Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [18,39,Mod(17,18)]
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([1]))
N = Newforms(chi, 39, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("18.17");
S:= CuspForms(chi, 39);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 18 = 2 \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 39 \) |
| Character orbit: | \([\chi]\) | \(=\) | 18.b (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(164.634244260\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + \cdots + 27\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{44}\cdot 3^{54}\cdot 5^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + \cdots + 27\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 7517583003 \nu^{6} + \cdots - 13\!\cdots\!00 ) / 10\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 48\!\cdots\!31 \nu^{6} + \cdots - 33\!\cdots\!00 ) / 30\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 62\!\cdots\!49 \nu^{6} + \cdots - 36\!\cdots\!00 ) / 18\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 14\!\cdots\!71 \nu^{7} + \cdots - 14\!\cdots\!00 \nu ) / 10\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 10\!\cdots\!13 \nu^{7} + \cdots + 44\!\cdots\!00 \nu ) / 11\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 31\!\cdots\!03 \nu^{7} + \cdots + 20\!\cdots\!00 \nu ) / 11\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 53\!\cdots\!97 \nu^{7} + \cdots - 10\!\cdots\!00 \nu ) / 15\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + 684100053273\beta_{4} ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -9958780\beta_{3} - 29437092280\beta_{2} + 1830679128618\beta _1 - 487873013301787900280058882 ) / 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 24\!\cdots\!60 \beta_{7} + \cdots + 18\!\cdots\!02 \beta_{4} ) / 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 85\!\cdots\!60 \beta_{3} + \cdots + 46\!\cdots\!44 ) / 9 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 26\!\cdots\!20 \beta_{7} + \cdots - 39\!\cdots\!84 \beta_{4} ) / 9 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 92\!\cdots\!20 \beta_{3} + \cdots - 47\!\cdots\!48 ) / 9 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 28\!\cdots\!40 \beta_{7} + \cdots + 57\!\cdots\!28 \beta_{4} ) / 9 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 17.1 |
|
− | 370728.i | 0 | −1.37439e11 | − | 3.09827e13i | 0 | 1.05726e16 | 5.09524e16i | 0 | −1.14862e19 | ||||||||||||||||||||||||||||||||||||||||
| 17.2 | − | 370728.i | 0 | −1.37439e11 | − | 4.63344e12i | 0 | −1.09850e16 | 5.09524e16i | 0 | −1.71774e18 | |||||||||||||||||||||||||||||||||||||||||
| 17.3 | − | 370728.i | 0 | −1.37439e11 | 9.98286e12i | 0 | −1.30465e16 | 5.09524e16i | 0 | 3.70092e18 | ||||||||||||||||||||||||||||||||||||||||||
| 17.4 | − | 370728.i | 0 | −1.37439e11 | 2.95032e13i | 0 | 2.09340e16 | 5.09524e16i | 0 | 1.09376e19 | ||||||||||||||||||||||||||||||||||||||||||
| 17.5 | 370728.i | 0 | −1.37439e11 | − | 2.95032e13i | 0 | 2.09340e16 | − | 5.09524e16i | 0 | 1.09376e19 | |||||||||||||||||||||||||||||||||||||||||
| 17.6 | 370728.i | 0 | −1.37439e11 | − | 9.98286e12i | 0 | −1.30465e16 | − | 5.09524e16i | 0 | 3.70092e18 | |||||||||||||||||||||||||||||||||||||||||
| 17.7 | 370728.i | 0 | −1.37439e11 | 4.63344e12i | 0 | −1.09850e16 | − | 5.09524e16i | 0 | −1.71774e18 | ||||||||||||||||||||||||||||||||||||||||||
| 17.8 | 370728.i | 0 | −1.37439e11 | 3.09827e13i | 0 | 1.05726e16 | − | 5.09524e16i | 0 | −1.14862e19 | ||||||||||||||||||||||||||||||||||||||||||
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 | 18.39.b.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 18.39.b.b | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 18.39.b.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 18.39.b.b | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + \cdots + 17\!\cdots\!00 \)
acting on \(S_{39}^{\mathrm{new}}(18, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 137438953472)^{4} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + \cdots + 17\!\cdots\!00 \)
|
| $7$ |
\( (T^{4} + \cdots + 31\!\cdots\!56)^{2} \)
|
| $11$ |
\( T^{8} + \cdots + 21\!\cdots\!76 \)
|
| $13$ |
\( (T^{4} + \cdots - 15\!\cdots\!24)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 14\!\cdots\!56 \)
|
| $19$ |
\( (T^{4} + \cdots + 12\!\cdots\!96)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 16\!\cdots\!76 \)
|
| $29$ |
\( T^{8} + \cdots + 15\!\cdots\!56 \)
|
| $31$ |
\( (T^{4} + \cdots + 91\!\cdots\!56)^{2} \)
|
| $37$ |
\( (T^{4} + \cdots - 21\!\cdots\!84)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 18\!\cdots\!16 \)
|
| $43$ |
\( (T^{4} + \cdots - 89\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 72\!\cdots\!00 \)
|
| $53$ |
\( T^{8} + \cdots + 85\!\cdots\!76 \)
|
| $59$ |
\( T^{8} + \cdots + 99\!\cdots\!76 \)
|
| $61$ |
\( (T^{4} + \cdots - 48\!\cdots\!00)^{2} \)
|
| $67$ |
\( (T^{4} + \cdots - 10\!\cdots\!44)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 95\!\cdots\!76 \)
|
| $73$ |
\( (T^{4} + \cdots - 70\!\cdots\!44)^{2} \)
|
| $79$ |
\( (T^{4} + \cdots - 13\!\cdots\!84)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 21\!\cdots\!56 \)
|
| $89$ |
\( T^{8} + \cdots + 33\!\cdots\!76 \)
|
| $97$ |
\( (T^{4} + \cdots - 23\!\cdots\!84)^{2} \)
|
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