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: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 2573495x^{3} + 1741012708x^{2} - 129847160472x - 10015787544672 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{32}\cdot 3^{4}\cdot 5^{3} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 2573495x^{3} + 1741012708x^{2} - 129847160472x - 10015787544672 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -6836\nu^{4} - 3908816\nu^{3} + 14601544432\nu^{2} - 4585366668032\nu - 299376756211169 ) / 31331406925 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 5911088 \nu^{4} - 6899923328 \nu^{3} + 9131501876656 \nu^{2} + \cdots - 33\!\cdots\!02 ) / 31331406925 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 20433484 \nu^{4} + 22243752304 \nu^{3} - 30154300615808 \nu^{2} + 7186178670208 \nu - 19\!\cdots\!89 ) / 31331406925 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 53838976 \nu^{4} - 9665232256 \nu^{3} + 129949935070112 \nu^{2} + \cdots + 42\!\cdots\!71 ) / 31331406925 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + 2\beta_{3} + 12\beta_{2} - 12274\beta _1 + 144997 ) / 368640 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -77\beta_{4} + 6\beta_{3} - 444\beta_{2} + 1008298\beta _1 + 15811767391 ) / 15360 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 457279\beta_{4} + 438038\beta_{3} + 3647628\beta_{2} - 5446202566\beta _1 - 47994305818037 ) / 46080 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -6989403\beta_{4} - 3164282\beta_{3} - 49474856\beta_{2} + 86609707738\beta _1 + 1056113704209681 ) / 384 \)
|
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 | −18379.5 | 0 | 390625. | 0 | −1.33655e7 | 0 | 2.08667e8 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −4455.28 | 0 | 390625. | 0 | 3.85479e6 | 0 | −1.09291e8 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 1318.46 | 0 | 390625. | 0 | 1.28833e7 | 0 | −1.27402e8 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 15641.7 | 0 | 390625. | 0 | −2.95100e7 | 0 | 1.15523e8 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 21258.6 | 0 | 390625. | 0 | 2.58245e7 | 0 | 3.22790e8 | 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.l | 5 | |
| 4.b | odd | 2 | 1 | 40.18.a.d | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.18.a.d | ✓ | 5 | 4.b | odd | 2 | 1 | |
| 80.18.a.l | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} - 15384 T_{3}^{4} - 409659888 T_{3}^{3} + 5136009804288 T_{3}^{2} + \cdots - 35\!\cdots\!00 \)
acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(80))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} + \cdots - 35\!\cdots\!00 \)
|
| $5$ |
\( (T - 390625)^{5} \)
|
| $7$ |
\( T^{5} + \cdots - 50\!\cdots\!12 \)
|
| $11$ |
\( T^{5} + \cdots - 56\!\cdots\!04 \)
|
| $13$ |
\( T^{5} + \cdots - 41\!\cdots\!76 \)
|
| $17$ |
\( T^{5} + \cdots + 91\!\cdots\!00 \)
|
| $19$ |
\( T^{5} + \cdots - 14\!\cdots\!44 \)
|
| $23$ |
\( T^{5} + \cdots - 23\!\cdots\!28 \)
|
| $29$ |
\( T^{5} + \cdots + 21\!\cdots\!28 \)
|
| $31$ |
\( T^{5} + \cdots - 22\!\cdots\!00 \)
|
| $37$ |
\( T^{5} + \cdots + 96\!\cdots\!16 \)
|
| $41$ |
\( T^{5} + \cdots - 43\!\cdots\!68 \)
|
| $43$ |
\( T^{5} + \cdots - 12\!\cdots\!44 \)
|
| $47$ |
\( T^{5} + \cdots - 85\!\cdots\!96 \)
|
| $53$ |
\( T^{5} + \cdots + 94\!\cdots\!84 \)
|
| $59$ |
\( T^{5} + \cdots + 11\!\cdots\!32 \)
|
| $61$ |
\( T^{5} + \cdots - 18\!\cdots\!00 \)
|
| $67$ |
\( T^{5} + \cdots + 11\!\cdots\!96 \)
|
| $71$ |
\( T^{5} + \cdots + 63\!\cdots\!00 \)
|
| $73$ |
\( T^{5} + \cdots + 54\!\cdots\!08 \)
|
| $79$ |
\( T^{5} + \cdots - 60\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots - 10\!\cdots\!92 \)
|
| $89$ |
\( T^{5} + \cdots - 43\!\cdots\!56 \)
|
| $97$ |
\( T^{5} + \cdots - 29\!\cdots\!96 \)
|
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