Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [230,6,Mod(1,230)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(230, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("230.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 230 = 2 \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 230.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: | \(36.8882785570\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 772x^{3} - 255x^{2} + 13416x + 10080 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 772x^{3} - 255x^{2} + 13416x + 10080 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -29\nu^{4} + 96\nu^{3} + 21494\nu^{2} - 64449\nu - 49872 ) / 3342 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -25\nu^{4} + 198\nu^{3} + 19336\nu^{2} - 138879\nu - 308394 ) / 3342 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -11\nu^{4} - 2\nu^{3} + 8441\nu^{2} + 3884\nu - 102006 ) / 557 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} - 4\beta_{2} - \beta _1 + 308 \)
|
| \(\nu^{3}\) | \(=\) |
\( -7\beta_{4} + 15\beta_{3} + 3\beta_{2} + 730\beta _1 + 147 \)
|
| \(\nu^{4}\) | \(=\) |
\( 718\beta_{4} + 1532\beta_{3} - 3070\beta_{2} - 547\beta _1 + 227048 \)
|
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 |
|
−4.00000 | −26.2974 | 16.0000 | −25.0000 | 105.190 | 152.657 | −64.0000 | 448.555 | 100.000 | |||||||||||||||||||||||||||||||||
| 1.2 | −4.00000 | −2.96404 | 16.0000 | −25.0000 | 11.8561 | −147.425 | −64.0000 | −234.214 | 100.000 | ||||||||||||||||||||||||||||||||||
| 1.3 | −4.00000 | 0.233965 | 16.0000 | −25.0000 | −0.935860 | 203.513 | −64.0000 | −242.945 | 100.000 | ||||||||||||||||||||||||||||||||||
| 1.4 | −4.00000 | 5.40191 | 16.0000 | −25.0000 | −21.6076 | −140.289 | −64.0000 | −213.819 | 100.000 | ||||||||||||||||||||||||||||||||||
| 1.5 | −4.00000 | 28.6256 | 16.0000 | −25.0000 | −114.502 | 61.5435 | −64.0000 | 576.424 | 100.000 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(1\) |
| \(23\) | \(-1\) |
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 | 230.6.a.g | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 230.6.a.g | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} - 5T_{3}^{4} - 762T_{3}^{3} + 2051T_{3}^{2} + 11615T_{3} - 2820 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(230))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{5} \)
|
| $3$ |
\( T^{5} - 5 T^{4} + \cdots - 2820 \)
|
| $5$ |
\( (T + 25)^{5} \)
|
| $7$ |
\( T^{5} + \cdots - 39544118760 \)
|
| $11$ |
\( T^{5} + \cdots + 647661944664 \)
|
| $13$ |
\( T^{5} + \cdots - 30567624422454 \)
|
| $17$ |
\( T^{5} + \cdots - 12\!\cdots\!80 \)
|
| $19$ |
\( T^{5} + \cdots + 893102873667144 \)
|
| $23$ |
\( (T - 529)^{5} \)
|
| $29$ |
\( T^{5} + \cdots - 11\!\cdots\!84 \)
|
| $31$ |
\( T^{5} + \cdots + 92\!\cdots\!85 \)
|
| $37$ |
\( T^{5} + \cdots - 23\!\cdots\!76 \)
|
| $41$ |
\( T^{5} + \cdots - 48\!\cdots\!95 \)
|
| $43$ |
\( T^{5} + \cdots - 11\!\cdots\!80 \)
|
| $47$ |
\( T^{5} + \cdots + 18\!\cdots\!00 \)
|
| $53$ |
\( T^{5} + \cdots - 81\!\cdots\!76 \)
|
| $59$ |
\( T^{5} + \cdots - 39\!\cdots\!80 \)
|
| $61$ |
\( T^{5} + \cdots - 71\!\cdots\!00 \)
|
| $67$ |
\( T^{5} + \cdots + 65\!\cdots\!20 \)
|
| $71$ |
\( T^{5} + \cdots + 16\!\cdots\!65 \)
|
| $73$ |
\( T^{5} + \cdots + 10\!\cdots\!48 \)
|
| $79$ |
\( T^{5} + \cdots + 44\!\cdots\!28 \)
|
| $83$ |
\( T^{5} + \cdots - 86\!\cdots\!16 \)
|
| $89$ |
\( T^{5} + \cdots + 18\!\cdots\!28 \)
|
| $97$ |
\( T^{5} + \cdots + 41\!\cdots\!76 \)
|
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