Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1296,3,Mod(593,1296)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1296, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1296.593");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1296 = 2^{4} \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1296.q (of order \(6\), degree \(2\), not 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: | \(35.3134422611\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-3})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{25}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | no (minimal twist has level 18) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{4} - 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{3} ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{3} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).
| \(n\) | \(325\) | \(1135\) | \(1217\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1 - \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 593.1 |
|
0 | 0 | 0 | −3.67423 | + | 2.12132i | 0 | −2.00000 | + | 3.46410i | 0 | 0 | 0 | ||||||||||||||||||||||||||
| 593.2 | 0 | 0 | 0 | 3.67423 | − | 2.12132i | 0 | −2.00000 | + | 3.46410i | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 1025.1 | 0 | 0 | 0 | −3.67423 | − | 2.12132i | 0 | −2.00000 | − | 3.46410i | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 1025.2 | 0 | 0 | 0 | 3.67423 | + | 2.12132i | 0 | −2.00000 | − | 3.46410i | 0 | 0 | 0 | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 9.c | even | 3 | 1 | inner |
| 9.d | odd | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1296, [\chi])\):
|
\( T_{5}^{4} - 18T_{5}^{2} + 324 \)
|
|
\( T_{7}^{2} + 4T_{7} + 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 18T^{2} + 324 \)
|
| $7$ |
\( (T^{2} + 4 T + 16)^{2} \)
|
| $11$ |
\( T^{4} - 288 T^{2} + 82944 \)
|
| $13$ |
\( (T^{2} + 8 T + 64)^{2} \)
|
| $17$ |
\( (T^{2} + 162)^{2} \)
|
| $19$ |
\( (T - 16)^{4} \)
|
| $23$ |
\( T^{4} - 288 T^{2} + 82944 \)
|
| $29$ |
\( T^{4} - 18T^{2} + 324 \)
|
| $31$ |
\( (T^{2} - 44 T + 1936)^{2} \)
|
| $37$ |
\( (T + 34)^{4} \)
|
| $41$ |
\( T^{4} - 2178 T^{2} + 4743684 \)
|
| $43$ |
\( (T^{2} + 40 T + 1600)^{2} \)
|
| $47$ |
\( T^{4} - 7200 T^{2} + 51840000 \)
|
| $53$ |
\( (T^{2} + 1458)^{2} \)
|
| $59$ |
\( T^{4} - 1152 T^{2} + 1327104 \)
|
| $61$ |
\( (T^{2} + 50 T + 2500)^{2} \)
|
| $67$ |
\( (T^{2} - 8 T + 64)^{2} \)
|
| $71$ |
\( (T^{2} + 2592)^{2} \)
|
| $73$ |
\( (T + 16)^{4} \)
|
| $79$ |
\( (T^{2} + 76 T + 5776)^{2} \)
|
| $83$ |
\( T^{4} - 14112 T^{2} + 199148544 \)
|
| $89$ |
\( (T^{2} + 162)^{2} \)
|
| $97$ |
\( (T^{2} + 176 T + 30976)^{2} \)
|
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