Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1849,2,Mod(1,1849)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1849, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1849.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1849 = 43^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1849.a (trivial) |
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: | \(14.7643393337\) |
| Analytic rank: | \(1\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\Q(\zeta_{44})^+\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 11x^{8} + 44x^{6} - 77x^{4} + 55x^{2} - 11 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of \(\nu = \zeta_{44} + \zeta_{44}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 4\nu^{2} + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 5\nu^{3} + 5\nu \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - 6\nu^{4} + 9\nu^{2} - 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{7} - 7\nu^{5} + 14\nu^{3} - 7\nu \)
|
| \(\beta_{8}\) | \(=\) |
\( \nu^{8} - 9\nu^{6} + 27\nu^{4} - 30\nu^{2} + 9 \)
|
| \(\beta_{9}\) | \(=\) |
\( \nu^{9} - 9\nu^{7} + 27\nu^{5} - 30\nu^{3} + 9\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 4\beta_{2} + 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 5\beta_{3} + 10\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} + 6\beta_{4} + 15\beta_{2} + 20 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{7} + 7\beta_{5} + 21\beta_{3} + 35\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( \beta_{8} + 9\beta_{6} + 27\beta_{4} + 57\beta_{2} + 69 \)
|
| \(\nu^{9}\) | \(=\) |
\( \beta_{9} + 9\beta_{7} + 36\beta_{5} + 84\beta_{3} + 126\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−1.97964 | −0.160379 | 1.91899 | 3.06092 | 0.317493 | −2.43240 | 0.160379 | −2.97428 | −6.05954 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.81926 | −1.25580 | 1.30972 | −0.160379 | 2.28463 | 4.31672 | 1.25580 | −1.42297 | 0.291772 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.51150 | −2.59278 | 0.284630 | 2.07496 | 3.91899 | 3.84858 | 2.59278 | 3.72251 | −3.13631 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.08128 | −3.06092 | −0.830830 | 2.59278 | 3.30972 | 0.985960 | 3.06092 | 6.36926 | −2.80353 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.563465 | −2.07496 | −1.68251 | −1.25580 | 1.16917 | 1.91459 | 2.07496 | 1.30548 | 0.707599 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.563465 | 2.07496 | −1.68251 | 1.25580 | 1.16917 | −1.91459 | −2.07496 | 1.30548 | 0.707599 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.08128 | 3.06092 | −0.830830 | −2.59278 | 3.30972 | −0.985960 | −3.06092 | 6.36926 | −2.80353 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.51150 | 2.59278 | 0.284630 | −2.07496 | 3.91899 | −3.84858 | −2.59278 | 3.72251 | −3.13631 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.81926 | 1.25580 | 1.30972 | 0.160379 | 2.28463 | −4.31672 | −1.25580 | −1.42297 | 0.291772 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.97964 | 0.160379 | 1.91899 | −3.06092 | 0.317493 | 2.43240 | −0.160379 | −2.97428 | −6.05954 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(43\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 43.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1849.2.a.m | ✓ | 10 |
| 43.b | odd | 2 | 1 | inner | 1849.2.a.m | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1849.2.a.m | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 1849.2.a.m | ✓ | 10 | 43.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} - 11T_{2}^{8} + 44T_{2}^{6} - 77T_{2}^{4} + 55T_{2}^{2} - 11 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1849))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 11 T^{8} + \cdots - 11 \)
|
| $3$ |
\( T^{10} - 22 T^{8} + \cdots - 11 \)
|
| $5$ |
\( T^{10} - 22 T^{8} + \cdots - 11 \)
|
| $7$ |
\( T^{10} - 44 T^{8} + \cdots - 5819 \)
|
| $11$ |
\( (T^{5} + 5 T^{4} - T^{3} + \cdots + 1)^{2} \)
|
| $13$ |
\( (T^{5} + 7 T^{4} - 9 T^{3} + \cdots - 23)^{2} \)
|
| $17$ |
\( (T^{5} + 8 T^{4} + \cdots + 1627)^{2} \)
|
| $19$ |
\( T^{10} - 44 T^{8} + \cdots - 20339 \)
|
| $23$ |
\( (T^{5} + 9 T^{4} + \cdots - 2617)^{2} \)
|
| $29$ |
\( T^{10} - 198 T^{8} + \cdots - 130691 \)
|
| $31$ |
\( (T^{5} + T^{4} - 81 T^{3} + \cdots + 463)^{2} \)
|
| $37$ |
\( T^{10} + \cdots - 3152129024 \)
|
| $41$ |
\( (T^{5} - 77 T^{3} + \cdots + 2189)^{2} \)
|
| $43$ |
\( T^{10} \)
|
| $47$ |
\( (T^{5} + 9 T^{4} - 16 T^{3} + \cdots + 23)^{2} \)
|
| $53$ |
\( (T^{5} - T^{4} - 48 T^{3} + \cdots - 661)^{2} \)
|
| $59$ |
\( (T^{5} - 7 T^{4} + \cdots + 419)^{2} \)
|
| $61$ |
\( T^{10} - 231 T^{8} + \cdots - 37525499 \)
|
| $67$ |
\( (T^{5} - 13 T^{4} + \cdots - 1297)^{2} \)
|
| $71$ |
\( T^{10} - 374 T^{8} + \cdots - 16668971 \)
|
| $73$ |
\( T^{10} - 121 T^{8} + \cdots - 20339 \)
|
| $79$ |
\( (T^{5} + 28 T^{4} + \cdots + 116401)^{2} \)
|
| $83$ |
\( (T^{5} + 19 T^{4} + \cdots + 593)^{2} \)
|
| $89$ |
\( T^{10} - 121 T^{8} + \cdots - 20339 \)
|
| $97$ |
\( (T^{5} - 17 T^{4} + \cdots - 3541)^{2} \)
|
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