Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [36,27,Mod(17,36)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(36, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 27, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("36.17");
S:= CuspForms(chi, 27);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 36 = 2^{2} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 27 \) |
| Character orbit: | \([\chi]\) | \(=\) | 36.c (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: | \(154.185451463\) |
| 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: |
\( x^{8} + 420085905943560 x^{6} + \cdots + 46\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{44}\cdot 3^{65}\cdot 5^{2} \) |
| 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
\( x^{8} + 420085905943560 x^{6} + \cdots + 46\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 135\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 96\!\cdots\!43 \nu^{6} + \cdots - 52\!\cdots\!00 ) / 14\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 16\!\cdots\!57 \nu^{6} + \cdots + 12\!\cdots\!00 ) / 79\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 89\!\cdots\!19 \nu^{6} + \cdots + 26\!\cdots\!00 ) / 79\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 54\!\cdots\!13 \nu^{7} + \cdots + 26\!\cdots\!00 \nu ) / 41\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 34\!\cdots\!79 \nu^{7} + \cdots - 99\!\cdots\!00 \nu ) / 56\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 28\!\cdots\!61 \nu^{7} + \cdots - 19\!\cdots\!00 \nu ) / 20\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 135 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -1460\beta_{4} - 754\beta_{3} - 2761334\beta_{2} - 382803281791069050 ) / 3645 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 51957916360 \beta_{7} + 317732375760 \beta_{6} + \cdots - 13\!\cdots\!46 \beta_1 ) / 98415 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 16\!\cdots\!96 \beta_{4} + \cdots + 39\!\cdots\!96 ) / 19683 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 36\!\cdots\!00 \beta_{7} + \cdots + 77\!\cdots\!12 \beta_1 ) / 2657205 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 14\!\cdots\!20 \beta_{4} + \cdots - 36\!\cdots\!00 ) / 885735 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 80\!\cdots\!20 \beta_{7} + \cdots - 14\!\cdots\!52 \beta_1 ) / 23914845 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).
| \(n\) | \(19\) | \(29\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 |
|
0 | 0 | 0 | − | 2.03306e9i | 0 | 1.10679e11 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 17.2 | 0 | 0 | 0 | − | 1.75963e9i | 0 | 5.11823e10 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 17.3 | 0 | 0 | 0 | − | 5.35534e8i | 0 | 4.76510e10 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 17.4 | 0 | 0 | 0 | − | 3.73709e8i | 0 | −1.34582e11 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 17.5 | 0 | 0 | 0 | 3.73709e8i | 0 | −1.34582e11 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 17.6 | 0 | 0 | 0 | 5.35534e8i | 0 | 4.76510e10 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 17.7 | 0 | 0 | 0 | 1.75963e9i | 0 | 5.11823e10 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 17.8 | 0 | 0 | 0 | 2.03306e9i | 0 | 1.10679e11 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
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 | 36.27.c.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 36.27.c.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 36.27.c.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 36.27.c.a | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{27}^{\mathrm{new}}(36, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + \cdots + 51\!\cdots\!00 \)
|
| $7$ |
\( (T^{4} + \cdots - 36\!\cdots\!04)^{2} \)
|
| $11$ |
\( T^{8} + \cdots + 67\!\cdots\!76 \)
|
| $13$ |
\( (T^{4} + \cdots - 10\!\cdots\!04)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 11\!\cdots\!76 \)
|
| $19$ |
\( (T^{4} + \cdots - 11\!\cdots\!24)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 18\!\cdots\!36 \)
|
| $29$ |
\( T^{8} + \cdots + 11\!\cdots\!96 \)
|
| $31$ |
\( (T^{4} + \cdots + 27\!\cdots\!36)^{2} \)
|
| $37$ |
\( (T^{4} + \cdots + 12\!\cdots\!56)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 16\!\cdots\!56 \)
|
| $43$ |
\( (T^{4} + \cdots - 65\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 18\!\cdots\!96 \)
|
| $53$ |
\( T^{8} + \cdots + 97\!\cdots\!00 \)
|
| $59$ |
\( T^{8} + \cdots + 28\!\cdots\!16 \)
|
| $61$ |
\( (T^{4} + \cdots + 72\!\cdots\!00)^{2} \)
|
| $67$ |
\( (T^{4} + \cdots - 13\!\cdots\!84)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 32\!\cdots\!56 \)
|
| $73$ |
\( (T^{4} + \cdots + 34\!\cdots\!36)^{2} \)
|
| $79$ |
\( (T^{4} + \cdots - 30\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 34\!\cdots\!16 \)
|
| $89$ |
\( T^{8} + \cdots + 16\!\cdots\!96 \)
|
| $97$ |
\( (T^{4} + \cdots + 24\!\cdots\!36)^{2} \)
|
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