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: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 5) |
| 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 |
|
−2.00000 | + | 3.46410i | 0 | −4.00000 | − | 6.92820i | −2.50000 | − | 4.33013i | 0 | −3.00000 | + | 5.19615i | 0 | 0 | 20.0000 | ||||||||||||||||
| 271.1 | −2.00000 | − | 3.46410i | 0 | −4.00000 | + | 6.92820i | −2.50000 | + | 4.33013i | 0 | −3.00000 | − | 5.19615i | 0 | 0 | 20.0000 | |||||||||||||||||
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.c | 2 | |
| 3.b | odd | 2 | 1 | 405.4.e.l | 2 | ||
| 9.c | even | 3 | 1 | 45.4.a.d | 1 | ||
| 9.c | even | 3 | 1 | inner | 405.4.e.c | 2 | |
| 9.d | odd | 6 | 1 | 5.4.a.a | ✓ | 1 | |
| 9.d | odd | 6 | 1 | 405.4.e.l | 2 | ||
| 36.f | odd | 6 | 1 | 720.4.a.u | 1 | ||
| 36.h | even | 6 | 1 | 80.4.a.d | 1 | ||
| 45.h | odd | 6 | 1 | 25.4.a.c | 1 | ||
| 45.j | even | 6 | 1 | 225.4.a.b | 1 | ||
| 45.k | odd | 12 | 2 | 225.4.b.c | 2 | ||
| 45.l | even | 12 | 2 | 25.4.b.a | 2 | ||
| 63.i | even | 6 | 1 | 245.4.e.g | 2 | ||
| 63.j | odd | 6 | 1 | 245.4.e.f | 2 | ||
| 63.l | odd | 6 | 1 | 2205.4.a.q | 1 | ||
| 63.n | odd | 6 | 1 | 245.4.e.f | 2 | ||
| 63.o | even | 6 | 1 | 245.4.a.a | 1 | ||
| 63.s | even | 6 | 1 | 245.4.e.g | 2 | ||
| 72.j | odd | 6 | 1 | 320.4.a.g | 1 | ||
| 72.l | even | 6 | 1 | 320.4.a.h | 1 | ||
| 99.g | even | 6 | 1 | 605.4.a.d | 1 | ||
| 117.n | odd | 6 | 1 | 845.4.a.b | 1 | ||
| 144.u | even | 12 | 2 | 1280.4.d.l | 2 | ||
| 144.w | odd | 12 | 2 | 1280.4.d.e | 2 | ||
| 153.i | odd | 6 | 1 | 1445.4.a.a | 1 | ||
| 171.l | even | 6 | 1 | 1805.4.a.h | 1 | ||
| 180.n | even | 6 | 1 | 400.4.a.m | 1 | ||
| 180.v | odd | 12 | 2 | 400.4.c.k | 2 | ||
| 315.z | even | 6 | 1 | 1225.4.a.k | 1 | ||
| 360.bd | even | 6 | 1 | 1600.4.a.s | 1 | ||
| 360.bh | odd | 6 | 1 | 1600.4.a.bi | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.4.a.a | ✓ | 1 | 9.d | odd | 6 | 1 | |
| 25.4.a.c | 1 | 45.h | odd | 6 | 1 | ||
| 25.4.b.a | 2 | 45.l | even | 12 | 2 | ||
| 45.4.a.d | 1 | 9.c | even | 3 | 1 | ||
| 80.4.a.d | 1 | 36.h | even | 6 | 1 | ||
| 225.4.a.b | 1 | 45.j | even | 6 | 1 | ||
| 225.4.b.c | 2 | 45.k | odd | 12 | 2 | ||
| 245.4.a.a | 1 | 63.o | even | 6 | 1 | ||
| 245.4.e.f | 2 | 63.j | odd | 6 | 1 | ||
| 245.4.e.f | 2 | 63.n | odd | 6 | 1 | ||
| 245.4.e.g | 2 | 63.i | even | 6 | 1 | ||
| 245.4.e.g | 2 | 63.s | even | 6 | 1 | ||
| 320.4.a.g | 1 | 72.j | odd | 6 | 1 | ||
| 320.4.a.h | 1 | 72.l | even | 6 | 1 | ||
| 400.4.a.m | 1 | 180.n | even | 6 | 1 | ||
| 400.4.c.k | 2 | 180.v | odd | 12 | 2 | ||
| 405.4.e.c | 2 | 1.a | even | 1 | 1 | trivial | |
| 405.4.e.c | 2 | 9.c | even | 3 | 1 | inner | |
| 405.4.e.l | 2 | 3.b | odd | 2 | 1 | ||
| 405.4.e.l | 2 | 9.d | odd | 6 | 1 | ||
| 605.4.a.d | 1 | 99.g | even | 6 | 1 | ||
| 720.4.a.u | 1 | 36.f | odd | 6 | 1 | ||
| 845.4.a.b | 1 | 117.n | odd | 6 | 1 | ||
| 1225.4.a.k | 1 | 315.z | even | 6 | 1 | ||
| 1280.4.d.e | 2 | 144.w | odd | 12 | 2 | ||
| 1280.4.d.l | 2 | 144.u | even | 12 | 2 | ||
| 1445.4.a.a | 1 | 153.i | odd | 6 | 1 | ||
| 1600.4.a.s | 1 | 360.bd | even | 6 | 1 | ||
| 1600.4.a.bi | 1 | 360.bh | odd | 6 | 1 | ||
| 1805.4.a.h | 1 | 171.l | even | 6 | 1 | ||
| 2205.4.a.q | 1 | 63.l | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(405, [\chi])\):
|
\( T_{2}^{2} + 4T_{2} + 16 \)
|
|
\( T_{7}^{2} + 6T_{7} + 36 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 4T + 16 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} + 5T + 25 \)
|
| $7$ |
\( T^{2} + 6T + 36 \)
|
| $11$ |
\( T^{2} - 32T + 1024 \)
|
| $13$ |
\( T^{2} - 38T + 1444 \)
|
| $17$ |
\( (T + 26)^{2} \)
|
| $19$ |
\( (T - 100)^{2} \)
|
| $23$ |
\( T^{2} + 78T + 6084 \)
|
| $29$ |
\( T^{2} + 50T + 2500 \)
|
| $31$ |
\( T^{2} - 108T + 11664 \)
|
| $37$ |
\( (T - 266)^{2} \)
|
| $41$ |
\( T^{2} - 22T + 484 \)
|
| $43$ |
\( T^{2} + 442T + 195364 \)
|
| $47$ |
\( T^{2} + 514T + 264196 \)
|
| $53$ |
\( (T + 2)^{2} \)
|
| $59$ |
\( T^{2} - 500T + 250000 \)
|
| $61$ |
\( T^{2} - 518T + 268324 \)
|
| $67$ |
\( T^{2} + 126T + 15876 \)
|
| $71$ |
\( (T + 412)^{2} \)
|
| $73$ |
\( (T + 878)^{2} \)
|
| $79$ |
\( T^{2} + 600T + 360000 \)
|
| $83$ |
\( T^{2} - 282T + 79524 \)
|
| $89$ |
\( (T - 150)^{2} \)
|
| $97$ |
\( T^{2} + 386T + 148996 \)
|
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