Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [630,4,Mod(361,630)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(630, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("630.361");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 630.k (of order \(3\), degree \(2\), 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: | \(37.1712033036\) |
| 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 210) |
| 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 \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(281\) | \(451\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 361.1 |
|
−1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | 2.50000 | + | 4.33013i | 0 | 10.0000 | + | 15.5885i | 8.00000 | 0 | 5.00000 | − | 8.66025i | ||||||||||||||
| 541.1 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | 2.50000 | − | 4.33013i | 0 | 10.0000 | − | 15.5885i | 8.00000 | 0 | 5.00000 | + | 8.66025i | |||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 630.4.k.e | 2 | |
| 3.b | odd | 2 | 1 | 210.4.i.f | ✓ | 2 | |
| 7.c | even | 3 | 1 | inner | 630.4.k.e | 2 | |
| 21.g | even | 6 | 1 | 1470.4.a.e | 1 | ||
| 21.h | odd | 6 | 1 | 210.4.i.f | ✓ | 2 | |
| 21.h | odd | 6 | 1 | 1470.4.a.o | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 210.4.i.f | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 210.4.i.f | ✓ | 2 | 21.h | odd | 6 | 1 | |
| 630.4.k.e | 2 | 1.a | even | 1 | 1 | trivial | |
| 630.4.k.e | 2 | 7.c | even | 3 | 1 | inner | |
| 1470.4.a.e | 1 | 21.g | even | 6 | 1 | ||
| 1470.4.a.o | 1 | 21.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(630, [\chi])\):
|
\( T_{11}^{2} - 33T_{11} + 1089 \)
|
|
\( T_{13} + 37 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 2T + 4 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} - 5T + 25 \)
|
| $7$ |
\( T^{2} - 20T + 343 \)
|
| $11$ |
\( T^{2} - 33T + 1089 \)
|
| $13$ |
\( (T + 37)^{2} \)
|
| $17$ |
\( T^{2} - 60T + 3600 \)
|
| $19$ |
\( T^{2} + 119T + 14161 \)
|
| $23$ |
\( T^{2} - 75T + 5625 \)
|
| $29$ |
\( (T - 144)^{2} \)
|
| $31$ |
\( T^{2} - 46T + 2116 \)
|
| $37$ |
\( T^{2} - 199T + 39601 \)
|
| $41$ |
\( (T - 135)^{2} \)
|
| $43$ |
\( (T - 260)^{2} \)
|
| $47$ |
\( T^{2} - 183T + 33489 \)
|
| $53$ |
\( T^{2} - 411T + 168921 \)
|
| $59$ |
\( T^{2} - 492T + 242064 \)
|
| $61$ |
\( T^{2} - 460T + 211600 \)
|
| $67$ |
\( T^{2} + 980T + 960400 \)
|
| $71$ |
\( (T - 306)^{2} \)
|
| $73$ |
\( T^{2} - 934T + 872356 \)
|
| $79$ |
\( T^{2} - 604T + 364816 \)
|
| $83$ |
\( (T + 108)^{2} \)
|
| $89$ |
\( T^{2} + 546T + 298116 \)
|
| $97$ |
\( (T - 1808)^{2} \)
|
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