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: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-5})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 5x^{2} + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| 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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} - 5x^{2} + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{3} ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( 5\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{3} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(71\) | \(127\) |
| \(\chi(n)\) | \(-1 + \beta_{2}\) | \(1\) | \(1\) |
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 | −10.8095 | − | 15.0385i | −8.00000 | −4.50000 | + | 7.79423i | −5.00000 | − | 8.66025i | ||||||||||||||||
| 121.2 | 1.00000 | − | 1.73205i | 1.50000 | + | 2.59808i | −2.00000 | − | 3.46410i | 2.50000 | − | 4.33013i | 6.00000 | 0.809475 | + | 18.5026i | −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 | −10.8095 | + | 15.0385i | −8.00000 | −4.50000 | − | 7.79423i | −5.00000 | + | 8.66025i | |||||||||||||||||
| 151.2 | 1.00000 | + | 1.73205i | 1.50000 | − | 2.59808i | −2.00000 | + | 3.46410i | 2.50000 | + | 4.33013i | 6.00000 | 0.809475 | − | 18.5026i | −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.j | ✓ | 4 |
| 3.b | odd | 2 | 1 | 630.4.k.k | 4 | ||
| 7.c | even | 3 | 1 | inner | 210.4.i.j | ✓ | 4 |
| 7.c | even | 3 | 1 | 1470.4.a.be | 2 | ||
| 7.d | odd | 6 | 1 | 1470.4.a.bj | 2 | ||
| 21.h | odd | 6 | 1 | 630.4.k.k | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 210.4.i.j | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 210.4.i.j | ✓ | 4 | 7.c | even | 3 | 1 | inner |
| 630.4.k.k | 4 | 3.b | odd | 2 | 1 | ||
| 630.4.k.k | 4 | 21.h | odd | 6 | 1 | ||
| 1470.4.a.be | 2 | 7.c | even | 3 | 1 | ||
| 1470.4.a.bj | 2 | 7.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{4} - 18T_{11}^{3} + 1458T_{11}^{2} + 20412T_{11} + 1285956 \)
acting on \(S_{4}^{\mathrm{new}}(210, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 4)^{2} \)
|
| $3$ |
\( (T^{2} - 3 T + 9)^{2} \)
|
| $5$ |
\( (T^{2} - 5 T + 25)^{2} \)
|
| $7$ |
\( T^{4} + 20 T^{3} + \cdots + 117649 \)
|
| $11$ |
\( T^{4} - 18 T^{3} + \cdots + 1285956 \)
|
| $13$ |
\( (T^{2} - 64 T - 191)^{2} \)
|
| $17$ |
\( T^{4} + 102 T^{3} + \cdots + 1920996 \)
|
| $19$ |
\( T^{4} - 62 T^{3} + \cdots + 1437601 \)
|
| $23$ |
\( T^{4} + 42 T^{3} + \cdots + 38118276 \)
|
| $29$ |
\( (T^{2} - 222 T + 5706)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 5949808225 \)
|
| $37$ |
\( T^{4} + \cdots + 5652182761 \)
|
| $41$ |
\( (T^{2} - 54 T - 486)^{2} \)
|
| $43$ |
\( (T^{2} + 248 T + 12001)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 2220671376 \)
|
| $53$ |
\( T^{4} + \cdots + 24343488576 \)
|
| $59$ |
\( T^{4} + \cdots + 65912346756 \)
|
| $61$ |
\( T^{4} + 64 T^{3} + \cdots + 58003456 \)
|
| $67$ |
\( T^{4} + \cdots + 7218031681 \)
|
| $71$ |
\( (T^{2} + 78 T - 271854)^{2} \)
|
| $73$ |
\( T^{4} - 464 T^{3} + \cdots + 549855601 \)
|
| $79$ |
\( T^{4} + \cdots + 892210595761 \)
|
| $83$ |
\( (T^{2} + 234 T - 961686)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 468176166756 \)
|
| $97$ |
\( (T^{2} - 832 T - 42944)^{2} \)
|
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