Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [64,12,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, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("64.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 64 = 2^{6} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| 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: | \(49.1739635558\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.1056820.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 331x - 1379 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{14} \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 331x - 1379 \)
:
| \(\beta_{1}\) | \(=\) |
\( 32\nu - 11 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 512\nu^{2} - 3840\nu - 111869 ) / 9 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 11 ) / 32 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 9\beta_{2} + 120\beta _1 + 113189 ) / 512 \)
|
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 | −342.797 | 0 | 6432.96 | 0 | 40843.8 | 0 | −59637.5 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −8.10720 | 0 | −9611.74 | 0 | −77473.2 | 0 | −177081. | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 790.904 | 0 | 2408.78 | 0 | 42661.4 | 0 | 448382. | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 64.12.a.m | 3 | |
| 4.b | odd | 2 | 1 | 64.12.a.l | 3 | ||
| 8.b | even | 2 | 1 | 32.12.a.d | ✓ | 3 | |
| 8.d | odd | 2 | 1 | 32.12.a.e | yes | 3 | |
| 16.e | even | 4 | 2 | 256.12.b.l | 6 | ||
| 16.f | odd | 4 | 2 | 256.12.b.m | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 32.12.a.d | ✓ | 3 | 8.b | even | 2 | 1 | |
| 32.12.a.e | yes | 3 | 8.d | odd | 2 | 1 | |
| 64.12.a.l | 3 | 4.b | odd | 2 | 1 | ||
| 64.12.a.m | 3 | 1.a | even | 1 | 1 | trivial | |
| 256.12.b.l | 6 | 16.e | even | 4 | 2 | ||
| 256.12.b.m | 6 | 16.f | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 440T_{3}^{2} - 274752T_{3} - 2198016 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(64))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} - 440 T^{2} + \cdots - 2198016 \)
|
| $5$ |
\( T^{3} + \cdots + 148939701400 \)
|
| $7$ |
\( T^{3} + \cdots + 134993299460096 \)
|
| $11$ |
\( T^{3} + \cdots + 40\!\cdots\!00 \)
|
| $13$ |
\( T^{3} + \cdots - 48\!\cdots\!00 \)
|
| $17$ |
\( T^{3} + \cdots - 20\!\cdots\!00 \)
|
| $19$ |
\( T^{3} + \cdots + 17\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots - 25\!\cdots\!12 \)
|
| $29$ |
\( T^{3} + \cdots + 37\!\cdots\!72 \)
|
| $31$ |
\( T^{3} + \cdots + 39\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots + 93\!\cdots\!00 \)
|
| $41$ |
\( T^{3} + \cdots - 16\!\cdots\!00 \)
|
| $43$ |
\( T^{3} + \cdots - 43\!\cdots\!56 \)
|
| $47$ |
\( T^{3} + \cdots - 61\!\cdots\!44 \)
|
| $53$ |
\( T^{3} + \cdots + 44\!\cdots\!00 \)
|
| $59$ |
\( T^{3} + \cdots + 77\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots - 21\!\cdots\!60 \)
|
| $67$ |
\( T^{3} + \cdots - 24\!\cdots\!68 \)
|
| $71$ |
\( T^{3} + \cdots + 41\!\cdots\!00 \)
|
| $73$ |
\( T^{3} + \cdots + 15\!\cdots\!00 \)
|
| $79$ |
\( T^{3} + \cdots - 14\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots + 10\!\cdots\!80 \)
|
| $89$ |
\( T^{3} + \cdots - 67\!\cdots\!76 \)
|
| $97$ |
\( T^{3} + \cdots - 12\!\cdots\!00 \)
|
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