Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [324,3,Mod(163,324)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(324, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("324.163");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 324 = 2^{2} \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 324.d (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: | \(8.82836056527\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.3636603.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 8x^{4} + 12x^{2} + 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 8x^{4} + 12x^{2} + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} + 7\nu^{3} + 7\nu^{2} + 7\nu + 5 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} - 9\nu^{3} + 5\nu^{2} - 17\nu + 1 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{5} - 8\nu^{3} + \nu^{2} - 12\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{5} + 2\nu^{4} - 6\nu^{3} + 13\nu^{2} - 2\nu + 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 5\nu^{5} - \nu^{4} + 37\nu^{3} - 5\nu^{2} + 33\nu - 1 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + 4\beta _1 + 1 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + \beta _1 - 5 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{5} + 4\beta_{4} + \beta_{3} + 2\beta_{2} - 13\beta _1 - 1 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{4} - 13\beta_{3} + 14\beta_{2} - 10\beta _1 + 49 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -28\beta_{5} - 26\beta_{4} - 5\beta_{3} - 7\beta_{2} + 83\beta _1 + 5 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).
| \(n\) | \(163\) | \(245\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 163.1 |
|
−1.45476 | − | 1.37247i | 0 | 0.232642 | + | 3.99323i | −4.37480 | 0 | − | 2.13525i | 5.14216 | − | 6.12848i | 0 | 6.36427 | + | 6.00429i | |||||||||||||||||||||||||||
| 163.2 | −1.45476 | + | 1.37247i | 0 | 0.232642 | − | 3.99323i | −4.37480 | 0 | 2.13525i | 5.14216 | + | 6.12848i | 0 | 6.36427 | − | 6.00429i | |||||||||||||||||||||||||||||
| 163.3 | 0.195350 | − | 1.99044i | 0 | −3.92368 | − | 0.777662i | 7.23805 | 0 | 6.88025i | −2.31438 | + | 7.65792i | 0 | 1.41395 | − | 14.4069i | |||||||||||||||||||||||||||||
| 163.4 | 0.195350 | + | 1.99044i | 0 | −3.92368 | + | 0.777662i | 7.23805 | 0 | − | 6.88025i | −2.31438 | − | 7.65792i | 0 | 1.41395 | + | 14.4069i | ||||||||||||||||||||||||||||
| 163.5 | 1.75941 | − | 0.951043i | 0 | 2.19104 | − | 3.34655i | −1.86325 | 0 | − | 11.3183i | 0.672219 | − | 7.97171i | 0 | −3.27823 | + | 1.77203i | ||||||||||||||||||||||||||||
| 163.6 | 1.75941 | + | 0.951043i | 0 | 2.19104 | + | 3.34655i | −1.86325 | 0 | 11.3183i | 0.672219 | + | 7.97171i | 0 | −3.27823 | − | 1.77203i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 324.3.d.f | yes | 6 |
| 3.b | odd | 2 | 1 | 324.3.d.e | ✓ | 6 | |
| 4.b | odd | 2 | 1 | inner | 324.3.d.f | yes | 6 |
| 9.c | even | 3 | 2 | 324.3.f.q | 12 | ||
| 9.d | odd | 6 | 2 | 324.3.f.r | 12 | ||
| 12.b | even | 2 | 1 | 324.3.d.e | ✓ | 6 | |
| 36.f | odd | 6 | 2 | 324.3.f.q | 12 | ||
| 36.h | even | 6 | 2 | 324.3.f.r | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 324.3.d.e | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 324.3.d.e | ✓ | 6 | 12.b | even | 2 | 1 | |
| 324.3.d.f | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 324.3.d.f | yes | 6 | 4.b | odd | 2 | 1 | inner |
| 324.3.f.q | 12 | 9.c | even | 3 | 2 | ||
| 324.3.f.q | 12 | 36.f | odd | 6 | 2 | ||
| 324.3.f.r | 12 | 9.d | odd | 6 | 2 | ||
| 324.3.f.r | 12 | 36.h | even | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} - T_{5}^{2} - 37T_{5} - 59 \)
acting on \(S_{3}^{\mathrm{new}}(324, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - T^{5} + \cdots + 64 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T^{3} - T^{2} - 37 T - 59)^{2} \)
|
| $7$ |
\( T^{6} + 180 T^{4} + \cdots + 27648 \)
|
| $11$ |
\( T^{6} + 516 T^{4} + \cdots + 1769472 \)
|
| $13$ |
\( (T^{3} - 3 T^{2} + \cdots + 991)^{2} \)
|
| $17$ |
\( (T^{3} + 5 T^{2} + \cdots - 233)^{2} \)
|
| $19$ |
\( T^{6} + 1476 T^{4} + \cdots + 20155392 \)
|
| $23$ |
\( T^{6} + 2868 T^{4} + \cdots + 425115648 \)
|
| $29$ |
\( (T^{3} + 11 T^{2} + \cdots - 32519)^{2} \)
|
| $31$ |
\( T^{6} + 5376 T^{4} + \cdots + 7077888 \)
|
| $37$ |
\( (T^{3} - 27 T^{2} + \cdots + 144607)^{2} \)
|
| $41$ |
\( (T^{3} - 46 T^{2} + \cdots + 21976)^{2} \)
|
| $43$ |
\( T^{6} + 6372 T^{4} + \cdots + 204484608 \)
|
| $47$ |
\( T^{6} + 8208 T^{4} + \cdots + 743620608 \)
|
| $53$ |
\( (T^{3} - 58 T^{2} + \cdots + 5128)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 15635054592 \)
|
| $61$ |
\( (T^{3} - 27 T^{2} + \cdots + 101191)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 12897487872 \)
|
| $71$ |
\( T^{6} + 8532 T^{4} + \cdots + 808455168 \)
|
| $73$ |
\( (T^{3} + 39 T^{2} + \cdots - 447851)^{2} \)
|
| $79$ |
\( T^{6} + 7764 T^{4} + \cdots + 113246208 \)
|
| $83$ |
\( T^{6} + \cdots + 900806787072 \)
|
| $89$ |
\( (T^{3} + 125 T^{2} + \cdots - 107777)^{2} \)
|
| $97$ |
\( (T^{3} + 102 T^{2} + \cdots - 369272)^{2} \)
|
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