Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2010,2,Mod(1609,2010)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2010, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2010.1609");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2010 = 2 \cdot 3 \cdot 5 \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2010.e (of order \(2\), degree \(1\), 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: | \(16.0499308063\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.31133896704.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 16x^{6} + 70x^{4} + 84x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 16x^{6} + 70x^{4} + 84x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + 8\nu^{2} + 3 ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 14\nu^{2} + 27 ) / 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} - 16\nu^{5} - 67\nu^{3} - 60\nu ) / 18 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} + 3\nu^{6} + 19\nu^{5} + 45\nu^{4} + 109\nu^{3} + 159\nu^{2} + 195\nu + 81 ) / 36 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - 3\nu^{6} + 19\nu^{5} - 45\nu^{4} + 109\nu^{3} - 159\nu^{2} + 195\nu - 81 ) / 36 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 4\nu^{7} + 61\nu^{5} + 244\nu^{3} + 249\nu ) / 18 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta_{2} - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{5} + 5\beta_{4} - 8\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -8\beta_{3} + 14\beta_{2} + 29 \)
|
| \(\nu^{5}\) | \(=\) |
\( -14\beta_{7} - 8\beta_{6} - 8\beta_{5} - 64\beta_{4} + 67\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -6\beta_{6} + 6\beta_{5} + 67\beta_{3} - 157\beta_{2} - 250 \)
|
| \(\nu^{7}\) | \(=\) |
\( 157\beta_{7} + 61\beta_{6} + 61\beta_{5} + 671\beta_{4} - 596\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2010\mathbb{Z}\right)^\times\).
| \(n\) | \(671\) | \(1141\) | \(1207\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1609.1 |
|
− | 1.00000i | 1.00000i | −1.00000 | −1.74389 | + | 1.39959i | 1.00000 | 4.14348i | 1.00000i | −1.00000 | 1.39959 | + | 1.74389i | |||||||||||||||||||||||||||||||||||||
| 1609.2 | − | 1.00000i | 1.00000i | −1.00000 | −1.24378 | − | 1.85823i | 1.00000 | 0.385557i | 1.00000i | −1.00000 | −1.85823 | + | 1.24378i | ||||||||||||||||||||||||||||||||||||||
| 1609.3 | − | 1.00000i | 1.00000i | −1.00000 | −0.109898 | + | 2.23337i | 1.00000 | 3.34326i | 1.00000i | −1.00000 | 2.23337 | + | 0.109898i | ||||||||||||||||||||||||||||||||||||||
| 1609.4 | − | 1.00000i | 1.00000i | −1.00000 | 2.09757 | − | 0.774733i | 1.00000 | − | 1.87230i | 1.00000i | −1.00000 | −0.774733 | − | 2.09757i | |||||||||||||||||||||||||||||||||||||
| 1609.5 | 1.00000i | − | 1.00000i | −1.00000 | −1.74389 | − | 1.39959i | 1.00000 | − | 4.14348i | − | 1.00000i | −1.00000 | 1.39959 | − | 1.74389i | ||||||||||||||||||||||||||||||||||||
| 1609.6 | 1.00000i | − | 1.00000i | −1.00000 | −1.24378 | + | 1.85823i | 1.00000 | − | 0.385557i | − | 1.00000i | −1.00000 | −1.85823 | − | 1.24378i | ||||||||||||||||||||||||||||||||||||
| 1609.7 | 1.00000i | − | 1.00000i | −1.00000 | −0.109898 | − | 2.23337i | 1.00000 | − | 3.34326i | − | 1.00000i | −1.00000 | 2.23337 | − | 0.109898i | ||||||||||||||||||||||||||||||||||||
| 1609.8 | 1.00000i | − | 1.00000i | −1.00000 | 2.09757 | + | 0.774733i | 1.00000 | 1.87230i | − | 1.00000i | −1.00000 | −0.774733 | + | 2.09757i | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2010.2.e.g | ✓ | 8 |
| 5.b | even | 2 | 1 | inner | 2010.2.e.g | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2010.2.e.g | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 2010.2.e.g | ✓ | 8 | 5.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2010, [\chi])\):
|
\( T_{7}^{8} + 32T_{7}^{6} + 296T_{7}^{4} + 716T_{7}^{2} + 100 \)
|
|
\( T_{11}^{4} + 10T_{11}^{3} + 4T_{11}^{2} - 192T_{11} - 436 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{4} \)
|
| $3$ |
\( (T^{2} + 1)^{4} \)
|
| $5$ |
\( T^{8} + 2 T^{7} + \cdots + 625 \)
|
| $7$ |
\( T^{8} + 32 T^{6} + \cdots + 100 \)
|
| $11$ |
\( (T^{4} + 10 T^{3} + \cdots - 436)^{2} \)
|
| $13$ |
\( T^{8} + 48 T^{6} + \cdots + 36 \)
|
| $17$ |
\( T^{8} + 56 T^{6} + \cdots + 1296 \)
|
| $19$ |
\( (T^{4} - 10 T^{3} + \cdots - 60)^{2} \)
|
| $23$ |
\( T^{8} + 128 T^{6} + \cdots + 669124 \)
|
| $29$ |
\( (T^{4} - 26 T^{3} + \cdots + 1350)^{2} \)
|
| $31$ |
\( (T^{4} + 2 T^{3} - 44 T^{2} + \cdots + 50)^{2} \)
|
| $37$ |
\( T^{8} + 320 T^{6} + \cdots + 12960000 \)
|
| $41$ |
\( (T^{4} + 12 T^{3} + \cdots + 20)^{2} \)
|
| $43$ |
\( (T^{2} + 4)^{4} \)
|
| $47$ |
\( T^{8} + 20 T^{6} + \cdots + 4 \)
|
| $53$ |
\( T^{8} + 64 T^{6} + \cdots + 2304 \)
|
| $59$ |
\( (T^{4} - 28 T^{3} + \cdots - 1340)^{2} \)
|
| $61$ |
\( (T^{4} + 12 T^{3} + \cdots - 734)^{2} \)
|
| $67$ |
\( (T^{2} + 1)^{4} \)
|
| $71$ |
\( (T^{4} - 164 T^{2} + \cdots + 30)^{2} \)
|
| $73$ |
\( T^{8} + 272 T^{6} + \cdots + 246016 \)
|
| $79$ |
\( (T^{4} + 14 T^{3} + \cdots + 3114)^{2} \)
|
| $83$ |
\( T^{8} + 192 T^{6} + \cdots + 102400 \)
|
| $89$ |
\( (T^{4} + 4 T^{3} + \cdots + 16000)^{2} \)
|
| $97$ |
\( T^{8} + 392 T^{6} + \cdots + 31315216 \)
|
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