Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [32,17,Mod(31,32)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(32, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 17, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("32.31");
S:= CuspForms(chi, 17);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 32 = 2^{5} \) |
| Weight: | \( k \) | \(=\) | \( 17 \) |
| Character orbit: | \([\chi]\) | \(=\) | 32.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: | \(51.9438540341\) |
| 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: |
\( x^{8} - 4 x^{7} - 38020 x^{6} - 901434 x^{5} + 424670139 x^{4} + 18462385230 x^{3} + \cdots + 938226905832125 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{90}\cdot 3^{2} \) |
| 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
\( x^{8} - 4 x^{7} - 38020 x^{6} - 901434 x^{5} + 424670139 x^{4} + 18462385230 x^{3} + \cdots + 938226905832125 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 11\!\cdots\!04 \nu^{7} + \cdots - 35\!\cdots\!00 ) / 92\!\cdots\!25 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 10\!\cdots\!56 \nu^{7} + \cdots - 11\!\cdots\!00 ) / 46\!\cdots\!25 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 42\!\cdots\!88 \nu^{7} + \cdots + 18\!\cdots\!00 ) / 57\!\cdots\!25 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 10\!\cdots\!64 \nu^{7} + \cdots - 44\!\cdots\!00 ) / 46\!\cdots\!25 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 40\!\cdots\!52 \nu^{7} + \cdots + 71\!\cdots\!00 ) / 92\!\cdots\!25 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 91\!\cdots\!56 \nu^{7} + \cdots + 22\!\cdots\!00 ) / 65\!\cdots\!75 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 29\!\cdots\!96 \nu^{7} + \cdots - 74\!\cdots\!00 ) / 92\!\cdots\!25 \)
|
| \(\nu\) | \(=\) |
\( ( -7\beta_{5} - 16\beta_{4} + 96\beta_{3} - 24\beta_{2} + 1572864 ) / 3145728 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 16 \beta_{7} - 240 \beta_{6} - 2012 \beta_{5} - 2624 \beta_{4} + 3075 \beta_{3} + \cdots + 119625744384 ) / 12582912 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 12336 \beta_{7} - 19152 \beta_{6} - 2045588 \beta_{5} - 1906880 \beta_{4} + \cdots + 9942406791168 ) / 25165824 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 241872 \beta_{7} - 1665072 \beta_{6} - 13495124 \beta_{5} - 20767808 \beta_{4} + \cdots + 475483363344384 ) / 3145728 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 434823440 \beta_{7} - 1226653680 \beta_{6} - 39355065484 \beta_{5} - 35228855104 \beta_{4} + \cdots + 31\!\cdots\!92 ) / 25165824 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 16335740944 \beta_{7} - 89360359152 \beta_{6} - 751896698458 \beta_{5} - 1000018775776 \beta_{4} + \cdots + 17\!\cdots\!72 ) / 6291456 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 3124846419856 \beta_{7} - 10876870242672 \beta_{6} - 186066422034886 \beta_{5} + \cdots + 19\!\cdots\!92 ) / 6291456 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(31\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
0 | − | 12434.9i | 0 | −559134. | 0 | 8.71663e6i | 0 | −1.11580e8 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 31.2 | 0 | − | 9841.60i | 0 | 46037.5 | 0 | − | 4.34992e6i | 0 | −5.38105e7 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 31.3 | 0 | − | 3525.59i | 0 | 408887. | 0 | − | 1.86517e6i | 0 | 3.06169e7 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 31.4 | 0 | − | 1686.86i | 0 | −456518. | 0 | 4.06969e6i | 0 | 4.02012e7 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 31.5 | 0 | 1686.86i | 0 | −456518. | 0 | − | 4.06969e6i | 0 | 4.02012e7 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 31.6 | 0 | 3525.59i | 0 | 408887. | 0 | 1.86517e6i | 0 | 3.06169e7 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 31.7 | 0 | 9841.60i | 0 | 46037.5 | 0 | 4.34992e6i | 0 | −5.38105e7 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 31.8 | 0 | 12434.9i | 0 | −559134. | 0 | − | 8.71663e6i | 0 | −1.11580e8 | 0 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 32.17.c.a | ✓ | 8 |
| 4.b | odd | 2 | 1 | inner | 32.17.c.a | ✓ | 8 |
| 8.b | even | 2 | 1 | 64.17.c.f | 8 | ||
| 8.d | odd | 2 | 1 | 64.17.c.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 32.17.c.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 32.17.c.a | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 64.17.c.f | 8 | 8.b | even | 2 | 1 | ||
| 64.17.c.f | 8 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 266758720 T_{3}^{6} + \cdots + 52\!\cdots\!56 \)
acting on \(S_{17}^{\mathrm{new}}(32, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + \cdots + 52\!\cdots\!56 \)
|
| $5$ |
\( (T^{4} + \cdots + 48\!\cdots\!00)^{2} \)
|
| $7$ |
\( T^{8} + \cdots + 82\!\cdots\!84 \)
|
| $11$ |
\( T^{8} + \cdots + 15\!\cdots\!00 \)
|
| $13$ |
\( (T^{4} + \cdots - 16\!\cdots\!32)^{2} \)
|
| $17$ |
\( (T^{4} + \cdots - 96\!\cdots\!40)^{2} \)
|
| $19$ |
\( T^{8} + \cdots + 39\!\cdots\!76 \)
|
| $23$ |
\( T^{8} + \cdots + 15\!\cdots\!00 \)
|
| $29$ |
\( (T^{4} + \cdots + 44\!\cdots\!00)^{2} \)
|
| $31$ |
\( T^{8} + \cdots + 78\!\cdots\!00 \)
|
| $37$ |
\( (T^{4} + \cdots - 34\!\cdots\!20)^{2} \)
|
| $41$ |
\( (T^{4} + \cdots + 94\!\cdots\!00)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 36\!\cdots\!00 \)
|
| $47$ |
\( T^{8} + \cdots + 51\!\cdots\!16 \)
|
| $53$ |
\( (T^{4} + \cdots - 28\!\cdots\!20)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 16\!\cdots\!64 \)
|
| $61$ |
\( (T^{4} + \cdots - 33\!\cdots\!96)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 86\!\cdots\!00 \)
|
| $71$ |
\( T^{8} + \cdots + 85\!\cdots\!24 \)
|
| $73$ |
\( (T^{4} + \cdots - 15\!\cdots\!76)^{2} \)
|
| $79$ |
\( T^{8} + \cdots + 28\!\cdots\!36 \)
|
| $83$ |
\( T^{8} + \cdots + 69\!\cdots\!00 \)
|
| $89$ |
\( (T^{4} + \cdots + 34\!\cdots\!00)^{2} \)
|
| $97$ |
\( (T^{4} + \cdots - 37\!\cdots\!16)^{2} \)
|
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