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: | \(1\) |
| 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: |
\( x^{5} - x^{4} - 60x^{3} - 128x^{2} + 90x + 224 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| 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
\( x^{5} - x^{4} - 60x^{3} - 128x^{2} + 90x + 224 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -4\nu^{4} + 15\nu^{3} + 216\nu^{2} - 13\nu - 859 ) / 69 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -3\nu^{4} + 17\nu^{3} + 116\nu^{2} - 234\nu - 144 ) / 46 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 4\nu^{4} - 15\nu^{3} - 216\nu^{2} + 151\nu + 790 ) / 69 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5\nu^{4} - 13\nu^{3} - 270\nu^{2} - 254\nu + 470 ) / 46 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{4} + \beta_{3} - 2\beta_{2} + 7\beta _1 + 49 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 16\beta_{4} + 47\beta_{3} + 77\beta _1 + 257 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 168\beta_{4} + 227\beta_{3} - 108\beta_{2} + 629\beta _1 + 3177 ) / 2 \)
|
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 |
|
−2.00000 | −8.94312 | 4.00000 | −19.0223 | 17.8862 | −7.00000 | −8.00000 | 52.9794 | 38.0446 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −6.02004 | 4.00000 | 14.9551 | 12.0401 | −7.00000 | −8.00000 | 9.24088 | −29.9102 | ||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | −3.67369 | 4.00000 | 7.38684 | 7.34737 | −7.00000 | −8.00000 | −13.5040 | −14.7737 | ||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 4.16678 | 4.00000 | −13.9244 | −8.33356 | −7.00000 | −8.00000 | −9.63796 | 27.8489 | ||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | 5.47007 | 4.00000 | 8.60478 | −10.9401 | −7.00000 | −8.00000 | 2.92167 | −17.2096 | ||||||||||||||||||||||||||||||||||
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.g | ✓ | 5 |
| 7.b | odd | 2 | 1 | 2254.4.a.k | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 322.4.a.g | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 2254.4.a.k | 5 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} + 9T_{3}^{4} - 48T_{3}^{3} - 426T_{3}^{2} + 574T_{3} + 4508 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(322))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{5} \)
|
| $3$ |
\( T^{5} + 9 T^{4} + \cdots + 4508 \)
|
| $5$ |
\( T^{5} + 2 T^{4} + \cdots - 251784 \)
|
| $7$ |
\( (T + 7)^{5} \)
|
| $11$ |
\( T^{5} - 52 T^{4} + \cdots - 564726544 \)
|
| $13$ |
\( T^{5} - 53 T^{4} + \cdots + 185639328 \)
|
| $17$ |
\( T^{5} + 242 T^{4} + \cdots - 817374592 \)
|
| $19$ |
\( T^{5} + \cdots + 16699552256 \)
|
| $23$ |
\( (T - 23)^{5} \)
|
| $29$ |
\( T^{5} + \cdots + 38484903944 \)
|
| $31$ |
\( T^{5} + \cdots + 3913375816 \)
|
| $37$ |
\( T^{5} + \cdots - 612036834368 \)
|
| $41$ |
\( T^{5} + \cdots - 1713680908144 \)
|
| $43$ |
\( T^{5} + \cdots - 721571907584 \)
|
| $47$ |
\( T^{5} + \cdots - 1170003721824 \)
|
| $53$ |
\( T^{5} + \cdots + 5948217496576 \)
|
| $59$ |
\( T^{5} + \cdots + 14051040718176 \)
|
| $61$ |
\( T^{5} + \cdots + 8774830644664 \)
|
| $67$ |
\( T^{5} + \cdots - 3441389221712 \)
|
| $71$ |
\( T^{5} + \cdots + 125630997295104 \)
|
| $73$ |
\( T^{5} + \cdots - 357663432464 \)
|
| $79$ |
\( T^{5} + \cdots + 45186910030528 \)
|
| $83$ |
\( T^{5} + \cdots - 4857139648256 \)
|
| $89$ |
\( T^{5} + \cdots + 928824047439856 \)
|
| $97$ |
\( T^{5} + \cdots + 7292438751712 \)
|
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