Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [64,15,Mod(63,64)]
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([1, 0]))
N = Newforms(chi, 15, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("64.63");
S:= CuspForms(chi, 15);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 64 = 2^{6} \) |
| Weight: | \( k \) | \(=\) | \( 15 \) |
| Character orbit: | \([\chi]\) | \(=\) | 64.c (of order \(2\), degree \(1\), not 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: | \(79.5705396172\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 88x^{4} - 1824x^{3} + 325632x^{2} + 21572352x + 982333440 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{60}\cdot 3^{3} \) |
| Twist minimal: | no (minimal twist has level 4) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} + 88x^{4} - 1824x^{3} + 325632x^{2} + 21572352x + 982333440 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + 253\nu^{4} - 14692\nu^{3} + 116112\nu^{2} - 487872\nu + 51826176 ) / 196608 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{5} - 119\nu^{4} - 596\nu^{3} + 110416\nu^{2} - 103104\nu + 33488384 ) / 2048 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -1025\nu^{5} + 29949\nu^{4} + 112284\nu^{3} + 39323024\nu^{2} - 928412096\nu - 11239764480 ) / 65536 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -113\nu^{5} - 979\nu^{4} - 15780\nu^{3} + 734352\nu^{2} + 92750400\nu - 2265491968 ) / 2048 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 22657\nu^{5} + 362627\nu^{4} + 6441828\nu^{3} + 330429040\nu^{2} + 35627094464\nu + 491777044992 ) / 65536 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + 8\beta_{4} - 10\beta_{3} - 40\beta_{2} + 417\beta _1 + 174752 ) / 1048576 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 17\beta_{5} + 72\beta_{4} + 854\beta_{3} + 7832\beta_{2} + 4017\beta _1 - 30580832 ) / 1048576 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 537\beta_{5} - 2168\beta_{4} + 16134\beta_{3} - 79272\beta_{2} - 12423495\beta _1 + 910311584 ) / 1048576 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 27745 \beta_{5} - 183352 \beta_{4} + 489526 \beta_{3} - 7889640 \beta_{2} + 97349121 \beta _1 - 223715506784 ) / 1048576 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 615913 \beta_{5} - 10078712 \beta_{4} - 9152282 \beta_{3} + 97489368 \beta_{2} + \cdots - 19266694856032 ) / 1048576 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/64\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(63\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 63.1 |
|
0 | − | 2948.24i | 0 | 81680.5 | 0 | − | 1.09921e6i | 0 | −3.90914e6 | 0 | ||||||||||||||||||||||||||||||||||
| 63.2 | 0 | − | 2369.41i | 0 | −113343. | 0 | 697667.i | 0 | −831139. | 0 | ||||||||||||||||||||||||||||||||||||
| 63.3 | 0 | − | 1241.79i | 0 | 27633.0 | 0 | 784634.i | 0 | 3.24093e6 | 0 | ||||||||||||||||||||||||||||||||||||
| 63.4 | 0 | 1241.79i | 0 | 27633.0 | 0 | − | 784634.i | 0 | 3.24093e6 | 0 | ||||||||||||||||||||||||||||||||||||
| 63.5 | 0 | 2369.41i | 0 | −113343. | 0 | − | 697667.i | 0 | −831139. | 0 | ||||||||||||||||||||||||||||||||||||
| 63.6 | 0 | 2948.24i | 0 | 81680.5 | 0 | 1.09921e6i | 0 | −3.90914e6 | 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 | 64.15.c.d | 6 | |
| 4.b | odd | 2 | 1 | inner | 64.15.c.d | 6 | |
| 8.b | even | 2 | 1 | 4.15.b.a | ✓ | 6 | |
| 8.d | odd | 2 | 1 | 4.15.b.a | ✓ | 6 | |
| 24.f | even | 2 | 1 | 36.15.d.c | 6 | ||
| 24.h | odd | 2 | 1 | 36.15.d.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4.15.b.a | ✓ | 6 | 8.b | even | 2 | 1 | |
| 4.15.b.a | ✓ | 6 | 8.d | odd | 2 | 1 | |
| 36.15.d.c | 6 | 24.f | even | 2 | 1 | ||
| 36.15.d.c | 6 | 24.h | odd | 2 | 1 | ||
| 64.15.c.d | 6 | 1.a | even | 1 | 1 | trivial | |
| 64.15.c.d | 6 | 4.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 15848256T_{3}^{4} + 70859216793600T_{3}^{2} + 75249278198568714240 \)
acting on \(S_{15}^{\mathrm{new}}(64, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + \cdots + 75\!\cdots\!40 \)
|
| $5$ |
\( (T^{3} + \cdots + 255824948369000)^{2} \)
|
| $7$ |
\( T^{6} + \cdots + 36\!\cdots\!00 \)
|
| $11$ |
\( T^{6} + \cdots + 11\!\cdots\!00 \)
|
| $13$ |
\( (T^{3} + \cdots + 42\!\cdots\!00)^{2} \)
|
| $17$ |
\( (T^{3} + \cdots + 10\!\cdots\!00)^{2} \)
|
| $19$ |
\( T^{6} + \cdots + 11\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots + 33\!\cdots\!40 \)
|
| $29$ |
\( (T^{3} + \cdots + 36\!\cdots\!08)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 13\!\cdots\!00 \)
|
| $37$ |
\( (T^{3} + \cdots + 16\!\cdots\!00)^{2} \)
|
| $41$ |
\( (T^{3} + \cdots - 22\!\cdots\!88)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 65\!\cdots\!00 \)
|
| $47$ |
\( T^{6} + \cdots + 72\!\cdots\!40 \)
|
| $53$ |
\( (T^{3} + \cdots + 40\!\cdots\!00)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 85\!\cdots\!00 \)
|
| $61$ |
\( (T^{3} + \cdots - 24\!\cdots\!32)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 62\!\cdots\!40 \)
|
| $71$ |
\( T^{6} + \cdots + 50\!\cdots\!00 \)
|
| $73$ |
\( (T^{3} + \cdots - 34\!\cdots\!00)^{2} \)
|
| $79$ |
\( T^{6} + \cdots + 30\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 10\!\cdots\!40 \)
|
| $89$ |
\( (T^{3} + \cdots - 12\!\cdots\!08)^{2} \)
|
| $97$ |
\( (T^{3} + \cdots - 12\!\cdots\!00)^{2} \)
|
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