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: |
\( x^{5} - x^{4} - 774x^{3} - 197x^{2} + 66287x + 154128 \)
|
| 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
\( x^{5} - x^{4} - 774x^{3} - 197x^{2} + 66287x + 154128 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{4} - 13\nu^{3} - 1235\nu^{2} + 6295\nu + 13806 ) / 897 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} + 18\nu^{3} + 698\nu^{2} - 10657\nu - 37102 ) / 598 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -13\nu^{4} + 50\nu^{3} + 9580\nu^{2} - 25565\nu - 555282 ) / 1794 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 4\beta_{2} + 4\beta _1 + 310 \)
|
| \(\nu^{3}\) | \(=\) |
\( -7\beta_{4} + 45\beta_{3} + 11\beta_{2} + 625\beta _1 + 456 \)
|
| \(\nu^{4}\) | \(=\) |
\( 572\beta_{4} + 910\beta_{3} + 2990\beta_{2} + 3385\beta _1 + 187486 \)
|
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 | −25.4526 | 16.0000 | 25.0000 | 101.810 | 169.151 | −64.0000 | 404.834 | −100.000 | |||||||||||||||||||||||||||||||||
| 1.2 | −4.00000 | −8.46823 | 16.0000 | 25.0000 | 33.8729 | −118.977 | −64.0000 | −171.289 | −100.000 | ||||||||||||||||||||||||||||||||||
| 1.3 | −4.00000 | −2.48374 | 16.0000 | 25.0000 | 9.93494 | −252.275 | −64.0000 | −236.831 | −100.000 | ||||||||||||||||||||||||||||||||||
| 1.4 | −4.00000 | 10.8366 | 16.0000 | 25.0000 | −43.3465 | 46.1561 | −64.0000 | −125.567 | −100.000 | ||||||||||||||||||||||||||||||||||
| 1.5 | −4.00000 | 26.5679 | 16.0000 | 25.0000 | −106.272 | 257.945 | −64.0000 | 462.854 | −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.f | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 230.6.a.f | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} - T_{3}^{4} - 774T_{3}^{3} - 197T_{3}^{2} + 66287T_{3} + 154128 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(230))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{5} \)
|
| $3$ |
\( T^{5} - T^{4} + \cdots + 154128 \)
|
| $5$ |
\( (T - 25)^{5} \)
|
| $7$ |
\( T^{5} + \cdots - 60445984556 \)
|
| $11$ |
\( T^{5} + \cdots + 216794838000 \)
|
| $13$ |
\( T^{5} + \cdots + 15847720477654 \)
|
| $17$ |
\( T^{5} + \cdots + 151348597947360 \)
|
| $19$ |
\( T^{5} + \cdots - 57\!\cdots\!92 \)
|
| $23$ |
\( (T + 529)^{5} \)
|
| $29$ |
\( T^{5} + \cdots - 75\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots - 15\!\cdots\!15 \)
|
| $37$ |
\( T^{5} + \cdots - 31\!\cdots\!08 \)
|
| $41$ |
\( T^{5} + \cdots + 378692981425953 \)
|
| $43$ |
\( T^{5} + \cdots + 45\!\cdots\!52 \)
|
| $47$ |
\( T^{5} + \cdots - 87\!\cdots\!76 \)
|
| $53$ |
\( T^{5} + \cdots + 76\!\cdots\!48 \)
|
| $59$ |
\( T^{5} + \cdots + 87\!\cdots\!32 \)
|
| $61$ |
\( T^{5} + \cdots + 39\!\cdots\!00 \)
|
| $67$ |
\( T^{5} + \cdots - 43\!\cdots\!16 \)
|
| $71$ |
\( T^{5} + \cdots - 11\!\cdots\!87 \)
|
| $73$ |
\( T^{5} + \cdots + 20\!\cdots\!48 \)
|
| $79$ |
\( T^{5} + \cdots - 21\!\cdots\!88 \)
|
| $83$ |
\( T^{5} + \cdots - 59\!\cdots\!24 \)
|
| $89$ |
\( T^{5} + \cdots + 21\!\cdots\!00 \)
|
| $97$ |
\( T^{5} + \cdots + 12\!\cdots\!56 \)
|
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