Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4016,2,Mod(1,4016)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4016, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4016.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4016 = 2^{4} \cdot 251 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4016.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: | \(32.0679214517\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.60853001.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} - 8x^{4} + 15x^{3} + 20x^{2} - 12x - 5 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 502) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} - 8x^{4} + 15x^{3} + 20x^{2} - 12x - 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 4\nu^{4} + 4\nu^{3} - 15\nu^{2} - 9\nu + 1 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 4\nu^{4} - 4\nu^{3} + 19\nu^{2} + 5\nu - 17 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 6\nu^{4} + 6\nu^{3} + 17\nu^{2} - 27\nu - 3 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 3\nu^{4} - 7\nu^{3} + 12\nu^{2} + 15\nu - 4 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta_{3} + 4\beta_{2} + 7\beta _1 + 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5\beta_{5} + 3\beta_{4} + 6\beta_{3} + 19\beta_{2} + 18\beta _1 + 33 \)
|
| \(\nu^{5}\) | \(=\) |
\( 24\beta_{5} + 16\beta_{4} + 13\beta_{3} + 73\beta_{2} + 76\beta _1 + 97 \)
|
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 |
|
0 | −3.20334 | 0 | 1.15583 | 0 | 3.99175 | 0 | 7.26139 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.70836 | 0 | −2.61208 | 0 | −1.43044 | 0 | 4.33524 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.28113 | 0 | 0.633107 | 0 | −3.19968 | 0 | −1.35870 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.63228 | 0 | 2.87032 | 0 | −4.63886 | 0 | −0.335646 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.68934 | 0 | 0.420498 | 0 | −0.389856 | 0 | −0.146119 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.87121 | 0 | −3.46767 | 0 | −0.332919 | 0 | 5.24384 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(251\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 4016.2.a.f | 6 | |
| 4.b | odd | 2 | 1 | 502.2.a.e | ✓ | 6 | |
| 12.b | even | 2 | 1 | 4518.2.a.x | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 502.2.a.e | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 4016.2.a.f | 6 | 1.a | even | 1 | 1 | trivial | |
| 4518.2.a.x | 6 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + T_{3}^{5} - 16T_{3}^{4} - 9T_{3}^{3} + 74T_{3}^{2} + 8T_{3} - 88 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4016))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + T^{5} + \cdots - 88 \)
|
| $5$ |
\( T^{6} + T^{5} - 14 T^{4} + \cdots + 8 \)
|
| $7$ |
\( T^{6} + 6 T^{5} + \cdots - 11 \)
|
| $11$ |
\( T^{6} + T^{5} + \cdots - 832 \)
|
| $13$ |
\( T^{6} + 5 T^{5} + \cdots + 392 \)
|
| $17$ |
\( T^{6} - 8 T^{5} + \cdots - 1 \)
|
| $19$ |
\( T^{6} + 3 T^{5} + \cdots + 232 \)
|
| $23$ |
\( T^{6} + 18 T^{5} + \cdots + 539 \)
|
| $29$ |
\( T^{6} - T^{5} + \cdots - 35656 \)
|
| $31$ |
\( T^{6} + 6 T^{5} + \cdots - 6259 \)
|
| $37$ |
\( T^{6} + 13 T^{5} + \cdots + 8 \)
|
| $41$ |
\( T^{6} - 4 T^{5} + \cdots - 2233 \)
|
| $43$ |
\( T^{6} - 5 T^{5} + \cdots - 21992 \)
|
| $47$ |
\( T^{6} + 8 T^{5} + \cdots + 16448 \)
|
| $53$ |
\( T^{6} + 3 T^{5} + \cdots - 3448 \)
|
| $59$ |
\( T^{6} - 5 T^{5} + \cdots + 59624 \)
|
| $61$ |
\( T^{6} + 61 T^{5} + \cdots + 805288 \)
|
| $67$ |
\( T^{6} - 13 T^{5} + \cdots + 408920 \)
|
| $71$ |
\( T^{6} + 22 T^{5} + \cdots - 54848 \)
|
| $73$ |
\( T^{6} - 6 T^{5} + \cdots - 796103 \)
|
| $79$ |
\( T^{6} + 28 T^{5} + \cdots - 3023 \)
|
| $83$ |
\( T^{6} + 14 T^{5} + \cdots + 799232 \)
|
| $89$ |
\( T^{6} - 18 T^{5} + \cdots + 33517 \)
|
| $97$ |
\( T^{6} - 16 T^{5} + \cdots - 330176 \)
|
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