Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [80,18,Mod(1,80)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(80, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 18, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("80.1");
S:= CuspForms(chi, 18);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 80 = 2^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 18 \) |
| Character orbit: | \([\chi]\) | \(=\) | 80.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: | \(146.577669876\) |
| 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: |
\( x^{4} - 2x^{3} - 10129161x^{2} - 14868432888x - 5637012379920 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{19}\cdot 3^{2}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 40) |
| 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
\( x^{4} - 2x^{3} - 10129161x^{2} - 14868432888x - 5637012379920 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 1358\nu^{2} - 8287713\nu - 4284672876 ) / 113568 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 401\nu^{3} - 282478\nu^{2} - 4252708593\nu - 3045014674956 ) / 56784 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -1525\nu^{3} + 1197350\nu^{2} + 13992143445\nu + 10957867843740 ) / 113568 \)
|
| \(\nu\) | \(=\) |
\( ( -3\beta_{3} - 5\beta_{2} - 565\beta _1 + 23040 ) / 46080 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -1773\beta_{3} - 1291\beta_{2} - 1668443\beta _1 + 38895985920 ) / 7680 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -13103181\beta_{3} - 17319211\beta_{2} - 4347939323\beta _1 + 171517722769920 ) / 15360 \)
|
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 | −15935.5 | 0 | 390625. | 0 | 3.23095e6 | 0 | 1.24801e8 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −5728.07 | 0 | 390625. | 0 | 2.43425e7 | 0 | −9.63293e7 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 2332.02 | 0 | 390625. | 0 | −2.00926e7 | 0 | −1.23702e8 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 14587.6 | 0 | 390625. | 0 | 4.47785e6 | 0 | 8.36577e7 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( -1 \) |
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 | 80.18.a.i | 4 | |
| 4.b | odd | 2 | 1 | 40.18.a.c | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.18.a.c | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 80.18.a.i | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 4744T_{3}^{3} - 241241328T_{3}^{2} - 807457102848T_{3} + 3105205562327040 \)
acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(80))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + \cdots + 31\!\cdots\!40 \)
|
| $5$ |
\( (T - 390625)^{4} \)
|
| $7$ |
\( T^{4} + \cdots - 70\!\cdots\!12 \)
|
| $11$ |
\( T^{4} + \cdots + 10\!\cdots\!04 \)
|
| $13$ |
\( T^{4} + \cdots + 84\!\cdots\!32 \)
|
| $17$ |
\( T^{4} + \cdots + 42\!\cdots\!00 \)
|
| $19$ |
\( T^{4} + \cdots - 11\!\cdots\!36 \)
|
| $23$ |
\( T^{4} + \cdots - 26\!\cdots\!32 \)
|
| $29$ |
\( T^{4} + \cdots + 59\!\cdots\!84 \)
|
| $31$ |
\( T^{4} + \cdots - 51\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots - 15\!\cdots\!56 \)
|
| $41$ |
\( T^{4} + \cdots + 11\!\cdots\!92 \)
|
| $43$ |
\( T^{4} + \cdots - 18\!\cdots\!12 \)
|
| $47$ |
\( T^{4} + \cdots + 13\!\cdots\!76 \)
|
| $53$ |
\( T^{4} + \cdots + 10\!\cdots\!96 \)
|
| $59$ |
\( T^{4} + \cdots + 31\!\cdots\!72 \)
|
| $61$ |
\( T^{4} + \cdots + 84\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots + 11\!\cdots\!32 \)
|
| $71$ |
\( T^{4} + \cdots - 80\!\cdots\!80 \)
|
| $73$ |
\( T^{4} + \cdots - 76\!\cdots\!52 \)
|
| $79$ |
\( T^{4} + \cdots - 57\!\cdots\!60 \)
|
| $83$ |
\( T^{4} + \cdots - 18\!\cdots\!48 \)
|
| $89$ |
\( T^{4} + \cdots + 19\!\cdots\!44 \)
|
| $97$ |
\( T^{4} + \cdots + 48\!\cdots\!56 \)
|
| show more | |
| show less |