Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2230,2,Mod(1,2230)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2230, 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("2230.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2230 = 2 \cdot 5 \cdot 223 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2230.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: | \(17.8066396507\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.67955408.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 11x^{4} + 30x^{2} + 24x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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
\( x^{6} - x^{5} - 11x^{4} + 30x^{2} + 24x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 8\nu^{3} + 7\nu^{2} + 16\nu + 6 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 11\nu^{3} + 2\nu^{2} + 28\nu + 12 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 9\nu^{3} + 9\nu^{2} + 21\nu + 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{5} + 2\nu^{4} + 9\nu^{3} - 8\nu^{2} - 22\nu - 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{5} + \beta_{4} + 2\beta_{2} + 7\beta _1 + 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( 11\beta_{5} + 8\beta_{4} + 2\beta_{3} + 4\beta_{2} + 14\beta _1 + 29 \)
|
| \(\nu^{5}\) | \(=\) |
\( 31\beta_{5} + 17\beta_{4} + 4\beta_{3} + 26\beta_{2} + 61\beta _1 + 64 \)
|
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 | −3.28573 | 1.00000 | −1.00000 | 3.28573 | −0.542087 | −1.00000 | 7.79605 | 1.00000 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.25031 | 1.00000 | −1.00000 | 2.25031 | −4.12907 | −1.00000 | 2.06389 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 0.233483 | 1.00000 | −1.00000 | −0.233483 | −3.17392 | −1.00000 | −2.94549 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0.784677 | 1.00000 | −1.00000 | −0.784677 | 1.82926 | −1.00000 | −2.38428 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 1.38339 | 1.00000 | −1.00000 | −1.38339 | −2.92782 | −1.00000 | −1.08624 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 2.13450 | 1.00000 | −1.00000 | −2.13450 | 2.94362 | −1.00000 | 1.55608 | 1.00000 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(1\) |
| \(223\) | \(-1\) |
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 | 2230.2.a.p | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2230.2.a.p | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2230))\):
|
\( T_{3}^{6} + T_{3}^{5} - 11T_{3}^{4} + 30T_{3}^{2} - 24T_{3} + 4 \)
|
|
\( T_{7}^{6} + 6T_{7}^{5} - 6T_{7}^{4} - 76T_{7}^{3} - 36T_{7}^{2} + 208T_{7} + 112 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{6} \)
|
| $3$ |
\( T^{6} + T^{5} - 11 T^{4} + \cdots + 4 \)
|
| $5$ |
\( (T + 1)^{6} \)
|
| $7$ |
\( T^{6} + 6 T^{5} + \cdots + 112 \)
|
| $11$ |
\( T^{6} - 14 T^{5} + \cdots - 2804 \)
|
| $13$ |
\( T^{6} - 9 T^{5} + \cdots + 436 \)
|
| $17$ |
\( T^{6} - 4 T^{5} + \cdots + 5264 \)
|
| $19$ |
\( T^{6} - T^{5} + \cdots + 676 \)
|
| $23$ |
\( T^{6} - 9 T^{5} + \cdots - 83 \)
|
| $29$ |
\( T^{6} + 9 T^{5} + \cdots + 25268 \)
|
| $31$ |
\( T^{6} - T^{5} + \cdots - 6859 \)
|
| $37$ |
\( T^{6} + 18 T^{5} + \cdots - 548 \)
|
| $41$ |
\( T^{6} - 11 T^{5} + \cdots + 671 \)
|
| $43$ |
\( T^{6} + 26 T^{5} + \cdots - 13132 \)
|
| $47$ |
\( T^{6} - 14 T^{5} + \cdots + 102464 \)
|
| $53$ |
\( T^{6} - 14 T^{5} + \cdots + 2228 \)
|
| $59$ |
\( T^{6} - 18 T^{5} + \cdots + 44528 \)
|
| $61$ |
\( T^{6} - 24 T^{5} + \cdots + 49628 \)
|
| $67$ |
\( T^{6} + 8 T^{5} + \cdots + 28672 \)
|
| $71$ |
\( T^{6} - 32 T^{5} + \cdots + 44272 \)
|
| $73$ |
\( T^{6} + 12 T^{5} + \cdots - 49456 \)
|
| $79$ |
\( T^{6} - 8 T^{5} + \cdots + 12848 \)
|
| $83$ |
\( T^{6} - 36 T^{5} + \cdots + 1087664 \)
|
| $89$ |
\( T^{6} - 11 T^{5} + \cdots + 4639 \)
|
| $97$ |
\( T^{6} + T^{5} + \cdots + 45179 \)
|
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