Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4014,2,Mod(1,4014)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4014, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("4014.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4014 = 2 \cdot 3^{2} \cdot 223 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4014.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: | \(32.0519513713\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.232773917.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 12x^{4} - x^{3} + 33x^{2} + 5x - 22 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1338) |
| 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
\( x^{6} - 12x^{4} - x^{3} + 33x^{2} + 5x - 22 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 10\nu^{3} + 9\nu^{2} + 15\nu - 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 10\nu^{3} + 10\nu^{2} + 15\nu - 15 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{5} - 3\nu^{4} - 31\nu^{3} + 28\nu^{2} + 51\nu - 36 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -5\nu^{5} + 7\nu^{4} + 51\nu^{3} - 66\nu^{2} - 81\nu + 86 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{4} + \beta_{3} + 2\beta_{2} + 6\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 10\beta_{3} - 9\beta_{2} + 26 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} - 19\beta_{4} + 11\beta_{3} + 21\beta_{2} + 45\beta _1 + 11 \)
|
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 |
|
−1.00000 | 0 | 1.00000 | −3.85387 | 0 | −2.71491 | −1.00000 | 0 | 3.85387 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | 0 | 1.00000 | −2.36937 | 0 | 3.68274 | −1.00000 | 0 | 2.36937 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 0 | 1.00000 | −2.29309 | 0 | 0.862607 | −1.00000 | 0 | 2.29309 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0 | 1.00000 | −0.137207 | 0 | 3.14283 | −1.00000 | 0 | 0.137207 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 0 | 1.00000 | 0.720124 | 0 | −2.15975 | −1.00000 | 0 | −0.720124 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 0 | 1.00000 | 1.93341 | 0 | 2.18648 | −1.00000 | 0 | −1.93341 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(223\) | \(-1\) |
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 | 4014.2.a.s | 6 | |
| 3.b | odd | 2 | 1 | 1338.2.a.i | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1338.2.a.i | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 4014.2.a.s | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4014))\):
|
\( T_{5}^{6} + 6T_{5}^{5} + 3T_{5}^{4} - 29T_{5}^{3} - 27T_{5}^{2} + 26T_{5} + 4 \)
|
|
\( T_{7}^{6} - 5T_{7}^{5} - 8T_{7}^{4} + 61T_{7}^{3} - 12T_{7}^{2} - 176T_{7} + 128 \)
|
|
\( T_{11}^{6} + T_{11}^{5} - 26T_{11}^{4} - 80T_{11}^{3} - 56T_{11}^{2} + 13T_{11} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{6} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + 6 T^{5} + \cdots + 4 \)
|
| $7$ |
\( T^{6} - 5 T^{5} + \cdots + 128 \)
|
| $11$ |
\( T^{6} + T^{5} - 26 T^{4} + \cdots + 4 \)
|
| $13$ |
\( T^{6} - 6 T^{5} + \cdots - 40 \)
|
| $17$ |
\( T^{6} + 10 T^{5} + \cdots - 32 \)
|
| $19$ |
\( T^{6} + 4 T^{5} + \cdots - 1084 \)
|
| $23$ |
\( T^{6} + 2 T^{5} + \cdots - 1144 \)
|
| $29$ |
\( T^{6} + 6 T^{5} + \cdots - 200 \)
|
| $31$ |
\( T^{6} + 5 T^{5} + \cdots - 2264 \)
|
| $37$ |
\( T^{6} - T^{5} + \cdots + 6976 \)
|
| $41$ |
\( T^{6} + 12 T^{5} + \cdots - 992 \)
|
| $43$ |
\( T^{6} + 11 T^{5} + \cdots + 10340 \)
|
| $47$ |
\( T^{6} + 5 T^{5} + \cdots - 1021 \)
|
| $53$ |
\( T^{6} + 4 T^{5} + \cdots + 2998 \)
|
| $59$ |
\( T^{6} + T^{5} + \cdots + 88 \)
|
| $61$ |
\( T^{6} + 8 T^{5} + \cdots - 580400 \)
|
| $67$ |
\( T^{6} + 6 T^{5} + \cdots - 2644 \)
|
| $71$ |
\( T^{6} + T^{5} + \cdots - 91520 \)
|
| $73$ |
\( T^{6} + 10 T^{5} + \cdots - 36667 \)
|
| $79$ |
\( T^{6} + 10 T^{5} + \cdots - 485479 \)
|
| $83$ |
\( T^{6} - 6 T^{5} + \cdots + 119152 \)
|
| $89$ |
\( T^{6} - 3 T^{5} + \cdots + 467920 \)
|
| $97$ |
\( T^{6} - 3 T^{5} + \cdots + 4096 \)
|
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