Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [322,4,Mod(1,322)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(322, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("322.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 322 = 2 \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 322.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: | \(18.9986150218\) |
| 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: |
\( x^{5} - 92x^{3} - 28x^{2} + 1593x - 1782 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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
\( x^{5} - 92x^{3} - 28x^{2} + 1593x - 1782 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{4} + 18\nu^{3} + 361\nu^{2} - 544\nu - 2997 ) / 171 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5\nu^{4} - 18\nu^{3} - 190\nu^{2} + 544\nu - 3330 ) / 171 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{4} - 27\nu^{3} + 247\nu^{2} + 1595\nu - 4482 ) / 171 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 37 \)
|
| \(\nu^{3}\) | \(=\) |
\( -5\beta_{4} + 3\beta_{3} + 5\beta_{2} + 53\beta _1 + 15 \)
|
| \(\nu^{4}\) | \(=\) |
\( -18\beta_{4} + 83\beta_{3} + 56\beta_{2} + 82\beta _1 + 2126 \)
|
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 |
|
2.00000 | −7.78488 | 4.00000 | 4.71723 | −15.5698 | −7.00000 | 8.00000 | 33.6044 | 9.43446 | |||||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | −2.52502 | 4.00000 | −15.4206 | −5.05004 | −7.00000 | 8.00000 | −20.6243 | −30.8412 | ||||||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | −0.260120 | 4.00000 | 15.2484 | −0.520240 | −7.00000 | 8.00000 | −26.9323 | 30.4969 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | 7.17693 | 4.00000 | 18.7050 | 14.3539 | −7.00000 | 8.00000 | 24.5083 | 37.4099 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.00000 | 8.39309 | 4.00000 | −1.25007 | 16.7862 | −7.00000 | 8.00000 | 43.4440 | −2.50013 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(7\) | \( +1 \) |
| \(23\) | \( -1 \) |
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 | 322.4.a.h | ✓ | 5 |
| 7.b | odd | 2 | 1 | 2254.4.a.l | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 322.4.a.h | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 2254.4.a.l | 5 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} - 5T_{3}^{4} - 82T_{3}^{3} + 294T_{3}^{2} + 1266T_{3} + 308 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(322))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{5} \)
|
| $3$ |
\( T^{5} - 5 T^{4} + \cdots + 308 \)
|
| $5$ |
\( T^{5} - 22 T^{4} + \cdots - 25936 \)
|
| $7$ |
\( (T + 7)^{5} \)
|
| $11$ |
\( T^{5} - 42 T^{4} + \cdots - 45699552 \)
|
| $13$ |
\( T^{5} - 107 T^{4} + \cdots + 12193328 \)
|
| $17$ |
\( T^{5} - 218 T^{4} + \cdots + 397516032 \)
|
| $19$ |
\( T^{5} + \cdots + 1463912448 \)
|
| $23$ |
\( (T - 23)^{5} \)
|
| $29$ |
\( T^{5} + \cdots - 7111896936 \)
|
| $31$ |
\( T^{5} + \cdots + 2926575828 \)
|
| $37$ |
\( T^{5} + \cdots - 397708480224 \)
|
| $41$ |
\( T^{5} + \cdots - 106493094736 \)
|
| $43$ |
\( T^{5} + \cdots + 21785240064 \)
|
| $47$ |
\( T^{5} + \cdots + 462685828948 \)
|
| $53$ |
\( T^{5} + \cdots - 19250018178336 \)
|
| $59$ |
\( T^{5} + \cdots - 508399233312 \)
|
| $61$ |
\( T^{5} + \cdots + 26399711613344 \)
|
| $67$ |
\( T^{5} + \cdots - 3215830804704 \)
|
| $71$ |
\( T^{5} + \cdots - 11314691334144 \)
|
| $73$ |
\( T^{5} + \cdots + 50769080496 \)
|
| $79$ |
\( T^{5} + \cdots - 126397213520256 \)
|
| $83$ |
\( T^{5} + \cdots + 152986734123776 \)
|
| $89$ |
\( T^{5} + \cdots + 7266574196256 \)
|
| $97$ |
\( T^{5} + \cdots + 32250389900736 \)
|
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