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: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 15) |
| 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 |
|
−1.50000 | + | 2.59808i | 0 | −0.500000 | − | 0.866025i | 2.50000 | + | 4.33013i | 0 | −10.0000 | + | 17.3205i | −21.0000 | 0 | −15.0000 | ||||||||||||||||
| 271.1 | −1.50000 | − | 2.59808i | 0 | −0.500000 | + | 0.866025i | 2.50000 | − | 4.33013i | 0 | −10.0000 | − | 17.3205i | −21.0000 | 0 | −15.0000 | |||||||||||||||||
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.d | 2 | |
| 3.b | odd | 2 | 1 | 405.4.e.k | 2 | ||
| 9.c | even | 3 | 1 | 15.4.a.b | ✓ | 1 | |
| 9.c | even | 3 | 1 | inner | 405.4.e.d | 2 | |
| 9.d | odd | 6 | 1 | 45.4.a.b | 1 | ||
| 9.d | odd | 6 | 1 | 405.4.e.k | 2 | ||
| 36.f | odd | 6 | 1 | 240.4.a.f | 1 | ||
| 36.h | even | 6 | 1 | 720.4.a.r | 1 | ||
| 45.h | odd | 6 | 1 | 225.4.a.g | 1 | ||
| 45.j | even | 6 | 1 | 75.4.a.a | 1 | ||
| 45.k | odd | 12 | 2 | 75.4.b.a | 2 | ||
| 45.l | even | 12 | 2 | 225.4.b.d | 2 | ||
| 63.l | odd | 6 | 1 | 735.4.a.i | 1 | ||
| 63.o | even | 6 | 1 | 2205.4.a.c | 1 | ||
| 72.n | even | 6 | 1 | 960.4.a.bi | 1 | ||
| 72.p | odd | 6 | 1 | 960.4.a.l | 1 | ||
| 99.h | odd | 6 | 1 | 1815.4.a.a | 1 | ||
| 180.p | odd | 6 | 1 | 1200.4.a.o | 1 | ||
| 180.x | even | 12 | 2 | 1200.4.f.m | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.4.a.b | ✓ | 1 | 9.c | even | 3 | 1 | |
| 45.4.a.b | 1 | 9.d | odd | 6 | 1 | ||
| 75.4.a.a | 1 | 45.j | even | 6 | 1 | ||
| 75.4.b.a | 2 | 45.k | odd | 12 | 2 | ||
| 225.4.a.g | 1 | 45.h | odd | 6 | 1 | ||
| 225.4.b.d | 2 | 45.l | even | 12 | 2 | ||
| 240.4.a.f | 1 | 36.f | odd | 6 | 1 | ||
| 405.4.e.d | 2 | 1.a | even | 1 | 1 | trivial | |
| 405.4.e.d | 2 | 9.c | even | 3 | 1 | inner | |
| 405.4.e.k | 2 | 3.b | odd | 2 | 1 | ||
| 405.4.e.k | 2 | 9.d | odd | 6 | 1 | ||
| 720.4.a.r | 1 | 36.h | even | 6 | 1 | ||
| 735.4.a.i | 1 | 63.l | odd | 6 | 1 | ||
| 960.4.a.l | 1 | 72.p | odd | 6 | 1 | ||
| 960.4.a.bi | 1 | 72.n | even | 6 | 1 | ||
| 1200.4.a.o | 1 | 180.p | odd | 6 | 1 | ||
| 1200.4.f.m | 2 | 180.x | even | 12 | 2 | ||
| 1815.4.a.a | 1 | 99.h | odd | 6 | 1 | ||
| 2205.4.a.c | 1 | 63.o | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(405, [\chi])\):
|
\( T_{2}^{2} + 3T_{2} + 9 \)
|
|
\( T_{7}^{2} + 20T_{7} + 400 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 3T + 9 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} - 5T + 25 \)
|
| $7$ |
\( T^{2} + 20T + 400 \)
|
| $11$ |
\( T^{2} - 24T + 576 \)
|
| $13$ |
\( T^{2} + 74T + 5476 \)
|
| $17$ |
\( (T - 54)^{2} \)
|
| $19$ |
\( (T + 124)^{2} \)
|
| $23$ |
\( T^{2} - 120T + 14400 \)
|
| $29$ |
\( T^{2} - 78T + 6084 \)
|
| $31$ |
\( T^{2} + 200T + 40000 \)
|
| $37$ |
\( (T + 70)^{2} \)
|
| $41$ |
\( T^{2} + 330T + 108900 \)
|
| $43$ |
\( T^{2} + 92T + 8464 \)
|
| $47$ |
\( T^{2} - 24T + 576 \)
|
| $53$ |
\( (T - 450)^{2} \)
|
| $59$ |
\( T^{2} + 24T + 576 \)
|
| $61$ |
\( T^{2} - 322T + 103684 \)
|
| $67$ |
\( T^{2} - 196T + 38416 \)
|
| $71$ |
\( (T + 288)^{2} \)
|
| $73$ |
\( (T + 430)^{2} \)
|
| $79$ |
\( T^{2} - 520T + 270400 \)
|
| $83$ |
\( T^{2} + 156T + 24336 \)
|
| $89$ |
\( (T - 1026)^{2} \)
|
| $97$ |
\( T^{2} - 286T + 81796 \)
|
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