Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1024,4,Mod(1,1024)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1024, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1024.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1024 = 2^{10} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1024.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: | \(60.4179558459\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.995269607424.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 28x^{6} + 217x^{4} - 624x^{2} + 576 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{12} \) |
| Twist minimal: | no (minimal twist has level 256) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 28x^{6} + 217x^{4} - 624x^{2} + 576 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -13\nu^{7} + 340\nu^{5} - 2149\nu^{3} + 1176\nu ) / 576 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5\nu^{6} - 116\nu^{4} + 509\nu^{2} - 408 ) / 24 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -13\nu^{7} + 340\nu^{5} - 2149\nu^{3} + 3480\nu ) / 288 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} + 28\nu^{4} - 193\nu^{2} + 300 ) / 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -23\nu^{6} + 572\nu^{4} - 3263\nu^{2} + 4968 ) / 96 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 89\nu^{7} - 2084\nu^{5} + 9617\nu^{3} - 8472\nu ) / 576 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 7\nu^{7} - 172\nu^{5} + 943\nu^{3} - 1368\nu ) / 24 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - 2\beta_1 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -4\beta_{5} + 2\beta_{4} - 3\beta_{2} + 56 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 11\beta_{7} - 12\beta_{6} + 41\beta_{3} - 22\beta_1 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -76\beta_{5} + 48\beta_{4} - 49\beta_{2} + 700 ) / 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 230\beta_{7} - 232\beta_{6} + 859\beta_{3} - 334\beta_1 ) / 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -1356\beta_{5} + 910\beta_{4} - 793\beta_{2} + 11192 ) / 8 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 4197\beta_{7} - 4084\beta_{6} + 15779\beta_{3} - 5634\beta_1 ) / 8 \)
|
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 | −7.96554 | 0 | −0.466324 | 0 | −17.2022 | 0 | 36.4499 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −7.96554 | 0 | 0.466324 | 0 | 17.2022 | 0 | 36.4499 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.60615 | 0 | −14.7903 | 0 | −10.8978 | 0 | −24.4203 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.60615 | 0 | 14.7903 | 0 | 10.8978 | 0 | −24.4203 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 4.50144 | 0 | −5.19053 | 0 | 26.2423 | 0 | −6.73704 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 4.50144 | 0 | 5.19053 | 0 | −26.2423 | 0 | −6.73704 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 5.07025 | 0 | −20.4472 | 0 | −31.2516 | 0 | −1.29254 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 5.07025 | 0 | 20.4472 | 0 | 31.2516 | 0 | −1.29254 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 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 | 1024.4.a.l | 8 | |
| 4.b | odd | 2 | 1 | 1024.4.a.k | 8 | ||
| 8.b | even | 2 | 1 | 1024.4.a.k | 8 | ||
| 8.d | odd | 2 | 1 | inner | 1024.4.a.l | 8 | |
| 16.e | even | 4 | 2 | 1024.4.b.l | 16 | ||
| 16.f | odd | 4 | 2 | 1024.4.b.l | 16 | ||
| 32.g | even | 8 | 2 | 256.4.e.c | ✓ | 16 | |
| 32.g | even | 8 | 2 | 256.4.e.d | yes | 16 | |
| 32.h | odd | 8 | 2 | 256.4.e.c | ✓ | 16 | |
| 32.h | odd | 8 | 2 | 256.4.e.d | yes | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 256.4.e.c | ✓ | 16 | 32.g | even | 8 | 2 | |
| 256.4.e.c | ✓ | 16 | 32.h | odd | 8 | 2 | |
| 256.4.e.d | yes | 16 | 32.g | even | 8 | 2 | |
| 256.4.e.d | yes | 16 | 32.h | odd | 8 | 2 | |
| 1024.4.a.k | 8 | 4.b | odd | 2 | 1 | ||
| 1024.4.a.k | 8 | 8.b | even | 2 | 1 | ||
| 1024.4.a.l | 8 | 1.a | even | 1 | 1 | trivial | |
| 1024.4.a.l | 8 | 8.d | odd | 2 | 1 | inner | |
| 1024.4.b.l | 16 | 16.e | even | 4 | 2 | ||
| 1024.4.b.l | 16 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1024))\):
|
\( T_{3}^{4} - 56T_{3}^{2} + 96T_{3} + 292 \)
|
|
\( T_{5}^{8} - 664T_{5}^{6} + 108760T_{5}^{4} - 2487648T_{5}^{2} + 535824 \)
|
|
\( T_{7}^{8} - 2080T_{7}^{6} + 1398304T_{7}^{4} - 337433088T_{7}^{2} + 23637217536 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{4} - 56 T^{2} + \cdots + 292)^{2} \)
|
| $5$ |
\( T^{8} - 664 T^{6} + \cdots + 535824 \)
|
| $7$ |
\( T^{8} + \cdots + 23637217536 \)
|
| $11$ |
\( (T^{4} - 64 T^{3} + \cdots - 7356)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 20270417143824 \)
|
| $17$ |
\( (T^{4} - 96 T^{3} + \cdots + 1811088)^{2} \)
|
| $19$ |
\( (T^{4} - 32 T^{3} + \cdots - 17270076)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 20\!\cdots\!76 \)
|
| $29$ |
\( T^{8} + \cdots + 94\!\cdots\!76 \)
|
| $31$ |
\( T^{8} + \cdots + 16\!\cdots\!44 \)
|
| $37$ |
\( T^{8} + \cdots + 77\!\cdots\!84 \)
|
| $41$ |
\( (T^{4} + 320 T^{3} + \cdots + 247734528)^{2} \)
|
| $43$ |
\( (T^{4} - 768 T^{3} + \cdots - 1898720028)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 34\!\cdots\!96 \)
|
| $53$ |
\( T^{8} + \cdots + 31\!\cdots\!04 \)
|
| $59$ |
\( (T^{4} - 672 T^{3} + \cdots - 1851284124)^{2} \)
|
| $61$ |
\( T^{8} + \cdots + 83\!\cdots\!96 \)
|
| $67$ |
\( (T^{4} - 512 T^{3} + \cdots + 83495315524)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 21\!\cdots\!24 \)
|
| $73$ |
\( (T^{4} - 8 T^{3} + \cdots + 38640192)^{2} \)
|
| $79$ |
\( T^{8} + \cdots + 41\!\cdots\!24 \)
|
| $83$ |
\( (T^{4} - 2080 T^{3} + \cdots - 309643842204)^{2} \)
|
| $89$ |
\( (T^{4} - 136 T^{3} + \cdots + 119895761472)^{2} \)
|
| $97$ |
\( (T^{4} - 1120 T^{3} + \cdots - 580260467568)^{2} \)
|
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