Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [80,20,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, 20, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("80.1");
S:= CuspForms(chi, 20);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 80 = 2^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 20 \) |
| 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: | \(183.053357245\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{4} - 2x^{3} - 5694737x^{2} - 4823342760x - 829238069200 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{18}\cdot 3^{4}\cdot 5^{2} \) |
| 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
\( x^{4} - 2x^{3} - 5694737x^{2} - 4823342760x - 829238069200 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 264\nu^{3} + 200412\nu^{2} - 1571034048\nu - 1527138482070 ) / 26908975 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -119088\nu^{3} + 56372196\nu^{2} + 679030379016\nu + 270966957722190 ) / 26908975 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 110064\nu^{3} - 92577888\nu^{2} - 495403262448\nu - 135246320375520 ) / 5381795 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 6\beta_{2} + 622\beta _1 + 11520 ) / 23040 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 101\beta_{3} + 2718\beta_{2} + 1015526\beta _1 + 32801696640 ) / 11520 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5797541\beta_{3} + 31578654\beta_{2} + 4508025638\beta _1 + 83544173049600 ) / 23040 \)
|
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 | −31558.8 | 0 | 1.95312e6 | 0 | 1.09577e8 | 0 | −1.66303e8 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −3592.41 | 0 | 1.95312e6 | 0 | −1.23406e8 | 0 | −1.14936e9 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 27215.3 | 0 | 1.95312e6 | 0 | 3.09073e6 | 0 | −4.21590e8 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 54087.9 | 0 | 1.95312e6 | 0 | −4.83353e7 | 0 | 1.76324e9 | 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 | 80.20.a.i | 4 | |
| 4.b | odd | 2 | 1 | 40.20.a.a | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.20.a.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 80.20.a.i | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 46152T_{3}^{3} - 1272516912T_{3}^{2} + 42525728891136T_{3} + 166886029710443520 \)
acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(80))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + \cdots + 16\!\cdots\!20 \)
|
| $5$ |
\( (T - 1953125)^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 20\!\cdots\!68 \)
|
| $11$ |
\( T^{4} + \cdots + 70\!\cdots\!44 \)
|
| $13$ |
\( T^{4} + \cdots + 54\!\cdots\!92 \)
|
| $17$ |
\( T^{4} + \cdots + 13\!\cdots\!00 \)
|
| $19$ |
\( T^{4} + \cdots + 26\!\cdots\!64 \)
|
| $23$ |
\( T^{4} + \cdots + 59\!\cdots\!28 \)
|
| $29$ |
\( T^{4} + \cdots + 21\!\cdots\!04 \)
|
| $31$ |
\( T^{4} + \cdots - 11\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots + 30\!\cdots\!44 \)
|
| $41$ |
\( T^{4} + \cdots - 82\!\cdots\!68 \)
|
| $43$ |
\( T^{4} + \cdots - 45\!\cdots\!92 \)
|
| $47$ |
\( T^{4} + \cdots + 25\!\cdots\!36 \)
|
| $53$ |
\( T^{4} + \cdots + 16\!\cdots\!96 \)
|
| $59$ |
\( T^{4} + \cdots - 56\!\cdots\!48 \)
|
| $61$ |
\( T^{4} + \cdots - 28\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots + 35\!\cdots\!72 \)
|
| $71$ |
\( T^{4} + \cdots - 21\!\cdots\!40 \)
|
| $73$ |
\( T^{4} + \cdots + 47\!\cdots\!88 \)
|
| $79$ |
\( T^{4} + \cdots + 61\!\cdots\!20 \)
|
| $83$ |
\( T^{4} + \cdots + 16\!\cdots\!52 \)
|
| $89$ |
\( T^{4} + \cdots - 13\!\cdots\!76 \)
|
| $97$ |
\( T^{4} + \cdots + 87\!\cdots\!96 \)
|
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