Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4020,2,Mod(1,4020)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4020, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("4020.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4020.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: | \(32.0998616126\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 28x^{4} - 12x^{3} + 209x^{2} + 360x + 144 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 28x^{4} - 12x^{3} + 209x^{2} + 360x + 144 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5\nu^{5} - 17\nu^{4} - 104\nu^{3} + 180\nu^{2} + 685\nu + 348 ) / 24 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -7\nu^{5} + 19\nu^{4} + 184\nu^{3} - 204\nu^{2} - 1415\nu - 924 ) / 48 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7\nu^{5} - 19\nu^{4} - 160\nu^{3} + 204\nu^{2} + 1055\nu + 492 ) / 24 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -7\nu^{5} + 19\nu^{4} + 160\nu^{3} - 180\nu^{2} - 1103\nu - 708 ) / 24 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 2\beta _1 + 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + 15\beta _1 + 18 \)
|
| \(\nu^{4}\) | \(=\) |
\( 10\beta_{5} + 18\beta_{4} + 6\beta_{3} - 7\beta_{2} + 45\beta _1 + 143 \)
|
| \(\nu^{5}\) | \(=\) |
\( -2\beta_{5} + 46\beta_{4} + 62\beta_{3} - 19\beta_{2} + 256\beta _1 + 467 \)
|
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 | −1.00000 | 0 | 1.00000 | 0 | −3.20623 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.00000 | 0 | 1.00000 | 0 | −2.21014 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.00000 | 0 | 1.00000 | 0 | −1.71821 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.00000 | 0 | 1.00000 | 0 | −0.620094 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −1.00000 | 0 | 1.00000 | 0 | 4.08027 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −1.00000 | 0 | 1.00000 | 0 | 4.67440 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(67\) | \(-1\) |
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 | 4020.2.a.f | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4020.2.a.f | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{6} - T_{7}^{5} - 28T_{7}^{4} - 12T_{7}^{3} + 209T_{7}^{2} + 360T_{7} + 144 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4020))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T + 1)^{6} \)
|
| $5$ |
\( (T - 1)^{6} \)
|
| $7$ |
\( T^{6} - T^{5} + \cdots + 144 \)
|
| $11$ |
\( T^{6} + 7 T^{5} + \cdots + 168 \)
|
| $13$ |
\( T^{6} - 7 T^{5} + \cdots + 218 \)
|
| $17$ |
\( T^{6} - 10 T^{5} + \cdots + 6174 \)
|
| $19$ |
\( T^{6} + 3 T^{5} + \cdots + 608 \)
|
| $23$ |
\( T^{6} + 4 T^{5} + \cdots + 192 \)
|
| $29$ |
\( T^{6} - 9 T^{5} + \cdots + 8628 \)
|
| $31$ |
\( T^{6} - 3 T^{5} + \cdots + 702 \)
|
| $37$ |
\( T^{6} - 16 T^{5} + \cdots - 4316 \)
|
| $41$ |
\( T^{6} - 7 T^{5} + \cdots + 256 \)
|
| $43$ |
\( T^{6} - 3 T^{5} + \cdots + 88 \)
|
| $47$ |
\( T^{6} - T^{5} + \cdots + 1368 \)
|
| $53$ |
\( T^{6} - 7 T^{5} + \cdots + 876 \)
|
| $59$ |
\( T^{6} - 5 T^{5} + \cdots - 1782 \)
|
| $61$ |
\( T^{6} - 15 T^{5} + \cdots + 2892 \)
|
| $67$ |
\( (T - 1)^{6} \)
|
| $71$ |
\( T^{6} + 12 T^{5} + \cdots - 486 \)
|
| $73$ |
\( T^{6} - 12 T^{5} + \cdots - 67356 \)
|
| $79$ |
\( T^{6} - 9 T^{5} + \cdots + 210114 \)
|
| $83$ |
\( T^{6} + 2 T^{5} + \cdots - 813384 \)
|
| $89$ |
\( T^{6} - 10 T^{5} + \cdots - 10108 \)
|
| $97$ |
\( T^{6} - 37 T^{5} + \cdots + 469434 \)
|
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