Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [320,8,Mod(129,320)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(320, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("320.129");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 320 = 2^{6} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 320.c (of order \(2\), degree \(1\), not minimal) |
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: | no |
| Analytic conductor: | \(99.9632081549\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{31})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 15x^{2} + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 10) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} - 15x^{2} + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu^{3} - 14\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( -6\nu^{3} + 122\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 40\nu^{2} + 7\nu - 300 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 3\beta_1 ) / 80 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{3} + \beta _1 + 600 ) / 80 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 7\beta_{2} + 61\beta_1 ) / 80 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).
| \(n\) | \(191\) | \(257\) | \(261\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 129.1 |
|
0 | − | 83.6776i | 0 | −237.711 | − | 147.033i | 0 | 185.744i | 0 | −4814.95 | 0 | |||||||||||||||||||||||||||
| 129.2 | 0 | − | 27.6776i | 0 | 207.711 | − | 187.033i | 0 | 593.744i | 0 | 1420.95 | 0 | ||||||||||||||||||||||||||||
| 129.3 | 0 | 27.6776i | 0 | 207.711 | + | 187.033i | 0 | − | 593.744i | 0 | 1420.95 | 0 | ||||||||||||||||||||||||||||
| 129.4 | 0 | 83.6776i | 0 | −237.711 | + | 147.033i | 0 | − | 185.744i | 0 | −4814.95 | 0 | ||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 320.8.c.f | 4 | |
| 4.b | odd | 2 | 1 | 320.8.c.g | 4 | ||
| 5.b | even | 2 | 1 | inner | 320.8.c.f | 4 | |
| 8.b | even | 2 | 1 | 10.8.b.a | ✓ | 4 | |
| 8.d | odd | 2 | 1 | 80.8.c.d | 4 | ||
| 20.d | odd | 2 | 1 | 320.8.c.g | 4 | ||
| 24.h | odd | 2 | 1 | 90.8.c.c | 4 | ||
| 40.e | odd | 2 | 1 | 80.8.c.d | 4 | ||
| 40.f | even | 2 | 1 | 10.8.b.a | ✓ | 4 | |
| 40.i | odd | 4 | 1 | 50.8.a.i | 2 | ||
| 40.i | odd | 4 | 1 | 50.8.a.j | 2 | ||
| 40.k | even | 4 | 1 | 400.8.a.t | 2 | ||
| 40.k | even | 4 | 1 | 400.8.a.bf | 2 | ||
| 120.i | odd | 2 | 1 | 90.8.c.c | 4 | ||
| 120.w | even | 4 | 1 | 450.8.a.bd | 2 | ||
| 120.w | even | 4 | 1 | 450.8.a.bi | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 10.8.b.a | ✓ | 4 | 8.b | even | 2 | 1 | |
| 10.8.b.a | ✓ | 4 | 40.f | even | 2 | 1 | |
| 50.8.a.i | 2 | 40.i | odd | 4 | 1 | ||
| 50.8.a.j | 2 | 40.i | odd | 4 | 1 | ||
| 80.8.c.d | 4 | 8.d | odd | 2 | 1 | ||
| 80.8.c.d | 4 | 40.e | odd | 2 | 1 | ||
| 90.8.c.c | 4 | 24.h | odd | 2 | 1 | ||
| 90.8.c.c | 4 | 120.i | odd | 2 | 1 | ||
| 320.8.c.f | 4 | 1.a | even | 1 | 1 | trivial | |
| 320.8.c.f | 4 | 5.b | even | 2 | 1 | inner | |
| 320.8.c.g | 4 | 4.b | odd | 2 | 1 | ||
| 320.8.c.g | 4 | 20.d | odd | 2 | 1 | ||
| 400.8.a.t | 2 | 40.k | even | 4 | 1 | ||
| 400.8.a.bf | 2 | 40.k | even | 4 | 1 | ||
| 450.8.a.bd | 2 | 120.w | even | 4 | 1 | ||
| 450.8.a.bi | 2 | 120.w | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(320, [\chi])\):
|
\( T_{3}^{4} + 7768T_{3}^{2} + 5363856 \)
|
|
\( T_{11}^{2} + 8904T_{11} + 19026704 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 7768 T^{2} + 5363856 \)
|
| $5$ |
\( T^{4} + \cdots + 6103515625 \)
|
| $7$ |
\( T^{4} + \cdots + 12162560656 \)
|
| $11$ |
\( (T^{2} + 8904 T + 19026704)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 249352396482816 \)
|
| $17$ |
\( T^{4} + \cdots + 15\!\cdots\!76 \)
|
| $19$ |
\( (T^{2} + 31800 T - 954255600)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 90\!\cdots\!56 \)
|
| $29$ |
\( (T^{2} + 84780 T - 19030326300)^{2} \)
|
| $31$ |
\( (T^{2} + 197056 T + 4879504384)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 26\!\cdots\!56 \)
|
| $41$ |
\( (T^{2} - 116244 T - 183681487516)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 31\!\cdots\!76 \)
|
| $47$ |
\( T^{4} + \cdots + 35\!\cdots\!16 \)
|
| $53$ |
\( T^{4} + \cdots + 25\!\cdots\!16 \)
|
| $59$ |
\( (T^{2} + 1046760 T - 159475001200)^{2} \)
|
| $61$ |
\( (T^{2} + \cdots + 1716951910564)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 58\!\cdots\!16 \)
|
| $71$ |
\( (T^{2} + \cdots - 4934358737856)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 87\!\cdots\!16 \)
|
| $79$ |
\( (T^{2} + \cdots - 3963839328000)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 13\!\cdots\!96 \)
|
| $89$ |
\( (T^{2} + \cdots + 66278240560100)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 14\!\cdots\!56 \)
|
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