Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [210,4,Mod(121,210)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(210, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("210.121");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 210.i (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: | \(12.3904011012\) |
| 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: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(71\) | \(127\) |
| \(\chi(n)\) | \(-\zeta_{6}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 121.1 |
|
−1.00000 | + | 1.73205i | −1.50000 | − | 2.59808i | −2.00000 | − | 3.46410i | −2.50000 | + | 4.33013i | 6.00000 | 17.5000 | − | 6.06218i | 8.00000 | −4.50000 | + | 7.79423i | −5.00000 | − | 8.66025i | ||||||||||
| 151.1 | −1.00000 | − | 1.73205i | −1.50000 | + | 2.59808i | −2.00000 | + | 3.46410i | −2.50000 | − | 4.33013i | 6.00000 | 17.5000 | + | 6.06218i | 8.00000 | −4.50000 | − | 7.79423i | −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 | 210.4.i.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | 630.4.k.j | 2 | ||
| 7.c | even | 3 | 1 | inner | 210.4.i.a | ✓ | 2 |
| 7.c | even | 3 | 1 | 1470.4.a.bb | 1 | ||
| 7.d | odd | 6 | 1 | 1470.4.a.q | 1 | ||
| 21.h | odd | 6 | 1 | 630.4.k.j | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 210.4.i.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 210.4.i.a | ✓ | 2 | 7.c | even | 3 | 1 | inner |
| 630.4.k.j | 2 | 3.b | odd | 2 | 1 | ||
| 630.4.k.j | 2 | 21.h | odd | 6 | 1 | ||
| 1470.4.a.q | 1 | 7.d | odd | 6 | 1 | ||
| 1470.4.a.bb | 1 | 7.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{2} - 4T_{11} + 16 \)
acting on \(S_{4}^{\mathrm{new}}(210, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 2T + 4 \)
|
| $3$ |
\( T^{2} + 3T + 9 \)
|
| $5$ |
\( T^{2} + 5T + 25 \)
|
| $7$ |
\( T^{2} - 35T + 343 \)
|
| $11$ |
\( T^{2} - 4T + 16 \)
|
| $13$ |
\( (T + 77)^{2} \)
|
| $17$ |
\( T^{2} - 26T + 676 \)
|
| $19$ |
\( T^{2} - 121T + 14641 \)
|
| $23$ |
\( T^{2} - 166T + 27556 \)
|
| $29$ |
\( (T - 6)^{2} \)
|
| $31$ |
\( T^{2} + 235T + 55225 \)
|
| $37$ |
\( T^{2} - 419T + 175561 \)
|
| $41$ |
\( (T + 128)^{2} \)
|
| $43$ |
\( (T + 291)^{2} \)
|
| $47$ |
\( T^{2} - 442T + 195364 \)
|
| $53$ |
\( T^{2} + 276T + 76176 \)
|
| $59$ |
\( T^{2} - 706T + 498436 \)
|
| $61$ |
\( T^{2} + 442T + 195364 \)
|
| $67$ |
\( T^{2} + 531T + 281961 \)
|
| $71$ |
\( (T - 1036)^{2} \)
|
| $73$ |
\( T^{2} + 427T + 182329 \)
|
| $79$ |
\( T^{2} + 1317 T + 1734489 \)
|
| $83$ |
\( (T + 1188)^{2} \)
|
| $89$ |
\( T^{2} - 38T + 1444 \)
|
| $97$ |
\( (T + 866)^{2} \)
|
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