Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [224,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("224.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 224 = 2^{5} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(35.9259756381\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 198x^{2} + 43x + 5999 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{7} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 198x^{2} + 43x + 5999 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{3} + 8\nu^{2} - 272\nu - 959 ) / 21 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{3} + 20\nu^{2} + 258\nu - 1820 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 3\beta_{2} + \beta _1 + 397 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{3} + 15\beta_{2} + 134\beta _1 + 165 ) / 2 \)
|
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 | −20.4188 | 0 | 21.4943 | 0 | 49.0000 | 0 | 173.928 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −8.17317 | 0 | −20.6340 | 0 | 49.0000 | 0 | −176.199 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 16.2651 | 0 | 69.5406 | 0 | 49.0000 | 0 | 21.5544 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 30.3268 | 0 | −100.401 | 0 | 49.0000 | 0 | 676.717 | 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 | 224.6.a.h | yes | 4 |
| 4.b | odd | 2 | 1 | 224.6.a.g | ✓ | 4 | |
| 8.b | even | 2 | 1 | 448.6.a.bc | 4 | ||
| 8.d | odd | 2 | 1 | 448.6.a.bd | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 224.6.a.g | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 224.6.a.h | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 448.6.a.bc | 4 | 8.b | even | 2 | 1 | ||
| 448.6.a.bd | 4 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 18T_{3}^{3} - 672T_{3}^{2} + 6328T_{3} + 82320 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(224))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 18 T^{3} + \cdots + 82320 \)
|
| $5$ |
\( T^{4} + 30 T^{3} + \cdots + 3096576 \)
|
| $7$ |
\( (T - 49)^{4} \)
|
| $11$ |
\( T^{4} + \cdots - 9325648640 \)
|
| $13$ |
\( T^{4} + \cdots - 40778482496 \)
|
| $17$ |
\( T^{4} + \cdots - 670305393104 \)
|
| $19$ |
\( T^{4} + \cdots - 9671002992 \)
|
| $23$ |
\( T^{4} + \cdots + 2509965337600 \)
|
| $29$ |
\( T^{4} + \cdots - 69185366122448 \)
|
| $31$ |
\( T^{4} + \cdots + 5102153647360 \)
|
| $37$ |
\( T^{4} + \cdots - 386774083587152 \)
|
| $41$ |
\( T^{4} + \cdots - 20\!\cdots\!12 \)
|
| $43$ |
\( T^{4} + \cdots - 14\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots + 11\!\cdots\!44 \)
|
| $53$ |
\( T^{4} + \cdots - 12\!\cdots\!88 \)
|
| $59$ |
\( T^{4} + \cdots - 80\!\cdots\!16 \)
|
| $61$ |
\( T^{4} + \cdots - 43\!\cdots\!20 \)
|
| $67$ |
\( T^{4} + \cdots - 14\!\cdots\!68 \)
|
| $71$ |
\( T^{4} + \cdots - 22\!\cdots\!40 \)
|
| $73$ |
\( T^{4} + \cdots + 60\!\cdots\!40 \)
|
| $79$ |
\( T^{4} + \cdots - 14\!\cdots\!20 \)
|
| $83$ |
\( T^{4} + \cdots - 34\!\cdots\!04 \)
|
| $89$ |
\( T^{4} + \cdots + 12\!\cdots\!40 \)
|
| $97$ |
\( T^{4} + \cdots + 56\!\cdots\!76 \)
|
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