Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [490,4,Mod(361,490)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(490, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([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("490.361");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 490 = 2 \cdot 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 490.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: | \(28.9109359028\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-59})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 14x^{2} - 15x + 225 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| 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
\( x^{4} - x^{3} - 14x^{2} - 15x + 225 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 14\nu^{2} - 14\nu - 225 ) / 210 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + \nu^{2} + 29\nu ) / 15 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 14\nu - 29 ) / 14 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} + 43\beta _1 + 44 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 28\beta_{3} - 14\beta_{2} + 14\beta _1 + 73 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/490\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(197\) |
| \(\chi(n)\) | \(\beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 361.1 |
|
1.00000 | + | 1.73205i | −4.57603 | + | 7.92592i | −2.00000 | + | 3.46410i | −2.50000 | − | 4.33013i | −18.3041 | 0 | −8.00000 | −28.3802 | − | 49.1559i | 5.00000 | − | 8.66025i | ||||||||||||||||||
| 361.2 | 1.00000 | + | 1.73205i | 2.07603 | − | 3.59580i | −2.00000 | + | 3.46410i | −2.50000 | − | 4.33013i | 8.30413 | 0 | −8.00000 | 4.88017 | + | 8.45270i | 5.00000 | − | 8.66025i | |||||||||||||||||||
| 471.1 | 1.00000 | − | 1.73205i | −4.57603 | − | 7.92592i | −2.00000 | − | 3.46410i | −2.50000 | + | 4.33013i | −18.3041 | 0 | −8.00000 | −28.3802 | + | 49.1559i | 5.00000 | + | 8.66025i | |||||||||||||||||||
| 471.2 | 1.00000 | − | 1.73205i | 2.07603 | + | 3.59580i | −2.00000 | − | 3.46410i | −2.50000 | + | 4.33013i | 8.30413 | 0 | −8.00000 | 4.88017 | − | 8.45270i | 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 | 490.4.e.t | 4 | |
| 7.b | odd | 2 | 1 | 490.4.e.x | 4 | ||
| 7.c | even | 3 | 1 | 490.4.a.u | yes | 2 | |
| 7.c | even | 3 | 1 | inner | 490.4.e.t | 4 | |
| 7.d | odd | 6 | 1 | 490.4.a.p | ✓ | 2 | |
| 7.d | odd | 6 | 1 | 490.4.e.x | 4 | ||
| 35.i | odd | 6 | 1 | 2450.4.a.ca | 2 | ||
| 35.j | even | 6 | 1 | 2450.4.a.bt | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 490.4.a.p | ✓ | 2 | 7.d | odd | 6 | 1 | |
| 490.4.a.u | yes | 2 | 7.c | even | 3 | 1 | |
| 490.4.e.t | 4 | 1.a | even | 1 | 1 | trivial | |
| 490.4.e.t | 4 | 7.c | even | 3 | 1 | inner | |
| 490.4.e.x | 4 | 7.b | odd | 2 | 1 | ||
| 490.4.e.x | 4 | 7.d | odd | 6 | 1 | ||
| 2450.4.a.bt | 2 | 35.j | even | 6 | 1 | ||
| 2450.4.a.ca | 2 | 35.i | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(490, [\chi])\):
|
\( T_{3}^{4} + 5T_{3}^{3} + 63T_{3}^{2} - 190T_{3} + 1444 \)
|
|
\( T_{11}^{4} - 5T_{11}^{3} + 1125T_{11}^{2} + 5500T_{11} + 1210000 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 4)^{2} \)
|
| $3$ |
\( T^{4} + 5 T^{3} + \cdots + 1444 \)
|
| $5$ |
\( (T^{2} + 5 T + 25)^{2} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} - 5 T^{3} + \cdots + 1210000 \)
|
| $13$ |
\( (T^{2} - 95 T + 1858)^{2} \)
|
| $17$ |
\( T^{4} - 25 T^{3} + \cdots + 12544 \)
|
| $19$ |
\( T^{4} - 120 T^{3} + \cdots + 8363664 \)
|
| $23$ |
\( T^{4} + 124 T^{3} + \cdots + 191988736 \)
|
| $29$ |
\( (T^{2} - 213 T - 16314)^{2} \)
|
| $31$ |
\( T^{4} + 70 T^{3} + \cdots + 1098304 \)
|
| $37$ |
\( T^{4} + \cdots + 1248632896 \)
|
| $41$ |
\( (T^{2} - 370 T - 29672)^{2} \)
|
| $43$ |
\( (T^{2} + 82 T - 2744)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 90921547024 \)
|
| $53$ |
\( T^{4} + \cdots + 59752869136 \)
|
| $59$ |
\( T^{4} + \cdots + 7578050704 \)
|
| $61$ |
\( T^{4} + \cdots + 1348945984 \)
|
| $67$ |
\( T^{4} + \cdots + 31046440000 \)
|
| $71$ |
\( (T^{2} - 372 T - 407904)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 238307996224 \)
|
| $79$ |
\( T^{4} + \cdots + 7867690000 \)
|
| $83$ |
\( (T^{2} - 2110 T + 1111432)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 1111759360000 \)
|
| $97$ |
\( (T^{2} - 675 T - 205800)^{2} \)
|
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