Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [512,3,Mod(511,512)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(512, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("512.511");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 512 = 2^{9} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 512.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: | \(13.9509895352\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.18939904.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{20} \) |
| 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
\( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu^{4} - 4\nu^{3} + 14\nu^{2} - 12\nu + 10 \)
|
| \(\beta_{2}\) | \(=\) |
\( -4\nu^{6} + 12\nu^{5} - 40\nu^{4} + 60\nu^{3} - 76\nu^{2} + 48\nu - 16 \)
|
| \(\beta_{3}\) | \(=\) |
\( -8\nu^{6} + 24\nu^{5} - 86\nu^{4} + 132\nu^{3} - 178\nu^{2} + 116\nu - 30 \)
|
| \(\beta_{4}\) | \(=\) |
\( 12\nu^{7} - 42\nu^{6} + 146\nu^{5} - 260\nu^{4} + 374\nu^{3} - 322\nu^{2} + 172\nu - 40 \)
|
| \(\beta_{5}\) | \(=\) |
\( -18\nu^{7} + 63\nu^{6} - 219\nu^{5} + 390\nu^{4} - 565\nu^{3} + 489\nu^{2} - 272\nu + 66 \)
|
| \(\beta_{6}\) | \(=\) |
\( 92\nu^{7} - 322\nu^{6} + 1130\nu^{5} - 2020\nu^{4} + 2974\nu^{3} - 2602\nu^{2} + 1500\nu - 376 \)
|
| \(\beta_{7}\) | \(=\) |
\( 134\nu^{7} - 469\nu^{6} + 1641\nu^{5} - 2930\nu^{4} + 4295\nu^{3} - 3747\nu^{2} + 2160\nu - 542 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} - \beta_{6} + 3\beta_{5} + \beta_{4} + 8 ) / 16 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} - \beta_{6} + 3\beta_{5} + \beta_{4} + \beta_{3} - 2\beta_{2} + 3\beta _1 - 24 ) / 16 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -4\beta_{7} + 4\beta_{6} - 20\beta_{5} - 16\beta_{4} + 3\beta_{3} - 6\beta_{2} + 9\beta _1 - 80 ) / 32 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -5\beta_{7} + 5\beta_{6} - 23\beta_{5} - 17\beta_{4} - 4\beta_{3} + 8\beta_{2} - 4\beta _1 + 56 ) / 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 6\beta_{7} - 3\beta_{6} + 58\beta_{5} + 43\beta_{4} - 25\beta_{3} + 50\beta_{2} - 35\beta _1 + 416 ) / 32 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 44\beta_{7} - 35\beta_{6} + 292\beta_{5} + 215\beta_{4} + 12\beta_{3} - 32\beta_{2} - 4\beta _1 - 96 ) / 32 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 14\beta_{7} - 19\beta_{6} + 18\beta_{5} + 19\beta_{4} + 133\beta_{3} - 294\beta_{2} + 119\beta _1 - 1888 ) / 32 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/512\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(511\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 511.1 |
|
0 | − | 3.95687i | 0 | −2.31788 | 0 | − | 6.82843i | 0 | −6.65685 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 511.2 | 0 | − | 3.95687i | 0 | 2.31788 | 0 | 6.82843i | 0 | −6.65685 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 511.3 | 0 | − | 2.08402i | 0 | −7.11529 | 0 | − | 1.17157i | 0 | 4.65685 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 511.4 | 0 | − | 2.08402i | 0 | 7.11529 | 0 | 1.17157i | 0 | 4.65685 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 511.5 | 0 | 2.08402i | 0 | −7.11529 | 0 | 1.17157i | 0 | 4.65685 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 511.6 | 0 | 2.08402i | 0 | 7.11529 | 0 | − | 1.17157i | 0 | 4.65685 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 511.7 | 0 | 3.95687i | 0 | −2.31788 | 0 | 6.82843i | 0 | −6.65685 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 511.8 | 0 | 3.95687i | 0 | 2.31788 | 0 | − | 6.82843i | 0 | −6.65685 | 0 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 8.b | even | 2 | 1 | inner |
| 8.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 512.3.c.g | ✓ | 8 |
| 4.b | odd | 2 | 1 | inner | 512.3.c.g | ✓ | 8 |
| 8.b | even | 2 | 1 | inner | 512.3.c.g | ✓ | 8 |
| 8.d | odd | 2 | 1 | inner | 512.3.c.g | ✓ | 8 |
| 16.e | even | 4 | 2 | 512.3.d.f | 8 | ||
| 16.f | odd | 4 | 2 | 512.3.d.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 512.3.c.g | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 512.3.c.g | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 512.3.c.g | ✓ | 8 | 8.b | even | 2 | 1 | inner |
| 512.3.c.g | ✓ | 8 | 8.d | odd | 2 | 1 | inner |
| 512.3.d.f | 8 | 16.e | even | 4 | 2 | ||
| 512.3.d.f | 8 | 16.f | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(512, [\chi])\):
|
\( T_{3}^{4} + 20T_{3}^{2} + 68 \)
|
|
\( T_{5}^{4} - 56T_{5}^{2} + 272 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{4} + 20 T^{2} + 68)^{2} \)
|
| $5$ |
\( (T^{4} - 56 T^{2} + 272)^{2} \)
|
| $7$ |
\( (T^{4} + 48 T^{2} + 64)^{2} \)
|
| $11$ |
\( (T^{4} + 436 T^{2} + 35972)^{2} \)
|
| $13$ |
\( (T^{4} - 632 T^{2} + 13328)^{2} \)
|
| $17$ |
\( (T^{2} + 8 T - 272)^{4} \)
|
| $19$ |
\( (T^{4} + 532 T^{2} + 65348)^{2} \)
|
| $23$ |
\( (T^{4} + 1584 T^{2} + 64)^{2} \)
|
| $29$ |
\( (T^{4} - 3448 T^{2} + 2885648)^{2} \)
|
| $31$ |
\( (T^{4} + 3776 T^{2} + 1721344)^{2} \)
|
| $37$ |
\( (T^{4} - 1720 T^{2} + 653072)^{2} \)
|
| $41$ |
\( (T^{2} - 12 T - 2012)^{4} \)
|
| $43$ |
\( (T^{4} + 1748 T^{2} + 424388)^{2} \)
|
| $47$ |
\( (T^{4} + 4352 T^{2} + 16384)^{2} \)
|
| $53$ |
\( (T^{4} - 4600 T^{2} + 5105168)^{2} \)
|
| $59$ |
\( (T^{4} + 11956 T^{2} + 34569092)^{2} \)
|
| $61$ |
\( (T^{4} - 12280 T^{2} + 29441552)^{2} \)
|
| $67$ |
\( (T^{4} + 1652 T^{2} + 538628)^{2} \)
|
| $71$ |
\( (T^{4} + 15664 T^{2} + 56911936)^{2} \)
|
| $73$ |
\( (T^{2} + 32 T + 224)^{4} \)
|
| $79$ |
\( (T^{4} + 7872 T^{2} + 5456896)^{2} \)
|
| $83$ |
\( (T^{4} + 20468 T^{2} + 104725508)^{2} \)
|
| $89$ |
\( (T^{2} - 128 T + 3808)^{4} \)
|
| $97$ |
\( (T^{2} - 72 T - 4112)^{4} \)
|
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