4 |
| 3.b | odd | 2 | 1 | 18.42.a.h | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 18.42.a.g | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 18.42.a.h | yes | 4 | 3.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 263537161770624 T_{5}^{3} + \cdots + 13\!\cdots\!00 \)
acting on \(S_{42}^{\mathrm{new}}(\Gamma_0(18))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1048576)^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 13\!\cdots\!00 \)
|
| $7$ |
\( T^{4} + \cdots - 34\!\cdots\!84 \)
|
| $11$ |
\( T^{4} + \cdots + 36\!\cdots\!76 \)
|
| $13$ |
\( T^{4} + \cdots + 19\!\cdots\!36 \)
|
| $17$ |
\( T^{4} + \cdots + 66\!\cdots\!16 \)
|
| $19$ |
\( T^{4} + \cdots + 70\!\cdots\!16 \)
|
| $23$ |
\( T^{4} + \cdots + 17\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots - 17\!\cdots\!04 \)
|
| $31$ |
\( T^{4} + \cdots + 38\!\cdots\!16 \)
|
| $37$ |
\( T^{4} + \cdots + 57\!\cdots\!96 \)
|
| $41$ |
\( T^{4} + \cdots + 26\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots - 56\!\cdots\!24 \)
|
| $47$ |
\( T^{4} + \cdots + 37\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots - 10\!\cdots\!44 \)
|
| $59$ |
\( T^{4} + \cdots + 12\!\cdots\!36 \)
|
| $61$ |
\( T^{4} + \cdots - 75\!\cdots\!44 \)
|
| $67$ |
\( T^{4} + \cdots + 34\!\cdots\!56 \)
|
| $71$ |
\( T^{4} + \cdots - 40\!\cdots\!44 \)
|
| $73$ |
\( T^{4} + \cdots + 15\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots + 26\!\cdots\!36 \)
|
| $83$ |
\( T^{4} + \cdots + 58\!\cdots\!36 \)
|
| $89$ |
\( T^{4} + \cdots + 74\!\cdots\!76 \)
|
| $97$ |
\( T^{4} + \cdots - 25\!\cdots\!04 \)
|
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