Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [405,4,Mod(136,405)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(405, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("405.136");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 405 = 3^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 405.e (of order \(3\), 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: | \(23.8957735523\) |
| 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: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 135) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).
| \(n\) | \(82\) | \(326\) |
| \(\chi(n)\) | \(1\) | \(-\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 136.1 |
|
0.500000 | − | 0.866025i | 0 | 3.50000 | + | 6.06218i | 2.50000 | + | 4.33013i | 0 | 3.00000 | − | 5.19615i | 15.0000 | 0 | 5.00000 | ||||||||||||||||
| 271.1 | 0.500000 | + | 0.866025i | 0 | 3.50000 | − | 6.06218i | 2.50000 | − | 4.33013i | 0 | 3.00000 | + | 5.19615i | 15.0000 | 0 | 5.00000 | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 405.4.e.h | 2 | |
| 3.b | odd | 2 | 1 | 405.4.e.f | 2 | ||
| 9.c | even | 3 | 1 | 135.4.a.b | ✓ | 1 | |
| 9.c | even | 3 | 1 | inner | 405.4.e.h | 2 | |
| 9.d | odd | 6 | 1 | 135.4.a.c | yes | 1 | |
| 9.d | odd | 6 | 1 | 405.4.e.f | 2 | ||
| 36.f | odd | 6 | 1 | 2160.4.a.f | 1 | ||
| 36.h | even | 6 | 1 | 2160.4.a.p | 1 | ||
| 45.h | odd | 6 | 1 | 675.4.a.c | 1 | ||
| 45.j | even | 6 | 1 | 675.4.a.h | 1 | ||
| 45.k | odd | 12 | 2 | 675.4.b.f | 2 | ||
| 45.l | even | 12 | 2 | 675.4.b.e | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 135.4.a.b | ✓ | 1 | 9.c | even | 3 | 1 | |
| 135.4.a.c | yes | 1 | 9.d | odd | 6 | 1 | |
| 405.4.e.f | 2 | 3.b | odd | 2 | 1 | ||
| 405.4.e.f | 2 | 9.d | odd | 6 | 1 | ||
| 405.4.e.h | 2 | 1.a | even | 1 | 1 | trivial | |
| 405.4.e.h | 2 | 9.c | even | 3 | 1 | inner | |
| 675.4.a.c | 1 | 45.h | odd | 6 | 1 | ||
| 675.4.a.h | 1 | 45.j | even | 6 | 1 | ||
| 675.4.b.e | 2 | 45.l | even | 12 | 2 | ||
| 675.4.b.f | 2 | 45.k | odd | 12 | 2 | ||
| 2160.4.a.f | 1 | 36.f | odd | 6 | 1 | ||
| 2160.4.a.p | 1 | 36.h | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(405, [\chi])\):
|
\( T_{2}^{2} - T_{2} + 1 \)
|
|
\( T_{7}^{2} - 6T_{7} + 36 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - T + 1 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} - 5T + 25 \)
|
| $7$ |
\( T^{2} - 6T + 36 \)
|
| $11$ |
\( T^{2} + 47T + 2209 \)
|
| $13$ |
\( T^{2} - 5T + 25 \)
|
| $17$ |
\( (T - 131)^{2} \)
|
| $19$ |
\( (T + 56)^{2} \)
|
| $23$ |
\( T^{2} - 3T + 9 \)
|
| $29$ |
\( T^{2} + 157T + 24649 \)
|
| $31$ |
\( T^{2} + 225T + 50625 \)
|
| $37$ |
\( (T + 70)^{2} \)
|
| $41$ |
\( T^{2} - 140T + 19600 \)
|
| $43$ |
\( T^{2} + 397T + 157609 \)
|
| $47$ |
\( T^{2} + 347T + 120409 \)
|
| $53$ |
\( (T + 4)^{2} \)
|
| $59$ |
\( T^{2} - 748T + 559504 \)
|
| $61$ |
\( T^{2} - 338T + 114244 \)
|
| $67$ |
\( T^{2} + 492T + 242064 \)
|
| $71$ |
\( (T + 32)^{2} \)
|
| $73$ |
\( (T - 970)^{2} \)
|
| $79$ |
\( T^{2} - 1257 T + 1580049 \)
|
| $83$ |
\( T^{2} + 102T + 10404 \)
|
| $89$ |
\( (T - 1488)^{2} \)
|
| $97$ |
\( T^{2} + 974T + 948676 \)
|
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