Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2673,2,Mod(1,2673)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2673, 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("2673.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2673 = 3^{5} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2673.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: | \(21.3440124603\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.864654912.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 8x^{4} + 14x^{3} + 14x^{2} - 16x - 9 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 8x^{4} + 14x^{3} + 14x^{2} - 16x - 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 7\nu^{2} + 5\nu + 7 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 8\nu^{3} + 5\nu^{2} + 11\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 7\beta_{2} + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 9\beta_{3} + 2\beta_{2} + 29\beta _1 + 8 \)
|
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 |
|
−2.39300 | 0 | 3.72646 | 2.72646 | 0 | −1.44554 | −4.13142 | 0 | −6.52442 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.10820 | 0 | −0.771888 | −1.77189 | 0 | 4.26854 | 3.07181 | 0 | 1.96361 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.480247 | 0 | −1.76936 | −2.76936 | 0 | −3.14826 | 1.81022 | 0 | 1.32998 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.35962 | 0 | −0.151441 | −1.15144 | 0 | −1.76201 | −2.92514 | 0 | −1.56552 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.93306 | 0 | 1.73670 | 0.736703 | 0 | 2.75195 | −0.508967 | 0 | 1.42409 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.68878 | 0 | 5.22953 | 4.22953 | 0 | −2.66468 | 8.68349 | 0 | 11.3723 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(11\) | \(-1\) |
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 | 2673.2.a.p | yes | 6 |
| 3.b | odd | 2 | 1 | 2673.2.a.j | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2673.2.a.j | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 2673.2.a.p | yes | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2673))\):
|
\( T_{2}^{6} - 2T_{2}^{5} - 8T_{2}^{4} + 14T_{2}^{3} + 14T_{2}^{2} - 16T_{2} - 9 \)
|
|
\( T_{5}^{6} - 2T_{5}^{5} - 17T_{5}^{4} + 14T_{5}^{3} + 77T_{5}^{2} + 8T_{5} - 48 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 2 T^{5} + \cdots - 9 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - 2 T^{5} + \cdots - 48 \)
|
| $7$ |
\( T^{6} + 2 T^{5} + \cdots + 251 \)
|
| $11$ |
\( (T - 1)^{6} \)
|
| $13$ |
\( T^{6} + 8 T^{5} + \cdots - 1 \)
|
| $17$ |
\( T^{6} - 2 T^{5} + \cdots - 132 \)
|
| $19$ |
\( T^{6} + 4 T^{5} + \cdots + 1647 \)
|
| $23$ |
\( T^{6} - 10 T^{5} + \cdots + 2256 \)
|
| $29$ |
\( T^{6} - 6 T^{5} + \cdots - 36 \)
|
| $31$ |
\( T^{6} + 8 T^{5} + \cdots + 281 \)
|
| $37$ |
\( T^{6} - 2 T^{5} + \cdots - 48903 \)
|
| $41$ |
\( T^{6} + 4 T^{5} + \cdots - 9216 \)
|
| $43$ |
\( T^{6} - 2 T^{5} + \cdots - 40368 \)
|
| $47$ |
\( T^{6} - 28 T^{5} + \cdots - 14304 \)
|
| $53$ |
\( T^{6} - 24 T^{5} + \cdots + 2772 \)
|
| $59$ |
\( T^{6} + 8 T^{5} + \cdots - 144576 \)
|
| $61$ |
\( T^{6} - 2 T^{5} + \cdots + 8076 \)
|
| $67$ |
\( T^{6} - 12 T^{5} + \cdots + 6352 \)
|
| $71$ |
\( T^{6} - 30 T^{5} + \cdots + 133812 \)
|
| $73$ |
\( T^{6} - 16 T^{5} + \cdots + 41024 \)
|
| $79$ |
\( T^{6} + 12 T^{5} + \cdots - 5273 \)
|
| $83$ |
\( T^{6} - 291 T^{4} + \cdots - 250452 \)
|
| $89$ |
\( T^{6} + 6 T^{5} + \cdots - 2124 \)
|
| $97$ |
\( T^{6} + 18 T^{5} + \cdots - 178064 \)
|
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