Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2031,1,Mod(2030,2031)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2031, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2031.2030");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2031 = 3 \cdot 677 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2031.d (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: | yes |
| Analytic conductor: | \(1.01360104066\) |
| Analytic rank: | \(0\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\Q(\zeta_{38})^+\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - x^{8} - 8x^{7} + 7x^{6} + 21x^{5} - 15x^{4} - 20x^{3} + 10x^{2} + 5x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{19}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{19} - \cdots)\) |
$q$-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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of \(\nu = \zeta_{38} + \zeta_{38}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 4\nu^{2} + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 5\nu^{3} + 5\nu \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - 6\nu^{4} + 9\nu^{2} - 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{7} - 7\nu^{5} + 14\nu^{3} - 7\nu \)
|
| \(\beta_{8}\) | \(=\) |
\( \nu^{8} - 8\nu^{6} + 20\nu^{4} - 16\nu^{2} + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 4\beta_{2} + 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 5\beta_{3} + 10\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} + 6\beta_{4} + 15\beta_{2} + 20 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{7} + 7\beta_{5} + 21\beta_{3} + 35\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( \beta_{8} + 8\beta_{6} + 28\beta_{4} + 56\beta_{2} + 70 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2031\mathbb{Z}\right)^\times\).
| \(n\) | \(679\) | \(1355\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2030.1 |
|
−1.89163 | −1.00000 | 2.57828 | 1.35456 | 1.89163 | 0 | −2.98553 | 1.00000 | −2.56234 | |||||||||||||||||||||||||||||||||||||||||||||
| 2030.2 | −1.57828 | −1.00000 | 1.49097 | 0.165159 | 1.57828 | 0 | −0.774890 | 1.00000 | −0.260667 | ||||||||||||||||||||||||||||||||||||||||||||||
| 2030.3 | −1.09390 | −1.00000 | 0.196609 | −1.57828 | 1.09390 | 0 | 0.878826 | 1.00000 | 1.72648 | ||||||||||||||||||||||||||||||||||||||||||||||
| 2030.4 | −0.490971 | −1.00000 | −0.758948 | 1.97272 | 0.490971 | 0 | 0.863592 | 1.00000 | −0.968550 | ||||||||||||||||||||||||||||||||||||||||||||||
| 2030.5 | 0.165159 | −1.00000 | −0.972723 | −1.09390 | −0.165159 | 0 | −0.325812 | 1.00000 | −0.180666 | ||||||||||||||||||||||||||||||||||||||||||||||
| 2030.6 | 0.803391 | −1.00000 | −0.354563 | −0.490971 | −0.803391 | 0 | −1.08824 | 1.00000 | −0.394442 | ||||||||||||||||||||||||||||||||||||||||||||||
| 2030.7 | 1.35456 | −1.00000 | 0.834841 | 1.75895 | −1.35456 | 0 | −0.223718 | 1.00000 | 2.38261 | ||||||||||||||||||||||||||||||||||||||||||||||
| 2030.8 | 1.75895 | −1.00000 | 2.09390 | −1.89163 | −1.75895 | 0 | 1.92411 | 1.00000 | −3.32729 | ||||||||||||||||||||||||||||||||||||||||||||||
| 2030.9 | 1.97272 | −1.00000 | 2.89163 | 0.803391 | −1.97272 | 0 | 3.73167 | 1.00000 | 1.58487 | ||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 2031.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-2031}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2031.1.d.b | yes | 9 |
| 3.b | odd | 2 | 1 | 2031.1.d.a | ✓ | 9 | |
| 677.b | even | 2 | 1 | 2031.1.d.a | ✓ | 9 | |
| 2031.d | odd | 2 | 1 | CM | 2031.1.d.b | yes | 9 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2031.1.d.a | ✓ | 9 | 3.b | odd | 2 | 1 | |
| 2031.1.d.a | ✓ | 9 | 677.b | even | 2 | 1 | |
| 2031.1.d.b | yes | 9 | 1.a | even | 1 | 1 | trivial |
| 2031.1.d.b | yes | 9 | 2031.d | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{9} - T_{2}^{8} - 8T_{2}^{7} + 7T_{2}^{6} + 21T_{2}^{5} - 15T_{2}^{4} - 20T_{2}^{3} + 10T_{2}^{2} + 5T_{2} - 1 \)
acting on \(S_{1}^{\mathrm{new}}(2031, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} - T^{8} - 8 T^{7} + \cdots - 1 \)
|
| $3$ |
\( (T + 1)^{9} \)
|
| $5$ |
\( T^{9} - T^{8} - 8 T^{7} + \cdots - 1 \)
|
| $7$ |
\( T^{9} \)
|
| $11$ |
\( T^{9} - T^{8} - 8 T^{7} + \cdots - 1 \)
|
| $13$ |
\( T^{9} + T^{8} - 8 T^{7} + \cdots + 1 \)
|
| $17$ |
\( T^{9} - T^{8} - 8 T^{7} + \cdots - 1 \)
|
| $19$ |
\( T^{9} \)
|
| $23$ |
\( T^{9} - T^{8} - 8 T^{7} + \cdots - 1 \)
|
| $29$ |
\( T^{9} - T^{8} - 8 T^{7} + \cdots - 1 \)
|
| $31$ |
\( T^{9} \)
|
| $37$ |
\( T^{9} + T^{8} - 8 T^{7} + \cdots + 1 \)
|
| $41$ |
\( T^{9} \)
|
| $43$ |
\( T^{9} \)
|
| $47$ |
\( T^{9} - T^{8} - 8 T^{7} + \cdots - 1 \)
|
| $53$ |
\( T^{9} - T^{8} - 8 T^{7} + \cdots - 1 \)
|
| $59$ |
\( T^{9} \)
|
| $61$ |
\( T^{9} \)
|
| $67$ |
\( T^{9} \)
|
| $71$ |
\( T^{9} \)
|
| $73$ |
\( T^{9} \)
|
| $79$ |
\( T^{9} + T^{8} - 8 T^{7} + \cdots + 1 \)
|
| $83$ |
\( T^{9} - T^{8} - 8 T^{7} + \cdots - 1 \)
|
| $89$ |
\( T^{9} - T^{8} - 8 T^{7} + \cdots - 1 \)
|
| $97$ |
\( T^{9} + T^{8} - 8 T^{7} + \cdots + 1 \)
|
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