Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [54,9,Mod(53,54)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(54, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("54.53");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 54 = 2 \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 54.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: | \(21.9984449433\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{79})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 80x^{2} + 1521 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 3^{6} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 80x^{2} + 1521 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 8\nu^{3} + 328\nu ) / 39 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 18\nu^{3} + 2142\nu ) / 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( 432\nu^{2} + 17280 \)
|
| \(\nu\) | \(=\) |
\( ( 4\beta_{2} - 27\beta_1 ) / 432 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 17280 ) / 432 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -164\beta_{2} + 3213\beta_1 ) / 432 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/54\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 53.1 |
|
− | 11.3137i | 0 | −128.000 | − | 441.182i | 0 | 3880.70 | 1448.15i | 0 | −4991.40 | ||||||||||||||||||||||||||||
| 53.2 | − | 11.3137i | 0 | −128.000 | 916.357i | 0 | −3798.70 | 1448.15i | 0 | 10367.4 | ||||||||||||||||||||||||||||||
| 53.3 | 11.3137i | 0 | −128.000 | − | 916.357i | 0 | −3798.70 | − | 1448.15i | 0 | 10367.4 | |||||||||||||||||||||||||||||
| 53.4 | 11.3137i | 0 | −128.000 | 441.182i | 0 | 3880.70 | − | 1448.15i | 0 | −4991.40 | ||||||||||||||||||||||||||||||
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 | 54.9.b.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 54.9.b.b | ✓ | 4 |
| 4.b | odd | 2 | 1 | 432.9.e.i | 4 | ||
| 9.c | even | 3 | 2 | 162.9.d.f | 8 | ||
| 9.d | odd | 6 | 2 | 162.9.d.f | 8 | ||
| 12.b | even | 2 | 1 | 432.9.e.i | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 54.9.b.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 54.9.b.b | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 162.9.d.f | 8 | 9.c | even | 3 | 2 | ||
| 162.9.d.f | 8 | 9.d | odd | 6 | 2 | ||
| 432.9.e.i | 4 | 4.b | odd | 2 | 1 | ||
| 432.9.e.i | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 1034352T_{5}^{2} + 163442318400 \)
acting on \(S_{9}^{\mathrm{new}}(54, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 128)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 163442318400 \)
|
| $7$ |
\( (T^{2} - 82 T - 14741615)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 15\!\cdots\!00 \)
|
| $13$ |
\( (T^{2} - 49822 T + 605814625)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 59\!\cdots\!16 \)
|
| $19$ |
\( (T^{2} - 256018 T + 9250548817)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 75\!\cdots\!76 \)
|
| $29$ |
\( T^{4} + \cdots + 35\!\cdots\!76 \)
|
| $31$ |
\( (T^{2} - 297772 T - 591389965340)^{2} \)
|
| $37$ |
\( (T^{2} + \cdots - 3696029027375)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 27\!\cdots\!00 \)
|
| $43$ |
\( (T^{2} + \cdots - 1002667718108)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 79\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 23\!\cdots\!44 \)
|
| $59$ |
\( T^{4} + \cdots + 56\!\cdots\!44 \)
|
| $61$ |
\( (T^{2} + \cdots - 44289948481535)^{2} \)
|
| $67$ |
\( (T^{2} + \cdots - 282923520334559)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 33\!\cdots\!84 \)
|
| $73$ |
\( (T^{2} + \cdots - 531549935014847)^{2} \)
|
| $79$ |
\( (T^{2} + \cdots - 149822587701983)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 22\!\cdots\!76 \)
|
| $89$ |
\( T^{4} + \cdots + 25\!\cdots\!44 \)
|
| $97$ |
\( (T^{2} + \cdots + 74\!\cdots\!45)^{2} \)
|
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