Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [64,20,Mod(1,64)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(64, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 20, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("64.1");
S:= CuspForms(chi, 20);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 64 = 2^{6} \) |
| Weight: | \( k \) | \(=\) | \( 20 \) |
| Character orbit: | \([\chi]\) | \(=\) | 64.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: | \(146.442685796\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 1308464x^{2} - 385175709x + 111844778211 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{30}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 32) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 1308464x^{2} - 385175709x + 111844778211 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 32\nu^{3} - 34272\nu^{2} + 13402720\nu + 13142876832 ) / 581329 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -1536\nu^{3} + 1645056\nu^{2} + 1142512128\nu - 631304548608 ) / 581329 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 29880736\nu^{3} - 12953279584\nu^{2} - 28268786684192\nu - 179803219873248 ) / 33135753 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 48\beta _1 + 768 ) / 3072 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 171\beta_{3} + 68512\beta_{2} + 487257\beta _1 + 64313647104 ) / 98304 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 183141\beta_{3} + 59973632\beta_{2} + 1664364375\beta _1 + 28494705131520 ) / 98304 \)
|
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 | −66002.5 | 0 | −1.92990e6 | 0 | 1.26041e8 | 0 | 3.19406e9 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −25179.1 | 0 | 4.59136e6 | 0 | −1.43963e8 | 0 | −5.28273e8 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 25179.1 | 0 | 4.59136e6 | 0 | 1.43963e8 | 0 | −5.28273e8 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 66002.5 | 0 | −1.92990e6 | 0 | −1.26041e8 | 0 | 3.19406e9 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
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 | 64.20.a.o | 4 | |
| 4.b | odd | 2 | 1 | inner | 64.20.a.o | 4 | |
| 8.b | even | 2 | 1 | 32.20.a.b | ✓ | 4 | |
| 8.d | odd | 2 | 1 | 32.20.a.b | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 32.20.a.b | ✓ | 4 | 8.b | even | 2 | 1 | |
| 32.20.a.b | ✓ | 4 | 8.d | odd | 2 | 1 | |
| 64.20.a.o | 4 | 1.a | even | 1 | 1 | trivial | |
| 64.20.a.o | 4 | 4.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 4990313472T_{3}^{2} + 2761861533259530240 \)
acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(64))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + \cdots + 27\!\cdots\!40 \)
|
| $5$ |
\( (T^{2} + \cdots - 8860867977500)^{2} \)
|
| $7$ |
\( T^{4} + \cdots + 32\!\cdots\!60 \)
|
| $11$ |
\( T^{4} + \cdots + 38\!\cdots\!60 \)
|
| $13$ |
\( (T^{2} + \cdots - 58\!\cdots\!72)^{2} \)
|
| $17$ |
\( (T^{2} + \cdots + 29\!\cdots\!64)^{2} \)
|
| $19$ |
\( T^{4} + \cdots + 50\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 14\!\cdots\!60 \)
|
| $29$ |
\( (T^{2} + \cdots - 13\!\cdots\!00)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 27\!\cdots\!40 \)
|
| $37$ |
\( (T^{2} + \cdots - 54\!\cdots\!40)^{2} \)
|
| $41$ |
\( (T^{2} + \cdots + 38\!\cdots\!60)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots + 15\!\cdots\!40 \)
|
| $53$ |
\( (T^{2} + \cdots - 14\!\cdots\!16)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 12\!\cdots\!40 \)
|
| $61$ |
\( (T^{2} + \cdots - 12\!\cdots\!20)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 42\!\cdots\!40 \)
|
| $71$ |
\( T^{4} + \cdots + 22\!\cdots\!60 \)
|
| $73$ |
\( (T^{2} + \cdots - 15\!\cdots\!44)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 11\!\cdots\!60 \)
|
| $83$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $89$ |
\( (T^{2} + \cdots - 14\!\cdots\!00)^{2} \)
|
| $97$ |
\( (T^{2} + \cdots + 95\!\cdots\!36)^{2} \)
|
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