Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [20,18,Mod(9,20)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(20, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 18, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("20.9");
S:= CuspForms(chi, 18);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 20 = 2^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 18 \) |
| Character orbit: | \([\chi]\) | \(=\) | 20.c (of order \(2\), degree \(1\), 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: | \(36.6444174689\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 10513788 x^{6} + 47438777752 x^{5} - 249513269598475 x^{4} + \cdots + 12\!\cdots\!68 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{35}\cdot 3^{4}\cdot 5^{12} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 10513788 x^{6} + 47438777752 x^{5} - 249513269598475 x^{4} + \cdots + 12\!\cdots\!68 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 16\!\cdots\!39 \nu^{7} + \cdots - 55\!\cdots\!52 ) / 27\!\cdots\!50 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 34\!\cdots\!39 \nu^{7} + \cdots + 12\!\cdots\!02 ) / 12\!\cdots\!50 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 45\!\cdots\!69 \nu^{7} + \cdots - 28\!\cdots\!42 ) / 12\!\cdots\!50 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 26\!\cdots\!59 \nu^{7} + \cdots - 91\!\cdots\!77 ) / 54\!\cdots\!75 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 11\!\cdots\!06 \nu^{7} + \cdots + 36\!\cdots\!33 ) / 13\!\cdots\!75 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 66\!\cdots\!61 \nu^{7} + \cdots - 14\!\cdots\!98 ) / 12\!\cdots\!50 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 48\!\cdots\!68 \nu^{7} + \cdots + 96\!\cdots\!99 ) / 45\!\cdots\!25 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + 6\beta_{2} - 524\beta _1 + 625 ) / 5000 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 125 \beta_{7} - 750 \beta_{6} - 24875 \beta_{5} - 10436 \beta_{4} + 3250 \beta_{3} + \cdots + 105137885000 ) / 40000 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 315875 \beta_{7} + 5651500 \beta_{6} - 262683375 \beta_{5} - 84814578 \beta_{4} + \cdots - 14\!\cdots\!00 ) / 80000 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 794969150 \beta_{7} + 728628975 \beta_{6} + 25977943650 \beta_{5} + 8251976656 \beta_{4} + \cdots + 12\!\cdots\!00 ) / 8000 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 76614562530125 \beta_{7} - 186588028874250 \beta_{6} + \cdots - 29\!\cdots\!00 ) / 160000 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 38\!\cdots\!25 \beta_{7} + \cdots - 38\!\cdots\!00 ) / 160000 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 15\!\cdots\!00 \beta_{7} + \cdots + 72\!\cdots\!00 ) / 80000 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) | \(17\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 9.1 |
|
0 | − | 20057.4i | 0 | 873406. | − | 10106.1i | 0 | − | 2.05068e7i | 0 | −2.73159e8 | 0 | ||||||||||||||||||||||||||||||||||||||
| 9.2 | 0 | − | 14851.7i | 0 | −846917. | + | 213710.i | 0 | − | 2.81651e6i | 0 | −9.14330e7 | 0 | |||||||||||||||||||||||||||||||||||||||
| 9.3 | 0 | − | 7565.06i | 0 | 672773. | + | 557061.i | 0 | 2.63351e7i | 0 | 7.19100e7 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 9.4 | 0 | − | 4988.24i | 0 | −60861.6 | − | 871341.i | 0 | 8.73679e6i | 0 | 1.04258e8 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 9.5 | 0 | 4988.24i | 0 | −60861.6 | + | 871341.i | 0 | − | 8.73679e6i | 0 | 1.04258e8 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 9.6 | 0 | 7565.06i | 0 | 672773. | − | 557061.i | 0 | − | 2.63351e7i | 0 | 7.19100e7 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 9.7 | 0 | 14851.7i | 0 | −846917. | − | 213710.i | 0 | 2.81651e6i | 0 | −9.14330e7 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 9.8 | 0 | 20057.4i | 0 | 873406. | + | 10106.1i | 0 | 2.05068e7i | 0 | −2.73159e8 | 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 | 20.18.c.a | ✓ | 8 |
| 4.b | odd | 2 | 1 | 80.18.c.c | 8 | ||
| 5.b | even | 2 | 1 | inner | 20.18.c.a | ✓ | 8 |
| 5.c | odd | 4 | 2 | 100.18.a.f | 8 | ||
| 20.d | odd | 2 | 1 | 80.18.c.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 20.18.c.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 20.18.c.a | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 80.18.c.c | 8 | 4.b | odd | 2 | 1 | ||
| 80.18.c.c | 8 | 20.d | odd | 2 | 1 | ||
| 100.18.a.f | 8 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{18}^{\mathrm{new}}(20, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + \cdots + 12\!\cdots\!96 \)
|
| $5$ |
\( T^{8} + \cdots + 33\!\cdots\!25 \)
|
| $7$ |
\( T^{8} + \cdots + 17\!\cdots\!76 \)
|
| $11$ |
\( (T^{4} + \cdots + 16\!\cdots\!00)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 10\!\cdots\!56 \)
|
| $17$ |
\( T^{8} + \cdots + 15\!\cdots\!96 \)
|
| $19$ |
\( (T^{4} + \cdots + 31\!\cdots\!76)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 76\!\cdots\!36 \)
|
| $29$ |
\( (T^{4} + \cdots + 11\!\cdots\!56)^{2} \)
|
| $31$ |
\( (T^{4} + \cdots - 17\!\cdots\!84)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 41\!\cdots\!96 \)
|
| $41$ |
\( (T^{4} + \cdots + 59\!\cdots\!56)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 60\!\cdots\!00 \)
|
| $47$ |
\( T^{8} + \cdots + 86\!\cdots\!96 \)
|
| $53$ |
\( T^{8} + \cdots + 41\!\cdots\!36 \)
|
| $59$ |
\( (T^{4} + \cdots + 24\!\cdots\!76)^{2} \)
|
| $61$ |
\( (T^{4} + \cdots - 16\!\cdots\!44)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 11\!\cdots\!56 \)
|
| $71$ |
\( (T^{4} + \cdots + 84\!\cdots\!56)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 12\!\cdots\!36 \)
|
| $79$ |
\( (T^{4} + \cdots - 34\!\cdots\!24)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 66\!\cdots\!36 \)
|
| $89$ |
\( (T^{4} + \cdots - 18\!\cdots\!44)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 70\!\cdots\!76 \)
|
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