Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [224,8,Mod(1,224)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(224, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("224.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 224 = 2^{5} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 224.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: | \(69.9742457084\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{5} - 2x^{4} - 1809x^{3} + 6482x^{2} + 488753x + 1733184 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{12} \) |
| 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
\( x^{5} - 2x^{4} - 1809x^{3} + 6482x^{2} + 488753x + 1733184 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -16\nu^{4} + 664\nu^{3} + 23020\nu^{2} - 954628\nu - 542862 ) / 8883 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -20\nu^{4} - 16\nu^{3} + 30044\nu^{2} - 125210\nu - 3474819 ) / 8883 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -8\nu^{4} + 144\nu^{3} + 15740\nu^{2} - 230094\nu - 3755118 ) / 987 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} - 2\beta_{3} - 2\beta_{2} - 5\beta _1 + 2895 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{4} - 90\beta_{3} + 99\beta_{2} + 5035\beta _1 - 12707 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 1501\beta_{4} - 4745\beta_{3} - 3044\beta_{2} - 22046\beta _1 + 3646467 ) / 4 \)
|
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 |
|
0 | −81.5387 | 0 | 494.197 | 0 | 343.000 | 0 | 4461.57 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −56.6900 | 0 | −323.003 | 0 | 343.000 | 0 | 1026.75 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.02201 | 0 | 280.632 | 0 | 343.000 | 0 | −2182.91 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 15.9041 | 0 | −336.131 | 0 | 343.000 | 0 | −1934.06 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 70.3466 | 0 | −31.6943 | 0 | 343.000 | 0 | 2761.65 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(7\) | \( -1 \) |
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 | 224.8.a.c | ✓ | 5 |
| 4.b | odd | 2 | 1 | 224.8.a.d | yes | 5 | |
| 8.b | even | 2 | 1 | 448.8.a.bb | 5 | ||
| 8.d | odd | 2 | 1 | 448.8.a.ba | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 224.8.a.c | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 224.8.a.d | yes | 5 | 4.b | odd | 2 | 1 | |
| 448.8.a.ba | 5 | 8.d | odd | 2 | 1 | ||
| 448.8.a.bb | 5 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} + 54T_{3}^{4} - 6076T_{3}^{3} - 256536T_{3}^{2} + 4678128T_{3} + 10456992 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(224))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} + 54 T^{4} + \cdots + 10456992 \)
|
| $5$ |
\( T^{5} + \cdots + 477235046400 \)
|
| $7$ |
\( (T - 343)^{5} \)
|
| $11$ |
\( T^{5} + \cdots + 26\!\cdots\!52 \)
|
| $13$ |
\( T^{5} + \cdots + 33\!\cdots\!88 \)
|
| $17$ |
\( T^{5} + \cdots + 13\!\cdots\!04 \)
|
| $19$ |
\( T^{5} + \cdots + 14\!\cdots\!84 \)
|
| $23$ |
\( T^{5} + \cdots + 91\!\cdots\!36 \)
|
| $29$ |
\( T^{5} + \cdots - 60\!\cdots\!64 \)
|
| $31$ |
\( T^{5} + \cdots + 13\!\cdots\!44 \)
|
| $37$ |
\( T^{5} + \cdots + 10\!\cdots\!88 \)
|
| $41$ |
\( T^{5} + \cdots + 22\!\cdots\!12 \)
|
| $43$ |
\( T^{5} + \cdots - 24\!\cdots\!12 \)
|
| $47$ |
\( T^{5} + \cdots + 12\!\cdots\!92 \)
|
| $53$ |
\( T^{5} + \cdots + 11\!\cdots\!12 \)
|
| $59$ |
\( T^{5} + \cdots + 39\!\cdots\!28 \)
|
| $61$ |
\( T^{5} + \cdots + 92\!\cdots\!08 \)
|
| $67$ |
\( T^{5} + \cdots - 35\!\cdots\!56 \)
|
| $71$ |
\( T^{5} + \cdots + 30\!\cdots\!36 \)
|
| $73$ |
\( T^{5} + \cdots - 16\!\cdots\!96 \)
|
| $79$ |
\( T^{5} + \cdots + 34\!\cdots\!76 \)
|
| $83$ |
\( T^{5} + \cdots - 44\!\cdots\!04 \)
|
| $89$ |
\( T^{5} + \cdots + 16\!\cdots\!24 \)
|
| $97$ |
\( T^{5} + \cdots - 76\!\cdots\!92 \)
|
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