Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [324,3,Mod(55,324)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(324, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 4]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("324.55");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 324 = 2^{2} \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 324.f (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: | \(8.82836056527\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 108) |
| 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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).
| \(n\) | \(163\) | \(245\) |
| \(\chi(n)\) | \(-1\) | \(-1 + \zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 55.1 |
|
−2.00000 | 0 | 4.00000 | 3.50000 | + | 6.06218i | 0 | −7.50000 | − | 4.33013i | −8.00000 | 0 | −7.00000 | − | 12.1244i | ||||||||||||||||||
| 271.1 | −2.00000 | 0 | 4.00000 | 3.50000 | − | 6.06218i | 0 | −7.50000 | + | 4.33013i | −8.00000 | 0 | −7.00000 | + | 12.1244i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 36.f | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 324.3.f.b | 2 | |
| 3.b | odd | 2 | 1 | 324.3.f.i | 2 | ||
| 4.b | odd | 2 | 1 | 324.3.f.h | 2 | ||
| 9.c | even | 3 | 1 | 108.3.d.b | yes | 2 | |
| 9.c | even | 3 | 1 | 324.3.f.h | 2 | ||
| 9.d | odd | 6 | 1 | 108.3.d.a | ✓ | 2 | |
| 9.d | odd | 6 | 1 | 324.3.f.c | 2 | ||
| 12.b | even | 2 | 1 | 324.3.f.c | 2 | ||
| 36.f | odd | 6 | 1 | 108.3.d.b | yes | 2 | |
| 36.f | odd | 6 | 1 | inner | 324.3.f.b | 2 | |
| 36.h | even | 6 | 1 | 108.3.d.a | ✓ | 2 | |
| 36.h | even | 6 | 1 | 324.3.f.i | 2 | ||
| 72.j | odd | 6 | 1 | 1728.3.g.a | 2 | ||
| 72.l | even | 6 | 1 | 1728.3.g.a | 2 | ||
| 72.n | even | 6 | 1 | 1728.3.g.f | 2 | ||
| 72.p | odd | 6 | 1 | 1728.3.g.f | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 108.3.d.a | ✓ | 2 | 9.d | odd | 6 | 1 | |
| 108.3.d.a | ✓ | 2 | 36.h | even | 6 | 1 | |
| 108.3.d.b | yes | 2 | 9.c | even | 3 | 1 | |
| 108.3.d.b | yes | 2 | 36.f | odd | 6 | 1 | |
| 324.3.f.b | 2 | 1.a | even | 1 | 1 | trivial | |
| 324.3.f.b | 2 | 36.f | odd | 6 | 1 | inner | |
| 324.3.f.c | 2 | 9.d | odd | 6 | 1 | ||
| 324.3.f.c | 2 | 12.b | even | 2 | 1 | ||
| 324.3.f.h | 2 | 4.b | odd | 2 | 1 | ||
| 324.3.f.h | 2 | 9.c | even | 3 | 1 | ||
| 324.3.f.i | 2 | 3.b | odd | 2 | 1 | ||
| 324.3.f.i | 2 | 36.h | even | 6 | 1 | ||
| 1728.3.g.a | 2 | 72.j | odd | 6 | 1 | ||
| 1728.3.g.a | 2 | 72.l | even | 6 | 1 | ||
| 1728.3.g.f | 2 | 72.n | even | 6 | 1 | ||
| 1728.3.g.f | 2 | 72.p | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(324, [\chi])\):
|
\( T_{5}^{2} - 7T_{5} + 49 \)
|
|
\( T_{7}^{2} + 15T_{7} + 75 \)
|
|
\( T_{11}^{2} + 15T_{11} + 75 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} - 7T + 49 \)
|
| $7$ |
\( T^{2} + 15T + 75 \)
|
| $11$ |
\( T^{2} + 15T + 75 \)
|
| $13$ |
\( T^{2} + 20T + 400 \)
|
| $17$ |
\( (T - 8)^{2} \)
|
| $19$ |
\( T^{2} + 108 \)
|
| $23$ |
\( T^{2} + 6T + 12 \)
|
| $29$ |
\( T^{2} - 10T + 100 \)
|
| $31$ |
\( T^{2} - 93T + 2883 \)
|
| $37$ |
\( (T + 10)^{2} \)
|
| $41$ |
\( T^{2} + 50T + 2500 \)
|
| $43$ |
\( T^{2} - 30T + 300 \)
|
| $47$ |
\( T^{2} + 150T + 7500 \)
|
| $53$ |
\( (T - 47)^{2} \)
|
| $59$ |
\( T^{2} + 60T + 1200 \)
|
| $61$ |
\( T^{2} - 64T + 4096 \)
|
| $67$ |
\( T^{2} + 150T + 7500 \)
|
| $71$ |
\( T^{2} \)
|
| $73$ |
\( (T + 55)^{2} \)
|
| $79$ |
\( T^{2} - 12T + 48 \)
|
| $83$ |
\( T^{2} + 51T + 867 \)
|
| $89$ |
\( (T + 10)^{2} \)
|
| $97$ |
\( T^{2} - 25T + 625 \)
|
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